# Publications

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- CJ Lustri and SJ Chapman (2013). Steady gravity waves due to a submerged source. Journal of Fluid Mechanics
**732**: 660-686. (doi:10.1017/jfm.2013.425) - CP Please, DP Mason, AJ Hutchinson, CM Khalique, JMT Ngnotchouye, JN van der Merwe and H Yilmaz (2013). Fracturing of an Euler-Bernoulli beam in coal mine pillar extraction. International Journal of Rock Mechanics and Mining Sciences
**64**: 132-138. (doi:10.1016/j.ijrmms.2013.08.001) (Abstract)The paper models the first stage of the process of pillar extraction in a coal mine. The problem of understanding how a coal mine roof collapses after secondary cutting of the supporting pillars to create small supporting snooks is considered. The fracture of the roof is considered when a set of snooks have failed and the roof must support itself between two pillars. Models that account for the relative importance of the overburden weight on the roof and the compressive stresses in the roof are examined using a simple strut and beam theory. © 2013 Elsevier Ltd.

- DS Bassett, NF Wymbs, ST Grafton, MP Rombach, MA Porter and PJ Mucha (2013). Task-Based Core-Periphery Organization of Human Brain Dynamics. Plos Computational Biology
**9**(9): e1003171. (doi:10.1371/journal.pcbi.1003171) (Abstract)As a person learns a new skill, distinct synapses, brain regions, and circuits are engaged and change over time. In this paper, we develop methods to examine patterns of correlated activity across a large set of brain regions. Our goal is to identify properties that enable robust learning of a motor skill. We measure brain activity during motor sequencing and characterize network properties based on coherent activity between brain regions. Using recently developed algorithms to detect time-evolving communities, we find that the complex reconfiguration patterns of the brain's putative functional modules that control learning can be described parsimoniously by the combined presence of a relatively stiff temporal core that is composed primarily of sensorimotor and visual regions whose connectivity changes little in time and a flexible temporal periphery that is composed primarily of multimodal association regions whose connectivity changes frequently. The separation between temporal core and periphery changes over the course of training and, importantly, is a good predictor of individual differences in learning success. The core of dynamically stiff regions exhibits dense connectivity, which is consistent with notions of core-periphery organization established previously in social networks. Our results demonstrate that core-periphery organization provides an insightful way to understand how putative functional modules are linked. This, in turn, enables the prediction of fundamental human capacities, including the production of complex goal-directed behavior. © 2013 Bassett et al.

- RM Lewis, S Brooks, MA Hanson, IP Crocker, J Glazier, ED Johnstone, CP Sibley, KL Widdows, CP Please, N Panitchob and BG Sengers (2013). Review: Modelling placental amino acid transfer - From transporters to placental function. Placenta
**34**(SUPPL): S46-S51. (doi:10.1016/j.placenta.2012.10.010) (Abstract)Amino acid transfer to the fetus is dependent on several different factors. While these factors can be understood in isolation, it is still not possible to predict the function of the system as a whole. In order to do this an integrated approach is required which incorporates the interactions between the different determinants of amino acid transfer. Computational modelling of amino acid transfer in the term human placenta provides a mechanism by which this integrated approach can be delivered. Such a model would be invaluable for understanding amino acid transfer in both normal and pathological pregnancies. In order to develop a computational model it is necessary to determine all the biological factors which are important contributors to net amino acid transfer and the ways in which they interact. For instance, how different classes of amino acid transporter must interact to transfer amino acids across the placenta. Mathematically, the kinetics of each type of transporter can be represented by separate equations that describe their transfer rate as a non-linear function of amino acid concentrations. These equations can then be combined in the model to predict the overall system behaviour. Testing these predictions experimentally will demonstrate the strengths and weaknesses of the model, which can then be refined with increasing complexity and retested in an iterative fashion. In this way we hope to develop a functional computational model which will allow exploration of the factors that determine amino acid transfer across the placenta. This model may also allow the development of strategies to optimise placental transfer in pathologies associated with impaired amino acid transfer such as fetal growth restriction.

- K Widdows, N Panitchob, I Crocker, S Brooks, C Please, C Sibley, E Johnstone, B Sengers, J Glazier and R Lewis (2013). MATHEMATICAL MODELS TO PREDICT EXCHANGE AND FACILITATED TRANSPORT ACROSS THE MICROVILLOUS PLASMA MEMBRANE (MVM) OF THE HUMAN PLACENTA. Placenta
**34**(9): A46-A46. (doi:10.1016/j.placenta.2013.06.141) - T Hoffmann, R Lambiotte and MA Porter (2013). Decentralized routing on spatial networks with stochastic edge weights. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**88**(2): 022815. (doi:10.1103/PhysRevE.88.022815) (Abstract)We investigate algorithms to find short paths in spatial networks with stochastic edge weights. Our formulation of the problem of finding short paths differs from traditional formulations because we specifically do not make two of the usual simplifying assumptions: (1) we allow edge weights to be stochastic rather than deterministic and (2) we do not assume that global knowledge of a network is available. We develop a decentralized routing algorithm that provides en route guidance for travelers on a spatial network with stochastic edge weights without the need to rely on global knowledge about the network. To guide a traveler, our algorithm uses an estimation function that evaluates cumulative arrival probability distributions based on distances between pairs of nodes. The estimation function carries a notion of proximity between nodes and thereby enables routing without global knowledge. In testing our decentralized algorithm, we define a criterion that makes it possible to discriminate among arrival probability distributions, and we test our algorithm and this criterion using both synthetic and real networks. © 2013 American Physical Society.

- PD Howell, J Robinson and HA Stone (2013). Gravity-driven thin-film flow on a flexible substrate. Journal of Fluid Mechanics
**732**: 190-213. (doi:10.1017/jfm.2013.404) (Abstract)We study the flow of a thin liquid film along a flexible substrate. The flow is modelled using lubrication theory, assuming that gravity is the dominant driving force. The substrate is modelled as an elastic beam that deforms in two dimensions. Steady solutions are found using numerical and perturbation methods, and several different asymptotic regimes are identified. We obtain a complete characterization of how the length and stiffness of the beam and the imposed liquid flux determine the profile of the liquid film and the resulting beam deformation. © 2013 Cambridge University Press.

- VS Zubkov, CJW Breward and EA Gaffney (2013). Meniscal Tear Film Fluid Dynamics Near Marx's Line. Bulletin of Mathematical Biology
**75**(9): 1524-1543. (doi:10.1007/s11538-013-9858-x) (Abstract)Extensive studies have explored the dynamics of the ocular surface fluid, though theoretical investigations are typically limited to the use of the lubrication approximation, which is not guaranteed to be uniformly valid a-priori throughout the tear meniscus. However, resolving tear film behaviour within the meniscus and especially its apices is required to characterise the flow dynamics where the tear film is especially thin, and thus most susceptible to evaporatively induced hyperosmolarity and subsequent epithelial damage. Hence, we have explored the accuracy of the standard lubrication approximation for the tear film by explicit comparisons with the 2D Navier-Stokes model, considering both stationary and moving eyelids. Our results demonstrate that the lubrication model is qualitatively accurate except in the vicinity of the eyelids. In particular, and in contrast to lubrication theory, the solution of the full Navier-Stokes equations predict a distinct absence of fluid flow, and thus convective mixing in the region adjacent to the tear film contact line. These observations not only support emergent hypotheses concerning the formation of Marx's line, a region of epithelial cell staining adjacent to the contact line on the eyelid, but also enhance our understanding of the pathophysiological consequences of the flow profile near the tear film contact line. © 2013 Society for Mathematical Biology.

- T Hoffmann, MA Porter and R Lambiotte (2013). Random walks on
stochastic temporal networks. Understanding Complex Systems:
Temporal Networks : 295-313. (doi:10.1007/978-3-642-36461-7_15) (Abstract)
In the study of dynamical processes on networks, there has been intense focus on network structure-i.e., the arrangement of edges and their associated weights-but the effects of the temporal patterns of edges remains poorly understood. In this chapter, we develop a mathematical framework for random walks on temporal networks using an approach that provides a compromise between abstract but unrealistic models and data-driven but non-mathematical approaches. To do this, we introduce a stochastic model for temporal networks in which we summarize the temporal and structural organization of a system using a matrix of waiting-time distributions. We show that random walks on stochastic temporal networks can be described exactly by an integro-differentialmaster equation and derive an analytical expression for its asymptotic steady state. We also discuss how our work might be useful to help build centrality measures for temporal networks. © Springer-Verlag Berlin Heidelberg 2013.

- D Mitra, JS Wettlaufer and A Brandenburg (2013).
CAN PLANETESIMALS FORM BY COLLISIONAL FUSION?. The Astrophysical Journal
**773**(2): 120-120. (doi:10.1088/0004-637X/773/2/120) - G Kozyreff and SJ Chapman (2013). Analytical Results for Front Pinning between an Hexagonal Pattern and a Uniform State in Pattern-Formation Systems. Physical Review Letters
**111**(5): 054501. (doi:10.1103/PhysRevLett.111.054501) - YM Lai and MA Porter (2013). Noise-induced synchronization, desynchronization, and clustering in globally coupled nonidentical oscillators.. Phys Rev E Stat Nonlin Soft Matter Phys
**88**(1-1): 012905. (Abstract)We study ensembles of globally coupled, nonidentical phase oscillators subject to correlated noise, and we identify several important factors that cause noise and coupling to synchronize or desynchronize a system. By introducing noise in various ways, we find an estimate for the onset of synchrony of a system in terms of the coupling strength, noise strength, and width of the frequency distribution of its natural oscillations. We also demonstrate that noise alone can be sufficient to synchronize nonidentical oscillators. However, this synchrony depends on the first Fourier mode of a phase-sensitivity function, through which we introduce common noise into the system. We show that higher Fourier modes can cause desynchronization due to clustering effects, and that this can reinforce clustering caused by different forms of coupling. Finally, we discuss the effects of noise on an ensemble in which antiferromagnetic coupling causes oscillators to form two clusters in the absence of noise.

- CJ Lustri, SJ Chapman and SW Mccue (2013). Exponential asymptotics of free surface flow due to a line source. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
**78**(4): 697-713. (doi:10.1093/imamat/hxt016) (Abstract)The steady problem of free surface flow due to a submerged line source is revisited for the case in which the fluid depth is finite and there is a stagnation point on the free surface directly above the source. Both the strength of the source and the speed of the fluid in the far field are measured by a dimensionless parameter - the Froude number. By applying techniques in exponential asymptotics, it is shown that there is a train of periodic waves on the surface of the fluid with an amplitude which is exponentially small in the limit that the Froude number vanishes. This study clarifies that periodic waves do form for flows due to a source, contrary to a suggestion by Chapman & Vanden-Broeck (2006, Exponential asymptotics and gravity waves. J. Fluid Mech., 567, 299-326). The exponentially small nature of the waves means that they appear beyond all orders of the original power-series expansion; this result explains why attempts at describing these flows using a finite number of terms in an algebraic power series incorrectly predict a flat free surface in the far field. © The authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

- YM Lai and MA Porter (2013). Noise-induced synchronization, desynchronization, and clustering in globally coupled nonidentical oscillators. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**88**(1): 012905. (doi:10.1103/PhysRevE.88.012905) (Abstract)We study ensembles of globally coupled, nonidentical phase oscillators subject to correlated noise, and we identify several important factors that cause noise and coupling to synchronize or desynchronize a system. By introducing noise in various ways, we find an estimate for the onset of synchrony of a system in terms of the coupling strength, noise strength, and width of the frequency distribution of its natural oscillations. We also demonstrate that noise alone can be sufficient to synchronize nonidentical oscillators. However, this synchrony depends on the first Fourier mode of a phase-sensitivity function, through which we introduce common noise into the system. We show that higher Fourier modes can cause desynchronization due to clustering effects, and that this can reinforce clustering caused by different forms of coupling. Finally, we discuss the effects of noise on an ensemble in which antiferromagnetic coupling causes oscillators to form two clusters in the absence of noise. © 2013 American Physical Society.

- TE Woolley, EA Gaffney, JM Oliver, RE Baker, SL Waters and A Goriely (2013). Cellular blebs: pressure-driven, axisymmetric, membrane protrusions. Biomechanics and Modeling in Mechanobiology : 1-14. (doi:10.1007/s10237-013-0509-9) (Abstract)
Blebs are cellular protrusions that are used by cells for multiple purposes including locomotion. A mechanical model for the problem of pressure-driven blebs based on force and moment balances of an axisymmetric shell model is proposed. The formation of a bleb is initiated by weakening the shell over a small region, and the deformation of the cellular membrane from the cortex is obtained during inflation. However, simply weakening the shell leads to an area increase of more than 4 %, which is physically unrealistic. Thus, the model is extended to include a reconfiguration process that allows large blebs to form with small increases in area. It is observed that both geometric and biomechanical constraints are important in this process. In particular, it is shown that although blebs are driven by a pressure difference across the cellular membrane, it is not the limiting factor in determining bleb size. © 2013 Springer-Verlag Berlin Heidelberg.

- MR Moore, SD Howison, JR Ockendon and JM Oliver (2013). A note on oblique water entry. Journal of Engineering Mathematics
**81**(1): 67-74. (doi:10.1007/s10665-012-9570-0) (Abstract)A minor error in Howison et al. (J. Eng. Math. 48:321-337, 2004) obscured the fact that the points at which the free surface turns over in the solution of the Wagner model for the oblique impact of a two-dimensional body are directly related to the turnover points in the equivalent normal impact problem. This note corrects some of the earlier results given in Howison et al. (J. Eng. Math. 48:321-337, 2004) and discusses the implications for the applicability of the Wagner model. © 2012 Springer Science+Business Media B.V.

- VS Zubkov, CJW Breward and EA Gaffney (2013). Meniscal Tear Film Fluid Dynamics Near Marx's Line. Bulletin of Mathematical Biology : 1-20. (doi:10.1007/s11538-013-9858-x) (Abstract)
Extensive studies have explored the dynamics of the ocular surface fluid, though theoretical investigations are typically limited to the use of the lubrication approximation, which is not guaranteed to be uniformly valid a-priori throughout the tear meniscus. However, resolving tear film behaviour within the meniscus and especially its apices is required to characterise the flow dynamics where the tear film is especially thin, and thus most susceptible to evaporatively induced hyperosmolarity and subsequent epithelial damage. Hence, we have explored the accuracy of the standard lubrication approximation for the tear film by explicit comparisons with the 2D Navier-Stokes model, considering both stationary and moving eyelids. Our results demonstrate that the lubrication model is qualitatively accurate except in the vicinity of the eyelids. In particular, and in contrast to lubrication theory, the solution of the full Navier-Stokes equations predict a distinct absence of fluid flow, and thus convective mixing in the region adjacent to the tear film contact line. These observations not only support emergent hypotheses concerning the formation of Marx's line, a region of epithelial cell staining adjacent to the contact line on the eyelid, but also enhance our understanding of the pathophysiological consequences of the flow profile near the tear film contact line. © 2013 Society for Mathematical Biology.

- M Perego, MD Gunzburger, PD Howell, JR Ockendon and JE Allen (2013). The expansion of a collisionless plasma into a plasma of lower density. Physics of Plasmas
**20**(5): 052101. (doi:10.1063/1.4802933) (Abstract)This paper considers the asymptotic and numerical solution of a simple model for the expansion of a collisionless plasma into a plasma of lower density. The dependence on the density ratio of qualitative and quantitative features of solutions of the well-known cold-ion model is explored. In the cold-ion limit, we find that a singularity develops in the ion density in finite time unless the density ratio is zero or close to unity. The classical cold-ion model may cease to be valid when such a singularity occurs and we then regularize the model by the finite ion-temperature Vlasov-Poisson system. Numerical evidence suggests the emergence of a multi-modal velocity distribution. © 2013 AIP Publishing LLC.

- J Knychala, OL Katsamenis, BG Sengers, N Bouropoulos, CJ Catt and CP Please (2013). Pore geometry regulates early stage human bone marrow cell tissue formation and organisation. Annals of Biomedical Engineering
**41**(5): 917-930. (doi:10.1007/s10439-013-0748-z) (Abstract)Porous architecture has a dramatic effect on tissue formation in porous biomaterials used in regenerative medicine. However, the wide variety of 3D structures used indicates there is a clear need for the optimal design of pore architecture to maximize tissue formation and ingrowth. Thus, the aim of this study was to characterize initial tissue growth solely as a function of pore geometry. We used an in vitro system with well-defined open pore slots of varying width, providing a 3D environment for neo-tissue formation while minimizing nutrient limitations. Results demonstrated that initial tissue formation was strongly influenced by pore geometry. Both velocity of tissue invasion and area of tissue formed increased as pores became narrower. This is associated with distinct patterns of actin organisation and alignment depending on pore width, indicating the role of active cell generated forces. A mathematical model based on curvature driven growth successfully predicted both shape of invasion front and constant rate of growth, which increased for narrower pores as seen in experiments. Our results provide further evidence for a front based, curvature driven growth mechanism depending on pore geometry and tissue organisation, which could provide important clues for 3D scaffold design. © 2013 Biomedical Engineering Society.

- Y Zhu, SJ Chapman and A Acharya (2013). Dislocation motion and instability. Journal of the Mechanics and Physics of Solids
**61**: 1835-1853. (doi:10.1016/j.jmps.2013.03.002) (Abstract)The Peach-Koehler expression for the stress generated by a single (non-planar) curvilinear dislocation is evaluated to calculate the dislocation self stress. This is combined with a law of motion to give the self-induced motion of a general dislocation curve. A stability analysis of a rectilinear, uniformly translating dislocation is then performed. The dislocation is found to be susceptible to a helical instability, with the maximum growth rate occurring when the dislocation is almost, but not exactly, pure screw. The non-linear evolution of the instability is determined numerically, and implications for slip band formation and non-Schmid behavior in yielding are discussed. © 2013 Elsevier Ltd. All rights reserved.

- AL Stewart and PJ Dellar (2013). Multilayer shallow water equations with complete Coriolis force. Part 3. Hyperbolicity and stability under shear. Journal of Fluid Mechanics
**723**: 289-317. (doi:10.1017/jfm.2013.121) (Abstract)We analyse the hyperbolicity of our multilayer shallow water equations that include the complete Coriolis force due to the Earth's rotation. Shallow water theory represents flows in which the vertical shear is concentrated into vortex sheets between layers of uniform velocity. Such configurations are subject to Kelvin-Helmholtz instabilities, with arbitrarily large growth rates for sufficiently short-wavelength disturbances. These instabilities manifest themselves through a loss of hyperbolicity in the shallow water equations, rendering them ill-posed for the solution of initial value problems. We show that, in the limit of vanishingly small density difference between the two layers, our two-layer shallow water equations remain hyperbolic when the velocity difference remains below the same threshold that also ensures the hyperbolicity of the standard shallow water equations. Direct calculation of the domain of hyperbolicity becomes much less tractable for three or more layers, so we demonstrate numerically that the threshold for the velocity differences, below which the three-layer equations remain hyperbolic, is also unchanged by the inclusion of the complete Coriolis force. In all cases, the shape of the domain of hyperbolicity, which extends outside the threshold, changes considerably. The standard shallow water equations only lose hyperbolicity due to shear parallel to the direction of wave propagation, but the complete Coriolis force introduces another mechanism for loss of hyperbolicity due to shear in the perpendicular direction. We demonstrate that this additional mechanism corresponds to the onset of a transverse shear instability driven by the non-traditional components of the Coriolis force in a three-dimensional continuously stratified fluid. © 2013 Cambridge University Press.

- ES Warneford and PJ Dellar (2013). The quasi-geostrophic theory of the thermal shallow water equations. Journal of Fluid Mechanics
**723**: 374-403. (doi:10.1017/jfm.2013.101) - SG O'Keeffe, DE Moulton, SL Waters and A Goriely (2013). Growth-induced axial buckling of a slender elastic filament embedded in an isotropic elastic matrix. International Journal of Non-Linear Mechanics
**56**: 94-104. (doi:10.1016/j.ijnonlinmec.2013.04.017) (Abstract)We investigate the problem of an axially loaded, isotropic, slender cylinder embedded in a soft, isotropic, outer elastic matrix. The cylinder undergoes uniform axial growth, whilst both the cylinder and the surrounding elastic matrix are confined between two rigid plates, so that this growth results in axial compression of the cylinder. We use two different modelling approaches to estimate the critical axial growth (that is, the amount of axial growth the cylinder is able to sustain before it buckles) and buckling wavelength of the cylinder. The first approach treats the filament and surrounding matrix as a single 3-dimensional elastic body undergoing large deformations, whilst the second approach treats the filament as a planar, elastic rod embedded in an infinite elastic foundation. By comparing the results of these two approaches, we obtain an estimate of the foundation modulus parameter, which characterises the strength of the foundation, in terms of the geometric and material properties of the system. © 2013 Elsevier Ltd. All rights reserved.

- SD Howison, C Reisinger and JH Witte (2013). The Effect of Non-Smooth Payoffs on the Penalty Approximation of American Options. SIAM Journal on Financial Mathematics : .
- M Bruna and SJ Chapman (2013). Diffusion of Finite-Size Particles in Confined Geometries. Bulletin of Mathematical Biology : 1-36. (doi:10.1007/s11538-013-9847-0) (Abstract)
The diffusion of finite-size hard-core interacting particles in two- or three-dimensional confined domains is considered in the limit that the confinement dimensions become comparable to the particle's dimensions. The result is a nonlinear diffusion equation for the one-particle probability density function, with an overall collective diffusion that depends on both the excluded-volume and the narrow confinement. By including both these effects, the equation is able to interpolate between severe confinement (for example, single-file diffusion) and unconfined diffusion. Numerical solutions of both the effective nonlinear diffusion equation and the stochastic particle system are presented and compared. As an application, the case of diffusion under a ratchet potential is considered, and the change in transport properties due to excluded-volume and confinement effects is examined. © 2013 Society for Mathematical Biology.

- PH Trinh and S Jonathan Chapman (2013). New gravity-capillary waves at low speeds. Part 1. Linear geometries. Journal of Fluid Mechanics
**724**: 367-391. (doi:10.1017/jfm.2013.110) (Abstract)When traditional linearized theory is used to study gravity-capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions invariably predict a waveless surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. By applying techniques in exponential asymptotics to this problem, we have discovered the existence of new classes of gravity-capillary waves, from which the usual linear solutions form but a special case. In this paper, we present the initial theory for deriving these waves through a study of gravity-capillary flow over a linearized step. This will be done using two approaches: in the first, we derive the surface waves using the standard method of Fourier transforms; in the second, we derive the same result using exponential asymptotics. Ultimately, these two methods give the same result, but conceptually, they offer different insights into the study of the low-Froude-number, low-Bond-number problem. ©2013 Cambridge University PressA.

- PH Trinh and SJ Chapman (2013). New gravity-capillary waves at low speeds. Part 2. Nonlinear geometries. Journal of Fluid Mechanics
**724**: 392-424. (doi:10.1017/jfm.2013.129) (Abstract)When traditional linearized theory is used to study gravity-capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions are waveless to every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. In Part 1 (Trinh & Chapman, J. Fluid Mech., vol. 724, 2013b, pp. 367-391), we showed how exponential asymptotics could be used to study the problem when the size of the obstruction is first linearized. In this paper, we extend the analysis to the nonlinear problem, thus allowing the full geometry to be considered at leading order. When applied to the classic problem of flow over a step, our analysis reveals the existence of six classes of gravity-capillary waves, two of which share a connection with the usual linearized solutions first discovered by Rayleigh. The new solutions arise due to the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects. ©2013 Cambridge University PressA.

- JS Wexler, PH Trinh, H Berthet, N Quennouz, O du Roure, HE Huppert, A Linder and HA Stone (2013).
Bending of elastic fibres in viscous flows: the influence of confinement. Journal of Fluid Mechanics
**720**: 517-544. (doi:10.1017/jfm.2013.49) - A Majumdar and A Goriely (2013). Static and dynamic stability results for a class of three-dimensional configurations of Kirchhoff elastic rods. Physica D: Nonlinear Phenomena : .
- S Ranković and MA Porter (2013). Two-particle circular billiards versus randomly perturbed one-particle circular billiards. Chaos
**23**(1): 013123. (doi:10.1063/1.4775756) (Abstract)We study a two-particle circular billiard containing two finite-size circular particles that collide elastically with the billiard boundary and with each other. Such a two-particle circular billiard provides a clean example of an "intermittent" system. This billiard system behaves chaotically, but the time scale on which chaos manifests can become arbitrarily long as the sizes of the confined particles become smaller. The finite-time dynamics of this system depends on the relative frequencies of (chaotic) particle-particle collisions versus (integrable) particle-boundary collisions, and investigating these dynamics is computationally intensive because of the long time scales involved. To help improve understanding of such two-particle dynamics, we compare the results of diagnostics used to measure chaotic dynamics for a two-particle circular billiard with those computed for two types of one-particle circular billiards in which a confined particle undergoes random perturbations. Importantly, such one-particle approximations are much less computationally demanding than the original two-particle system, and we expect them to yield reasonable estimates of the extent of chaotic behavior in the two-particle system when the sizes of confined particles are small. Our computations of recurrence-rate coefficients, finite-time Lyapunov exponents, and autocorrelation coefficients support this hypothesis and suggest that studying randomly perturbed one-particle billiards has the potential to yield insights into the aggregate properties of two-particle billiards, which are difficult to investigate directly without enormous computation times (especially when the sizes of the confined particles are small). © 2013 American Institute of Physics.

- NC Pearson, RJ Shipley, SL Waters and JM Oliver (2013). Multiphase modelling of the influence of fluid flow and chemical concentration on tissue growth in a hollow fibre membrane bioreactor.. Math Med Biol : . (doi:10.1093/imammb/dqt015) (Abstract)
A 2D model is developed for fluid flow, mass transport and cell distribution in a hollow fibre membrane bioreactor. The geometry of the modelling region is simplified by excluding the exit ports at either end and focusing on the upper half of the central section of the bioreactor. Cells are seeded on a porous scaffold throughout the extracapillary space (ECS), and fluid pumped through the bioreactor via the lumen inlet and/or exit ports. In the fibre lumen and porous fibre wall, flow is described using Stokes and Darcy governing equations, respectively, while in the ECS porous mixture theory is used to model the cells, culture medium and scaffold. Reaction-advection-diffusion equations govern the concentration of a solute of interest in each region. The governing equations are reduced by exploiting the small aspect ratio of the bioreactor. This yields a coupled system for the cell volume fraction, solute concentration and ECS water pressure which is solved numerically for a variety of experimentally relevant case studies. The model is used to identify different regimes of cell behaviour, and results indicate how the flow rate can be controlled experimentally to generate a uniform cell distribution under regimes relevant to nutrient- and/or chemotactic-driven behaviours.

- SJ Chapman, PH Trinh and TP Witelski (2013). Exponential asymptotics for thin film rupture. SIAM Journal on Applied Mathematics
**73**(1): 232-253. (doi:10.1137/120872012) (Abstract)The formation of singularities in models of many physical systems can be described using self-similar solutions. One particular example is the finite-time rupture of a thin film of viscous fluid which coats a solid substrate. Previous studies have suggested the existence of a discrete, countably infinite number of distinct solutions of the nonlinear differential equation which describes the self-similar behavior. However, no analytical mechanism for determining these solutions was identified. In this paper, we use techniques in exponential asymptotics to construct the analytical selection condition for the infinite sequence of similarity solutions, confirming the conjectures of earlier numerical studies. © 2013 Society for Industrial and Applied Mathematics.

- RM Lewis, S Brooks, MA Hanson, IP Crocker, J Glazier, ED Johnstone, CP Sibley, KL Widdows, CP Please, N Panitchob and BG Sengers (2013). Review: Modelling placental amino acid transfer - From transporters to placental function. Placenta
**34**(SUPPL): S46-S51. (doi:10.1016/j.placenta.2012.10.010) (Abstract)Amino acid transfer to the fetus is dependent on several different factors. While these factors can be understood in isolation, it is still not possible to predict the function of the system as a whole. In order to do this an integrated approach is required which incorporates the interactions between the different determinants of amino acid transfer. Computational modelling of amino acid transfer in the term human placenta provides a mechanism by which this integrated approach can be delivered. Such a model would be invaluable for understanding amino acid transfer in both normal and pathological pregnancies. In order to develop a computational model it is necessary to determine all the biological factors which are important contributors to net amino acid transfer and the ways in which they interact. For instance, how different classes of amino acid transporter must interact to transfer amino acids across the placenta. Mathematically, the kinetics of each type of transporter can be represented by separate equations that describe their transfer rate as a non-linear function of amino acid concentrations. These equations can then be combined in the model to predict the overall system behaviour. Testing these predictions experimentally will demonstrate the strengths and weaknesses of the model, which can then be refined with increasing complexity and retested in an iterative fashion. In this way we hope to develop a functional computational model which will allow exploration of the factors that determine amino acid transfer across the placenta. This model may also allow the development of strategies to optimise placental transfer in pathologies associated with impaired amino acid transfer such as fetal growth restriction.

- S Melnik, MA Porter, JP Gleeson and JA Ward (2013). Multi-stage complex contagions. Chaos
**23**(1): 013124. (doi:10.1063/1.4790836) (Abstract)The spread of ideas across a social network can be studied using complex contagion models, in which agents are activated by contact with multiple activated neighbors. The investigation of complex contagions can provide crucial insights into social influence and behavior-adoption cascades on networks. In this paper, we introduce a model of a multi-stage complex contagion on networks. Agents at different stages-which could, for example, represent differing levels of support for a social movement or differing levels of commitment to a certain product or idea-exert different amounts of influence on their neighbors. We demonstrate that the presence of even one additional stage introduces novel dynamical behavior, including interplay between multiple cascades, which cannot occur in single-stage contagion models. We find that cascades-and hence collective action-can be driven not only by high-stage influencers but also by low-stage influencers. © 2013 American Institute of Physics.

- DS Bassett, JM Carlson, MA Porter, NF Wymbs, ST Grafton and PJ Mucha (2013). Robust detection of dynamic community structure in networks. Chaos
**23**(1): 013142. (doi:10.1063/1.4790830) (Abstract)We describe techniques for the robust detection of community structure in some classes of time-dependent networks. Specifically, we consider the use of statistical null models for facilitating the principled identification of structural modules in semi-decomposable systems. Null models play an important role both in the optimization of quality functions such as modularity and in the subsequent assessment of the statistical validity of identified community structure. We examine the sensitivity of such methods to model parameters and show how comparisons to null models can help identify system scales. By considering a large number of optimizations, we quantify the variance of network diagnostics over optimizations ("optimization variance") and over randomizations of network structure ("randomization variance"). Because the modularity quality function typically has a large number of nearly degenerate local optima for networks constructed using real data, we develop a method to construct representative partitions that uses a null model to correct for statistical noise in sets of partitions. To illustrate our results, we employ ensembles of time-dependent networks extracted from both nonlinear oscillators and empirical neuroscience data. © 2013 American Institute of Physics.

- M Shakeel, PC Matthews, RS Graham and SL Waters (2013). A continuum model of cell proliferation and nutrient transport in a perfusion bioreactor. Mathematical Medicine and Biology
**30**(1): 21-44. (doi:10.1093/imammb/dqr022) (Abstract)Tissue engineering aims to regenerate, repair or replace organs or defective tissues. This tissue regeneration often occurs in a bioreactor. Important challenges in tissue engineering include ensuring adequate nutrient supply, maintaining the desired cell distribution and achieving sufficiently high cell yield. To put laboratory experiments into a theoretical framework, mathematical modelling of the physical and biochemical processes involved in tissue growth is a useful tool. In this work, we derive and solve a model for a cell-seeded porous scaffold placed in a perfusion bioreactor in which fluid delivers nutrients to the cells. The model describes the key features, including fluid flow, nutrient delivery, cell proliferation and consequent variation of scaffold porosity. Fluid flow through the porous scaffold is modelled by Darcy's law, and nutrient delivery is described by a reaction-advection-diffusion equation. A reaction-diffusion equation describes the evolution of cell density, in which cell proliferation is modelled via logistic growth and cell spreading via non-linear diffusion, which depends on cell density. The effect of shear stress on nutrient consumption and cell proliferation is also included in the model. COMSOL (a commercial finite element solver) is used to solve the model numerically. The results reveal the dependence of the cell distribution and total cell yield on the initial cell density and scaffold porosity. We suggest various seeding strategies and scaffold designs to improve the cell distribution and total cell yield in the engineered tissue construct. © The Author 2011. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

- E Aumaitre, P Cicuta, S Knoche and D Vella (2013). Wrinkling in the deflation of elastic bubbles. European Physical Journal E
**36**(3): . (doi:10.1140/epje/i2013-13022-3) (Abstract)The protein hydrophobin HFBII self-assembles into very elastic films at the surface of water; these films wrinkle readily upon compression. We demonstrate and study this wrinkling instability in the context of non-planar interfaces by forming HFBII layers at the surface of bubbles whose interfaces are then compressed by deflation of the bubble. By varying the initial concentration of the hydrophobin solutions, we are able to show that buckling occurs at a critical packing fraction of protein molecules on the surface. Independent experiments show that at this packing fraction the interface has a finite positive surface tension, and not zero surface tension as is usually assumed at buckling. We attribute this non-zero wrinkling tension to the finite elasticity of these interfaces. We develop a simple geometrical model for the evolution of the wrinkle length with further deflation and show that wrinkles grow rapidly near the needle (used for deflation) towards the mid-plane of the bubble. This geometrical model yields predictions for the length of wrinkles in good agreement with experiments independently of the rheological properties of the adsorbed layer. © 2013 EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.

- R Chirat, DE Moulton and A Goriely (2013). Mechanical basis of morphogenesis and convergent evolution of spiny seashells. Proceedings of the National Academy of Sciences of the United States of America
**110**(15): 6015-6020. (doi:10.1073/pnas.1220443110) (Abstract)Convergent evolution is a phenomenon whereby similar traits evolved independently in not closely related species, and is often interpreted in functional terms. Spines in mollusk seashells are classically interpreted as having repeatedly evolved as a defense in response to shell-crushing predators. Here we consider the morphogenetic process that shapes these structures and underlies their repeated emergence. We develop a mathematical model for spine morphogenesis based on the mechanical interaction between the secreting mantle edge and the calcified shell edge to which the mantle adheres during shell growth. It is demonstrated that a large diversity of spine structures can be accounted for through small variations in control parameters of this natural mechanical process. This physical mechanism suggests that convergent evolution of spines can be understood through a generic morphogenetic process, and provides unique perspectives in understanding the phenotypic evolution of this second largest phylum in the animal kingdom.

- C Schoof and I Hewitt (2013).
Ice-Sheet Dynamics. Annual Review of Fluid Mechanics
**45**(1): 217-239. (doi:10.1146/annurev-fluid-011212-140632) - NJ Balmforth and IJ Hewitt (2013). Viscoplastic sheets and threads. Journal of Non-Newtonian Fluid Mechanics
**193**: 28-42. (doi:10.1016/j.jnnfm.2012.05.007) (Abstract)A theory is presented for the dynamics of slender sheets of viscoplastic fluid. First a model suitable for relatively small curvature is presented and applied to the fall of a liquid bridge supported at its two ends. Second, order-one curvatures are considered, along with the sagging of a beam with a free end that is either emplaced or extruded horizontally. We then present a theory combining both limits of curvature, and consider the buckling of a nearly vertical column. Analogous models for a circular viscoplastic thread are also discussed. © 2012 Elsevier B.V.

- AJ Wells, JS Wettlaufer and SA Orszag (2013). Nonlinear mushy-layer convection with chimneys: Stability and optimal solute fluxes. Journal of Fluid Mechanics
**716**: 203-227. (doi:10.1017/jfm.2012.541) (Abstract)We model buoyancy-driven convection with chimneys-channels of zero solid fraction-in a mushy layer formed during directional solidification of a binary alloy in two dimensions. A large suite of numerical simulations is combined with scaling analysis in order to study the parametric dependence of the flow. Stability boundaries are calculated for states of finite-amplitude convection with chimneys, which for a narrow domain can be interpreted in terms of a modified Rayleigh number criterion based on the domain width and mushy-layer permeability. For solidification in a wide domain with multiple chimneys, it has previously been hypothesized that the chimney spacing will adjust to optimize the rate of removal of potential energy from the system. For a wide variety of initial liquid concentration conditions, we consider the detailed flow structure in this optimal state and derive scaling laws for how the flow evolves as the strength of convection increases. For moderate mushy-layer Rayleigh numbers these flow properties support a solute flux that increases linearly with Rayleigh number. This behaviour does not persist indefinitely, however, with porosity-dependent flow saturation resulting in sublinear growth of the solute flux for sufficiently large Rayleigh numbers. Finally, we consider the influence of the porosity dependence of permeability, with a cubic function and a Carman-Kozeny permeability yielding qualitatively similar system dynamics and flow profiles for the optimal states. © 2013 Cambridge University Press.

- C Wang, MA Porter, KJH Law and PG Kevrekidis (2013). Dark solitary waves in a class of collisionally inhomogeneous Bose-Einstein condensates. Physical Review A - Atomic, Molecular, and Optical Physics
**87**(2): 023621. (doi:10.1103/PhysRevA.87.023621) (Abstract)We study the structure, stability, and dynamics of dark solitary waves in parabolically trapped, collisionally inhomogeneous Bose-Einstein condensates (BECs) with spatially periodic variations of the scattering length. This collisional inhomogeneity yields a nonlinear lattice, which we tune from a small-amplitude, approximately sinusoidal structure to a periodic sequence of densely spaced spikes. We start by investigating time-independent inhomogeneities, and we subsequently examine the dynamical response when one starts with a collisionally homogeneous BEC and then switches on an inhomogeneity either adiabatically or nonadiabatically. Using Bogoliubov-de Gennes linearization as well as direct numerical simulations of the Gross-Pitaevskii equation, we observe dark solitary waves, which can become unstable through oscillatory or exponential instabilities. We find a critical wavelength of the nonlinear lattice that is comparable to the healing length. Near this value, the fundamental eigenmode responsible for the stability of the dark solitary wave changes its direction of movement as a function of the strength of the nonlinearity. When it increases, it collides with other eigenmodes, leading to oscillatory instabilities; when it decreases, it collides with the origin and becomes imaginary, illustrating that the instability mechanism is fundamentally different in wide-well versus narrow-well lattices. When starting from a collisionally homogeneous setup and switching on inhomogeneities, we find that dark solitary waves are preserved generically for aligned lattices. We briefly examine the time scales for the onset of solitary-wave oscillations in this scenario. © 2013 American Physical Society.

- JV Pohlmeyer, SL Waters and LJ Cummings (2013). Mathematical Model of Growth Factor Driven Haptotaxis and Proliferation in a Tissue Engineering Scaffold. Bulletin of Mathematical Biology : 1-35.
- J Knychala, N Bouropoulos, CJ Catt, OL Katsamenis, CP Please and BG Sengers (2013). Pore Geometry Regulates Early Stage Human Bone Marrow Cell Tissue Formation and Organisation. Annals of Biomedical Engineering : 1-14.
- A Yavari and A Goriely (2013). Riemann-Cartan geometry of nonlinear disclination mechanics. Mathematics and Mechanics of Solids
**18**(1): 91-102. (doi:10.1177/1081286511436137) (Abstract)In the continuous theory of defects in nonlinear elastic solids, it is known that a distribution of disclinations leads, in general, to a non-trivial residual stress field. To study this problem, we consider the particular case of determining the residual stress field of a cylindrically symmetric distribution of parallel wedge disclinations. We first use the tools of differential geometry to construct a Riemannian material manifold in which the body is stress-free. This manifold is metric compatible, has zero torsion, but has non-vanishing curvature. The problem then reduces to embedding this manifold in Euclidean 3-space following the procedure of a classical nonlinear elastic problem. We show that this embedding can be elegantly accomplished by using Cartan's method of moving frames and compute explicitly the residual stress field for various distributions in the case of a neo-Hookean material. © 2012 The Author(s).

- CG Bell, PD Howell and HA Stone (2013). Time-dependent chronoamperometric response of dual inlaid disk electrodes. Journal of Electroanalytical Chemistry
**689**: 303-313. (doi:10.1016/j.jelechem.2012.10.017) (Abstract)Dual-disk electrodes allow for novel electrochemical investigations, particularly when used in generator-collector mode. However analytical understanding of how the current response of each individual electrode is affected by the presence of the neighbouring electrode is limited. In this article, we derive an approximate analytical solution for the time-dependent chronoamperometric current response of a pair of electrodes, each subjected to a different initial potential step. The solution is valid when the centre-to-centre distance between the disks is much larger than the disk radii, and for times such that the diffusion layers generated at each electrode have started to interact. It accommodates quasi-reversible reactions and unequal diffusion coefficients of the oxidant and the reductant. The generator-collector system is a specific case, and we deduce simple expressions for the time-dependent current and steady-state collection efficiencies at each electrode. The analytical solution facilitates analysis of shielding and feedback effects, and shows the explicit dependence of the currents on the underlying electrochemical parameters. © 2012 Elsevier B.V. All rights reserved.

- PJ Dellar (2013). Lattice Boltzmann magnetohydrodynamics with current-dependent resistivity. Journal of Computational Physics
**237**: 115-131. (doi:10.1016/j.jcp.2012.11.021) (Abstract)Lattice Boltzmann magnetohydrodynamics is extended to allow the resistivity to be a prescribed function of the local current density. Current-dependent resistivities are used to model the so-called anomalous resistivity caused by unresolved small-scale processes, such as current-driven plasma microturbulence, that are excluded by the magnetohydrodynamics approximation. These models closely resemble the Smagorinsky eddy viscosity model used in large eddy simulations of hydrodynamic turbulence. Lattice Boltzmann implementations of the Smagorinsky model adjust the collision time in proportion to the local rate of strain, as obtained from the non-equilibrium parts of the hydrodynamic distribution functions. This works successfully even with a single relaxation time collision operator. However, the existing lattice Boltzmann magnetohydrodynamic implementation contains a spurious term in the evolution equation for the magnetic field that violates the divergence-free condition when the relaxation time varies in space. A correct implementation requires a matrix collision operator for the magnetic distribution functions. The relaxation time imposed on the antisymmetric component of the electric field tensor is calculated locally from the non-equilibrium part of the magnetic distribution functions, which determine the current, while the symmetric component remains subject to a uniform relaxation time to suppress the spurious term. The resulting numerical solutions are shown to converge to independent spectral solutions of the magnetohydrodynamic equations, and to preserve the divergence-free condition up to floating point round-off error. © 2012.

- CG Bell, HM Byrne, SL Waters and JP Whiteley (2013). Heat or mass transfer from a sphere in Stokes flow at low Péclet number. Applied Mathematics Letters
**26**(4): 392-396. (doi:10.1016/j.aml.2012.10.010) (Abstract)We consider the low Péclet number, Pe≪1, asymptotic solution for steady-state heat or mass transfer from a sphere immersed in Stokes flow with a Robin boundary condition on its surface, representing Newton cooling or a first-order chemical reaction. The application of Van Dyke's rule up to terms of O(Pe) shows that the O(PelogPe) terms in the expression for the average Nusselt/Sherwood number are twice those previously derived in the literature. Inclusion of the O(Pe) terms is shown to increase the range of validity of the expansion. © 2012 Elsevier Ltd. All rights reserved.

- TJW Wagner and D Vella (2013). The 'Sticky Elastica': Delamination blisters beyond small deformations. Soft Matter
**9**(4): 1025-1030. (doi:10.1039/c2sm26916c) (Abstract)We consider the form of an elastic loop adhered to a rigid substrate: the 'Sticky Elastica'. In contrast to previous studies of the shape of delamination 'blisters', the theory developed accounts for deflections with large slope (i.e. geometrically nonlinear). Starting from the classical Euler Elastica we provide numerical results for the dimensions of such blisters for a variety of end-end confinements and develop asymptotic expressions that reproduce these results well, even up to the point of self-contact. Interestingly, we find that the width of such blisters does not grow monotonically with increased confinement. Our theoretical predictions are confirmed by simple desktop experiments and suggest a new method for the measurement of the elastocapillary length for deformations that cannot be considered small. We discuss the implications of our results for applications such as flexible electronics. This journal is © 2013 The Royal Society of Chemistry.

- IM Griffiths, PD Howell and RJ Shipley (2013). Control and optimization of solute transport in a thin porous tube. Physics of Fluids
**25**(3): 033101. (doi:10.1063/1.4795545) (Abstract)Predicting the distribution of solutes or particles in flows within porous-walled tubes is essential to inform the design of devices that rely on cross-flow filtration, such as those used in water purification, irrigation devices, field-flow fractionation, and hollow-fibre bioreactors for tissue-engineering applications. Motivated by these applications, a radially averaged model for fluid and solute transport in a tube with thin porous walls is derived by developing the classical ideas of Taylor dispersion. The model includes solute diffusion and advection via both radial and axial flow components, and the advection, diffusion, and uptake coefficients in the averaged equation are explicitly derived. The effect of wall permeability, slip, and pressure differentials upon the dispersive solute behaviour are investigated. The model is used to explore the control of solute transport across the membrane walls via the membrane permeability, and a parametric expression for the permeability required to generate a given solute distribution is derived. The theory is applied to the specific example of a hollow-fibre membrane bioreactor, where a uniform delivery of nutrient across the membrane walls to the extra-capillary space is required to promote spatially uniform cell growth. © 2013 American Institute of Physics.

- MR Moore, JR Ockendon and JM Oliver (2013). Air-cushioning in impact problems. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
**78**(4): 818-838. (doi:10.1093/imamat/hxt026) (Abstract)This paper concerns the displacement potential formulation of the post-impact influence of an air-cushioning layer on the 2D impact of a liquid half-space by a rigid body. The liquid and air are both ideal and incompressible and attention is focussed on cases when the density ratio between the air and liquid is small. In particular, the correction to classical Wagner theory is analysed in detail for the impact of circular cylinders and wedges. © The Authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

- LA Mihai and A Goriely (2013). Numerical simulation of shear and the Poynting effects by the finite element method: An application of the generalised empirical inequalities in non-linear elasticity. International Journal of Non-Linear Mechanics
**49**: 1-14. (doi:10.1016/j.ijnonlinmec.2012.09.001) (Abstract)Finite element simulations of different shear deformations in non-linear elasticity are presented. We pay particular attention to the Poynting effects in hyperelastic materials, complementing recent theoretical findings by showing these effects manifested by specific models. As the finite element method computes uniform deformations exactly, for simple shear deformation and pure shear stress, the Poynting effect is represented exactly, while for the generalised shear and simple torsion, where the deformation is non-uniform, the solution is approximated efficiently and guaranteed computational bounds on the magnitude of the Poynting effect are obtained. The numerical results further indicate that, for a given elastic material, the same sign effect occurs under different shearing mechanisms, showing the genericity of the Poynting effect under a variety of shearing loads. In order to derive numerical models that exhibit either the positive or the negative Poynting effect, the so-called generalised empirical inequalities, which are less restrictive than the usual empirical inequalities involving material parameters, are assumed. © 2012 Elsevier Ltd.

- AC Fowler (2013). The motion of ice stream margins. Journal of Fluid Mechanics
**714**: 1-4. (doi:10.1017/jfm.2012.504) (Abstract)The recent article by Schoof (J. Fluid Mech., vol. 712, 2012, pp. 552-578) provides a technically demanding solution to the problem of determining ice-stream margin evolution. It is important in opening the way to the future theoretical description of how the ice sheets will melt and sea level will rise as the climate warms. But the sophistication of the mathematics should not operate as a mask to an examination of the credibility of the model. © 2013 Cambridge University Press.

- CR Stokes, AC Fowler, CD Clark, RCA Hindmarsh and M Spagnolo (2013). The instability theory of drumlin formation and its explanation of their varied composition and internal structure. Quaternary Science Reviews
**62**: 77-96. (doi:10.1016/j.quascirev.2012.11.011) (Abstract)Despite their importance in understanding glaciological processes and constraining large-scale flow patterns in palaeo-glaciology, there is little consensus as to how drumlins are formed. Attempts to solve the 'drumlin problem' often fail to address how they are created from an initially flat surface in the absence of obvious cores or obstacles. This is a key strength of the instability theory, which has been described in a suite of physically-based mathematical models and proposes that the coupled flow of ice and till causes spontaneous formation of relief in the till surface. Encouragingly, model predictions of bedform height and length are consistent with observations and, furthermore, the theory has been applied to a range of subglacial bedforms and not just drumlins. However, it has yet to confront the myriad observations relating to the composition and internal structure of drumlins and this could be seen as a major deficiency. This paper is a first attempt to assess whether the instability theory is compatible with the incredible diversity of sediments and structures found within drumlins. We summarise the underlying principles of the theory and then describe and attempt to explain the main types of drumlin composition (e.g. bedrock, till, glaciofluvial sediments, and combinations thereof). Contrary to a view which suggests that the presence of some sedimentary sequences (e.g. horizontally stratified cores) is inconsistent with the theory, we suggest that one would actually expect a diverse range of constituents depending on the inheritance of sediments that pre-date drumlin formation, the duration and variability of ice flow, and the balance between erosion and deposition (till continuity) at the ice-bed interface. We conclude that the instability theory is compatible with (and potentially strengthened by) what is known about drumlin composition and, as such, offers the most complete and promising solution to the drumlin problem to date. © 2012 Elsevier Ltd.

- MG Hennessy and A Münch (2013). A multiple-scale analysis of evaporation induced Marangoni convection. SIAM Journal on Applied Mathematics
**73**(2): 974-1001. (doi:10.1137/110849006) (Abstract)This paper considers the stability of thin liquid layers of binary mixtures of a volatile (solvent) species and a nonvolatile (polymer) species. Evaporation leads to a depletion of the solvent near the liquid surface. If surface tension increases for lower solvent concentrations, sufficiently strong compositional gradients can lead to Bénard-Marangoni-type convection that is similar to the kind which is observed in films that are heated from below. The onset of the instability is investigated by a linear stability analysis. Due to evaporation, the base state is time dependent, thus leading to a nonautonomous linearized system which impedes the use of normal modes. However, the time scale for the solvent loss due to evaporation is typically long compared to the diffusive time scale, so a systematic multiple scales expansion can be sought for a finite-dimensional approximation of the linearized problem. This is determined to leading and to next order. The corrections indicate that the validity of the expansion does not depend on the magnitude of the individual eigenvalues of the linear operator, but it requires these eigenvalues to be well separated. The approximations are applied to analyze experiments by Bassou and Rharbi with polystyrene/toluene mixtures [Langmuir, 25 (2009), pp. 624-632]. © 2013 Society for Industrial and Applied Mathematics.

- M Taroni, CJW Breward, IM Griffiths and LJ Cummings (2013). Asymptotic solutions of glass temperature profiles during steady optical fibre drawing. Journal of Engineering Mathematics
**80**(1): 1-20. (doi:10.1007/s10665-013-9623-z) (Abstract)In this paper we derive realistic simplified models for the high-speed drawing of glass optical fibres via the downdraw method that capture the fluid dynamics and heat transport in the fibre via conduction, convection and radiative heating. We exploit the small aspect ratio of the fibre and the relative orders of magnitude of the dimensionless parameters that characterize the heat transfer to reduce the problem to one- or two-dimensional systems via asymptotic analysis. The resulting equations may be readily solved numerically and in many cases admit exact analytic solutions. The systematic asymptotic breakdown presented is used to elucidate the relative importance of furnace temperature profile, convection, surface radiation and conduction in each portion of the furnace and the role of each in controlling the glass temperature. The models derived predict many of the qualitative features observed in real industrial processes, such as the glass temperature profile within the furnace and the sharp transition in fibre thickness. The models thus offer a desirable route to quick scenario testing, providing valuable practical information about the dependencies of the solution on the parameters and the dominant heat-transport mechanism. © 2013 Springer Science+Business Media Dordrecht.

- IM Griffiths, CJW Breward, PJ Dellar, PD Howell, DM Colegate and CD Bain (2013). A new pathway for the re-equilibration of micellar surfactant solutions. Soft Matter
**9**(3): 853-863. (doi:10.1039/c2sm27154k) (Abstract)Micellar surfactant solutions are generally assumed to undergo restructuring via stepwise monomer loss following a dilution. This process is captured by the Becker-Döring equations, an infinite-dimensional system of ordinary differential equations for the concentration of each aggregate in solution. We reveal certain classes of surfactants, such as the non-ionic family CE, for which the predicted re-equilibration times via stepwise monomer loss are far greater than those observed experimentally. We investigate two alternative pathways for re-equilibration, first allowing for micelles to break down into two aggregate fragments rather than stepwise monomer release, and secondly by allowing aggregates to merge together to form large super-micelles that exceed the size of a proper micelle. While the former shows no discernible difference in the predicted time to re-equilibration, the latter provides an alternative pathway to re-equilibration: the formation of unstable super-micelles that break down to proper micelles via a cascade of stepwise monomer release. The new theory is shown to describe the re-equilibration of any surfactant system, with the conventional Becker-Döring theory forming a subset of the model that describes the behaviour of a small range of surfactant systems with high critical micelle concentrations and low aggregation numbers. The pathway proposed provides an essential mechanistic route to equilibrium. This journal is © 2013 The Royal Society of Chemistry.

- HM Arnold, TN Palmer and IM Moroz (2013). Stochastic parametrizations and model uncertainty in the lorenz '96 system. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
**371**(1991): 20110479-20110479. (doi:10.1098/rsta.2011.0479) (Abstract)Simple chaotic systems are useful tools for testing methods for use in numerical weather simulations owing to their transparency and computational cheapness. The Lorenz system was used here; the full system was defined as 'truth', whereas a truncated version was used as a testbed for parametrization schemes. Several stochastic parametrization schemes were investigated, including additive and multiplicative noise. The forecasts were started from perfect initial conditions, eliminating initial condition uncertainty. The stochastically generated ensembles were compared with perturbed parameter ensembles and deterministic schemes. The stochastic parametrizations showed an improvement in weather and climate forecasting skill over deterministic parametrizations. Including a temporal autocorrelation resulted in a significant improvement over white noise, challenging the standard idea that a parametrization should only represent subgridscale variability. The skill of the ensemble at representing model uncertainty was tested; the stochastic ensembles gave better estimates of model uncertainty than the perturbed parameter ensembles. The forecasting skill of the parametrizations was found to be linked to their ability to reproduce the climatology of the full model. This is important in a seamless prediction system, allowing the reliability of short-term forecasts to provide a quantitative constraint on the accuracy of climate predictions from the same system. © 2013 The Author(s) Published by the Royal Society.

- B Franz, MB Flegg, S Jonathan Chapman and R Erban (2013). Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics. SIAM Journal on Applied Mathematics
**73**(3): 1224-1247. (doi:10.1137/120882469) (Abstract)Two algorithms that combine Brownian dynami cs (BD) simulations with mean-field partial differential equations (PDEs) are presented. This PDE-assisted Brownian dynamics (PBD) methodology provides exact particle tracking data in parts of the domain, whilst making use of a mean-field reaction-diffusion PDE description elsewhere. The first PBD algorithm couples BD simulations with PDEs by randomly creating new particles close to the interface, which partitions the domain, and by reincorporating particles into the continuum PDE-description when they cross the interface. The second PBD algorithm introduces an overlap region, where both descriptions exist in parallel. It is shown that the overlap region is required to accurately compute variances using PBD simulations. Advantages of both PBD approaches are discussed and illustrative numerical examples are presented. © 2013 Society for Industrial and Applied Mathematics.

- LR Band, JM Oliver, SL Waters and DS Riley (2013). Steady symmetric low-Reynolds-number flow past a film-coated cylinder. European Journal of Applied Mathematics
**24**(1): 1-24. (doi:10.1017/S0956792512000289) - LS Kimpton, JP Whiteley, SL Waters, JR King and JM Oliver (2013). Multiple travelling-wave solutions in a minimal model for cell motility.. Math Med Biol
**30**(3): 241-272. (doi:10.1093/imammb/dqs023) (Abstract)Two-phase flow models have been used previously to model cell motility. In order to reduce the complexity inherent with describing the many physical processes, we formulate a minimal model. Here we demonstrate that even the simplest 1D, two-phase, poroviscous, reactive flow model displays various types of behaviour relevant to cell crawling. We present stability analyses that show that an asymmetric perturbation is required to cause a spatially uniform, stationary strip of cytoplasm to move, which is relevant to cell polarization. Our numerical simulations identify qualitatively distinct families of travelling-wave solutions that coexist at certain parameter values. Within each family, the crawling speed of the strip has a bell-shaped dependence on the adhesion strength. The model captures the experimentally observed behaviour that cells crawl quickest at intermediate adhesion strengths, when the substrate is neither too sticky nor too slippy.

- SL Cotter, T Vejchodský and R Erban (2013). Adaptive finite element method assisted by stochastic simulation of chemical systems. SIAM Journal on Scientific Computing
**35**(1): B107-B131. (doi:10.1137/120877374) (Abstract)Stochastic models of chemical systems are often analyzed by solving the corresponding Fokker-Planck equation, which is a drift-diffusion partial differential equation for the probability distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with nonnegligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the stationary probability density. Numerical examples demonstrate that the presented method is competitive with existing a posteriori methods. © 2013 Society for Industrial and Applied Mathematics.

- MD Gould, MA Porter, SD Howison, S Williams, M Mcdonald and DJ Fenn (2013). Limit order books. Quantitative Finance : 1-34. (doi:10.1080/14697688.2013.803148) (Abstract)
Limit order books (LOBs) match buyers and sellers in more than half of the world's financial markets. This survey highlights the insights that have emerged from the wealth of empirical and theoretical studies of LOBs. We examine the findings reported by statistical analyses of historical LOB data and discuss how several LOB models provide insight into certain aspects of the mechanism. We also illustrate that many such models poorly resemble real LOBs and that several well-established empirical facts have yet to be reproduced satisfactorily. Finally, we identify several key unresolved questions about LOBs. © 2013 Copyright Taylor and Francis Group, LLC.

### 2012

- SD Howison, JR Ockendon and AA Lacey (2012). Partial Differential Equations that are hard to classify. Journal of Partial Differential Equations
**25**: 41-65. - IJ Hewitt and NJ Balmforth (2012). Viscoplastic lubrication theory with application to bearings and the washboard instability of a planing plate. Journal of Non-Newtonian Fluid Mechanics
**169-170**: 74-90. (doi:10.1016/j.jnnfm.2011.11.008) (Abstract)We consider lubrication theory for a two-dimensional viscoplastic fluid confined between rigid moving boundaries. A general formulation is presented which allows the flow field and pressure to be calculated given an arbitrary rheological model; the Herschel-Bulkley law is used for illustration. The theory is first applied to a (full) viscoplastic journal bearing with arbitrary motions allowed for the inner cylinder (either prescribed, or arising from an imposed load and torque). Conditions are derived determining when motion is arrested by the yield stress. We next apply the theory to a slider bearing filled with Bingham fluid, computing the lift force on the bearing and the fluid flux through it. The results are then extended to model an inclined plate that is towed at constant horizontal speed over a shallow viscoplastic layer but is able to move vertically. Steady planing solutions are stable at low towing speeds, but give way to unstable vertical oscillations of the plate at higher speed; the yield stress has a relatively weak effect on this instability. The pattern imprinted on the fluid layer by the oscillations provides an analogue of the washboard phenomenon on gravel roads. © 2011 Elsevier B.V.

- IJ Hewitt, NJ Balmforth and JN Mcelwaine (2012). Granular and fluid washboards. Journal of Fluid Mechanics
**692**: 446-463. (doi:10.1017/jfm.2011.523) (Abstract)Abstract We investigate the dynamics of an object towed over the surface of an initially flat, deformable layer. Using a combination of simple laboratory experiments and a theoretical model, we demonstrate that an inclined plate, pivoted so as to move up and down, may be towed steadily over a substrate at low speed, but become unstable to vertical oscillations above a threshold speed. That threshold depends upon the weight of the plate and the physical properties of the substrate, but arises whether the substrate is a viscous fluid, a viscoplastic fluid, or a granular medium. For the latter two materials, the unstable oscillations imprint a permanent rippled pattern on the layer, suggesting that the phenomenon of the 'washboard road' can arise from the passage of a single vehicle (i.e. the absolute instability of a flat bed). We argue that the mechanism behind the instability originates from the mound of material that is pushed forward ahead of the object: the extent of the mound determines the resultant force, whereas its growth is controlled by the object's height relative to the undisturbed surface, allowing for an unstable coupling between the vertical motion and the substrate deformation. © 2012 Cambridge University Press.

- IJ Hewitt, C Schoof and MA Werder (2012). Flotation and free surface flow in a model for subglacial drainage. Part 2. Channel flow. Journal of Fluid Mechanics
**702**: 157-187. (doi:10.1017/jfm.2012.166) (Abstract)We present a new model of subglacial drainage incorporating flow in a network of channels and a porous sheet, with water exchange between the two determined by pressure gradients. The sheet represents the average effect of many linked cavities, whilst the channels emerge from individual cavities that enlarge due to dissipation-induced melting. The model distinguishes cases when the water pressure drops to zero, in which case it allows for the drainage space to be only partially filled with water (free surface flow), and when the pressure reaches the ice overburden pressure, in which case it allows for uplift of the ice to whatever extent is needed to accommodate the water (flotation). Numerical solutions are found for a one-dimensional flow-line version of the model. The results capture typically observed or inferred features of subglacial drainage systems, including open channel flow at the ice margin, seasonal channel evolution, and high water pressures and uplift of the ice surface driven by rapid changes in water supply. © 2012 Cambridge University Press.

- C Schoof, IJ Hewitt and MA Werder (2012). Flotation and free surface flow in a model for subglacial drainage. Part 1. Distributed drainage. Journal of Fluid Mechanics
**702**: 126-156. (doi:10.1017/jfm.2012.165) (Abstract)We present a continuum model for melt water drainage through a spatially distributed system of connected subglacial cavities, and consider in this context the complications introduced when effective pressure or water pressure drops to zero. Instead of unphysically allowing water pressure to become negative, we model the formation of a partially vapour-or air-filled space between ice and bed. Likewise, instead of allowing sustained negative effective pressures, we allow ice to separate from the bed at zero effective pressure. The resulting model is a free boundary problem in which an elliptic obstacle problem determines hydraulic potential, and therefore also determines regions of zero effective pressure and zero water pressure. This is coupled with a transport problem for stored water, and the coupled system bears some similarities with Hele-Shaw and squeeze-film models. We present a numerical method for computing time-dependent solutions, and find close agreement with semi-analytical travelling wave and steady-state solutions. As may be expected, we find that ice-bed separation is favoured by high fluxes and low ice surface slopes and low bed slopes, while partially filled cavities are favoured by low fluxes and high slopes. At the boundaries of regions with zero water or effective pressure, discontinuities in water level are frequently present, either in the form of propagating shocks or as stationary hydraulic jumps accompanied by discontinuities in potential gradient. © 2012 Cambridge University Press.

- Y Van Gennip, H Hu, B Hunter and MA Porter (2012). Geosocial graph-based community detection. Proceedings - 12th Ieee International Conference on Data Mining Workshops, Icdmw 2012 : 754-758. (doi:10.1109/ICDMW.2012.63) (Abstract)
We apply spectral clustering and multislice modularity optimization to a Los Angeles Police Department field interview card data set. To detect communities (i.e., cohesive groups of vertices), we use both geographic and social information about stops involving street gang members in the LAPD district of Hollenbeck. We then compare the algorithmically detected communities with known gang identifications and argue that discrepancies are due to sparsity of social connections in the data as well as complex underlying sociological factors that blur distinctions between communities. © 2012 IEEE.

- H Hu, Y Van Gennip, B Hunter, AL Bertozzi and MA Porter (2012). Multislice modularity optimization in community detection and image segmentation. Proceedings - 12th Ieee International Conference on Data Mining Workshops, Icdmw 2012 : 934-936. (doi:10.1109/ICDMW.2012.72) (Abstract)
Because networks can be used to represent many complex systems, they have attracted considerable attention in physics, computer science, sociology, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups (i.e., "communities") of nodes. In this paper, we algorithmically detect communities in social networks and image data by optimizing multislice modularity. A key advantage of modularity optimization is that it does not require prior knowledge of the number or sizes of communities, and it is capable of finding network partitions that are composed of communities of different sizes. By optimizing multislice modularity and subsequently calculating diagnostics on the resulting network partitions, it is thereby possible to obtain information about network structure across multiple system scales. We illustrate this method on data from both social networks and images, and we find that optimization of multislice modularity performs well on these two tasks without the need for extensive problem specific adaptation. However, improving the computational speed of this method remains a challenging open problem. © 2012 IEEE.

- T Reis and PJ Dellar (2012). Lattice Boltzmann simulations of pressure-driven flows in microchannels using Navier-Maxwell slip boundary conditions. Physics of Fluids
**24**(11): 112001. (doi:10.1063/1.4764514) (Abstract)We present lattice Boltzmann simulations of rarefied flows driven by pressure drops along two-dimensional microchannels. Rarefied effects lead to non-zero cross-channel velocities, nonlinear variations in the pressure along the channel. Both effects are absent in flows driven by uniform body forces. We obtain second-order accuracy for the two components of velocity the pressure relative to asymptotic solutions of the compressible Navier-Stokes equations with slip boundary conditions. Since the common lattice Boltzmann formulations cannot capture Knudsen boundary layers, we replace the usual discrete analogs of the specular diffuse reflection conditions from continuous kinetic theory with a moment-based implementation of the first-order Navier-Maxwell slip boundary conditions that relate the tangential velocity to the strain rate at the boundary. We use these conditions to solve for the unknown distribution functions that propagate into the domain across the boundary. We achieve second-order accuracy by reformulating these conditions for the second set of distribution functions that arise in the derivation of the lattice Boltzmann method by an integration along characteristics. Our moment formalism is also valuable for analysing the existing boundary conditions. It reveals the origin of numerical slip in the bounce-back other common boundary conditions that impose conditions on the higher moments, not on the local tangential velocity itself. © 2012 American Institute of Physics.

- G Richardson, C Please, J Foster and J Kirkpatrick (2012). Asymptotic solution of a model for bilayer organic diodes and solar cells. SIAM Journal on Applied Mathematics
**72**(6): 1792-1817. (doi:10.1137/110825807) (Abstract)Organic diodes and solar cells are constructed by placing together two organic semiconducting materials with dissimilar electron affinities and ionization potentials. The electrical behavior of such devices has been successfully modeled numerically using conventional drift diffusion together with recombination (which is usually assumed to be bimolecular) and thermal generation. Here a particular model is considered and the dark current-voltage curve and the spatial structure of the solution across the device is extracted analytically using asymptotic methods. We concentrate on the case of Shockley-Read-Hall recombination but note the extension to other recombination mechanisms. We find that there are three regimes of behavior, dependent on the total current. For small currents-i.e., at reverse bias or moderate forward bias-the structure of the solution is independent of the total current. For large currents-i.e., at strong forward bias-the current varies linearly with the voltage and is primarily controlled by drift of charges in the organic layers. There is then a narrow range of currents where the behavior undergoes a transition between the two regimes. The magnitude of the parameter that quantifies the interfacial recombination rate is critical in determining where the transition occurs. The extension of the theory to organic solar cells generating current under illumination is discussed as is the analogous current-voltage curves derived where the photo current is small. Finally, by comparing the analytic results to real experimental data, we show how the model parameters can be extracted from the shape of current-voltage curves measured in the dark. © 2012 Society for Industrial and Applied Mathematics.

- T Székely Jr, K Burrage, R Erban and KC Zygalakis (2012). A higher-order numerical framework for stochastic simulation of chemical reaction systems. Bmc Systems Biology
**6**: . (doi:10.1186/1752-0509-6-85) (Abstract)Background: In this paper, we present a framework for improving the accuracy of fixed-step methods for Monte Carlo simulation of discrete stochastic chemical kinetics. Stochasticity is ubiquitous in many areas of cell biology, for example in gene regulation, biochemical cascades and cell-cell interaction. However most discrete stochastic simulation techniques are slow. We apply Richardson extrapolation to the moments of three fixed-step methods, the Euler, midpoint and θ-trapezoidal τ-leap methods, to demonstrate the power of stochastic extrapolation. The extrapolation framework can increase the order of convergence of any fixed-step discrete stochastic solver and is very easy to implement; the only condition for its use is knowledge of the appropriate terms of the global error expansion of the solver in terms of its stepsize. In practical terms, a higher-order method with a larger stepsize can achieve the same level of accuracy as a lower-order method with a smaller one, potentially reducing the computational time of the system.Results: By obtaining a global error expansion for a general weak first-order method, we prove that extrapolation can increase the weak order of convergence for the moments of the Euler and the midpoint τ-leap methods, from one to two. This is supported by numerical simulations of several chemical systems of biological importance using the Euler, midpoint and θ-trapezoidal τ-leap methods. In almost all cases, extrapolation results in an improvement of accuracy. As in the case of ordinary and stochastic differential equations, extrapolation can be repeated to obtain even higher-order approximations.Conclusions: Extrapolation is a general framework for increasing the order of accuracy of any fixed-step stochastic solver. This enables the simulation of complicated systems in less time, allowing for more realistic biochemical problems to be solved. © 2012 Székely et al.; licensee BioMed Central Ltd.

- DS Bassett, ET Owens, KE Daniels and MA Porter (2012). Influence of network topology on sound propagation in granular materials.. Phys Rev E Stat Nonlin Soft Matter Phys
**86**(4 Pt 1): 041306. (Abstract)Granular media, whose features range from the particle scale to the force-chain scale and the bulk scale, are usually modeled as either particulate or continuum materials. In contrast with each of these approaches, network representations are natural for the simultaneous examination of microscopic, mesoscopic, and macroscopic features. In this paper, we treat granular materials as spatially embedded networks in which the nodes (particles) are connected by weighted edges obtained from contact forces. We test a variety of network measures to determine their utility in helping to describe sound propagation in granular networks and find that network diagnostics can be used to probe particle-, curve-, domain-, and system-scale structures in granular media. In particular, diagnostics of mesoscale network structure are reproducible across experiments, are correlated with sound propagation in this medium, and can be used to identify potentially interesting size scales. We also demonstrate that the sensitivity of network diagnostics depends on the phase of sound propagation. In the injection phase, the signal propagates systemically, as indicated by correlations with the network diagnostic of global efficiency. In the scattering phase, however, the signal is better predicted by mesoscale community structure, suggesting that the acoustic signal scatters over local geographic neighborhoods. Collectively, our results demonstrate how the force network of a granular system is imprinted on transmitted waves.

- T Hoffmann, MA Porter and R Lambiotte (2012). Generalized master equations for non-Poisson dynamics on networks.. Phys Rev E Stat Nonlin Soft Matter Phys
**86**(4 Pt 2): 046102. (Abstract)The traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static weighted network. This implicitly assumes that the edges are governed by Poisson processes, which is not typically the case in empirical temporal networks. Accordingly, we examine the effects of non-Poisson inter-event statistics on the dynamics of edges, and we apply the concept of a generalized master equation to the study of continuous-time random walks on networks. We show that this equation reduces to the standard rate equations when the underlying process is Poissonian and that its stationary solution is determined by an effective transition matrix whose leading eigenvector is easy to calculate. We conduct numerical simulations and also derive analytical results for the stationary solution under the assumption that all edges have the same waiting-time distribution. We discuss the implications of our work for dynamical processes on temporal networks and for the construction of network diagnostics that take into account their nontrivial stochastic nature.

- D Friedrich, T Melvin and CP Please (2012). The design of microfluidic affinity chromatography systems for the separation of bioanalytes. Journal of Chromatography B: Analytical Technologies in the Biomedical and Life Sciences
**910**: 163-171. (doi:10.1016/j.jchromb.2012.09.041) (Abstract)The analytical (numerical) design of planar microfluidic affinity chromatography devices, which consist of multiple separation lanes and multiple, different surface-immobilised receptor patterns in each lane, is described. The model is based on the analytical solution of the transport-reaction equations in microfluidic systems of low Gratz number and for injection of small analyte plugs. The results reveal a simple approach for the design of microfluidic affinity chromatography devices tailored to the separation of bioanalytes, where receptors with high binding affinity are available. These devices have been designed for bioanalytical applications in mind, most notably for the proteomics field; the results are illustrated with an example using β-Amyloid binding peptides. © 2012 Elsevier B.V.

- PD Howell, H Ockendon and JR Ockendon (2012). Mathematical modelling of elastoplasticity at high stress. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**468**(2148): 3842-3863. (doi:10.1098/rspa.2012.0269) (Abstract)This study describes a simple mathematical model for one-dimensional elastoplastic wave propagation in a metal in the regime where the applied stress greatly exceeds the yield stress. Attention is focused on the increasing ductility that occurs in the overdriven limit when the plastic wave speed approaches the elastic wave speed. Our model predicts that a plastic compression wave is unable to travel faster than the elastic wave speed, and instead splits into a compressive elastoplastic shock followed by a plastic expansion wave. © 2012 The Royal Society.

- RM Lewis, S Brooks, IP Crocker, J Glazier, MA Hanson, ED Johnstone, N Panitchob, CP Please, CP Sibley, KL Widdows and BG Sengers (2012). Review: Modelling placental amino acid transfer - From transporters to placental function. Placenta : .
- M Bruna and SJ Chapman (2012). Diffusion of multiple species with excluded-volume effects. Journal of Chemical Physics
**137**(20): 204116. (doi:10.1063/1.4767058) (Abstract)Stochastic models of diffusion with excluded-volume effects are used to model many biological and physical systems at a discrete level. The average properties of the population may be described by a continuum model based on partial differential equations. In this paper we consider multiple interacting subpopulations/species and study how the inter-species competition emerges at the population level. Each individual is described as a finite-size hard core interacting particle undergoing Brownian motion. The link between the discrete stochastic equations of motion and the continuum model is considered systematically using the method of matched asymptotic expansions. The system for two species leads to a nonlinear cross-diffusion system for each subpopulation, which captures the enhancement of the effective diffusion rate due to excluded-volume interactions between particles of the same species, and the diminishment due to particles of the other species. This model can explain two alternative notions of the diffusion coefficient that are often confounded, namely collective diffusion and self-diffusion. Simulations of the discrete system show good agreement with the analytic results. © 2012 American Institute of Physics.

- S Howison and D Schwarz (2012).
Risk-Neutral Pricing of Financial Instruments in Emission Markets: A Structural Approach. SIAM Journal on Financial Mathematics
**3**(1): 709-739. (doi:10.1137/100815219) - CG Bell, HM Byrne, JP Whiteley and SL Waters (2012). Heat or mass transfer from a sphere in Stokes flow at low Péclet number. Applied Mathematics Letters : .
- DE Moulton and A Goriely (2012). Surface growth kinematics via local curve evolution. Journal of Mathematical Biology
**4**(2): 1-28. (doi:10.1007/s00285-012-0625-7) (Abstract)A mathematical framework is developed to model the kinematics of surface growth for objects that can be generated by evolving a curve in space, such as seashells and horns. Growth is dictated by a growth velocity vector field defined at every point on a generating curve. A local orthonormal basis is attached to each point of the generating curve and the velocity field is given in terms of the local coordinate directions, leading to a fully local and elegant mathematical structure. Several examples of increasing complexity are provided, and we demonstrate how biologically relevant structures such as logarithmic shells and horns emerge as analytical solutions of the kinematics equations with a small number of parameters that can be linked to the underlying growth process. Direct access to cell tracks and local orientation enables for connections to be made to the underlying growth process. © 2012 Springer-Verlag Berlin Heidelberg.

- DS Bassett, ET Owens, KE Daniels and MA Porter (2012). Influence of network topology on sound propagation in granular materials. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**86**(4): 041306. (doi:10.1103/PhysRevE.86.041306) (Abstract)Granular media, whose features range from the particle scale to the force-chain scale and the bulk scale, are usually modeled as either particulate or continuum materials. In contrast with each of these approaches, network representations are natural for the simultaneous examination of microscopic, mesoscopic, and macroscopic features. In this paper, we treat granular materials as spatially embedded networks in which the nodes (particles) are connected by weighted edges obtained from contact forces. We test a variety of network measures to determine their utility in helping to describe sound propagation in granular networks and find that network diagnostics can be used to probe particle-, curve-, domain-, and system-scale structures in granular media. In particular, diagnostics of mesoscale network structure are reproducible across experiments, are correlated with sound propagation in this medium, and can be used to identify potentially interesting size scales. We also demonstrate that the sensitivity of network diagnostics depends on the phase of sound propagation. In the injection phase, the signal propagates systemically, as indicated by correlations with the network diagnostic of global efficiency. In the scattering phase, however, the signal is better predicted by mesoscale community structure, suggesting that the acoustic signal scatters over local geographic neighborhoods. Collectively, our results demonstrate how the force network of a granular system is imprinted on transmitted waves. © 2012 American Physical Society.

- DE Moulton, T Lessinnes and A Goriely (2012). Morphoelastic rods. Part I: A single growing elastic rod. Journal of the Mechanics and Physics of Solids : .
- A Yavari and A Goriely (2012). Weyl geometry and the nonlinear mechanics of distributed point defects. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**468**(2148): 3902-3922. (doi:10.1098/rspa.2012.0342) (Abstract)The residual stress field of a nonlinear elastic solid with a spherically symmetric distribution of point defects is obtained explicitly using methods from differential geometry. The material manifold of a solid with distributed point defects-where the body is stress-free-is a flat Weyl manifold, i.e. a manifold with an affine connection that has non-metricity with vanishing traceless part, but both its torsion and curvature tensors vanish. Given a spherically symmetric point defect distribution, we construct its Weyl material manifold using the method of Cartan's moving frames. Having the material manifold, the anelasticity problem is transformed to a nonlinear elasticity problem and reduces the problem of computing the residual stresses to finding an embedding into the Euclidean ambient space. In the case of incompressible neo-Hookean solids, we calculate explicitly this residual stress field. We consider the example of a finite ball and a point defect distribution uniform in a smaller ball and vanishing elsewhere. We show that the residual stress field inside the smaller ball is uniform and hydrostatic. We also prove a nonlinear analogue of Eshelby's celebrated inclusion problem for a spherical inclusion in an isotropic incompressible nonlinear solid. © 2012 The Royal Society.

- MD Korzec, B Wagner and A Münch (2012). Anisotropic surface energy formulations and their effect on stability of a growing thin film. Interfaces and Free Boundaries
**14**(4): 545-567. (doi:10.4171/IFB/291) (Abstract)In this paper we revisit models for the description of the evolution of crystalline films with anisotropic surface energies.We prove equivalences of symmetry properties of anisotropic surface energy models commonly used in the literature. Then we systematically develop a framework for the derivation of surface diffusion models for the self-assembly of quantum dots during Stranski-Krastanov growth that include surface energies also with large anisotropy as well as the effect of wetting energy, elastic energy and a randomly perturbed atomic deposition flux. A linear stability analysis for the resulting sixth-order semilinear evolution equation for the thin film surface shows that that the new model allows for large anisotropy and gives rise to the formation of anisotropic quantum dots. The nonlinear three-dimensional evolution is investigated via numerical solutions. These suggest that increasing anisotropy stabilizes the faceted surfaces and may lead to a dramatic slow-down of the coarsening of the dots. © European Mathematical Society 2012.

- J Schmidt, W Peukert, R Prignitz, E Bänsch, D Peschka, A Münch and B Wagner (2012). Conductivity in nonpolar media: Experimental and numerical studies on sodium AOT-hexadecane, lecithin-hexadecane and aluminum(III)-3,5-diisopropyl salicylate-hexadecane systems. Journal of Colloid and Interface Science
**386**(1): 240-251. (doi:10.1016/j.jcis.2012.07.051) (Abstract)The conductivity behavior of doped hydrocarbon systems is studied by applying impedance spectroscopy. In the case of 3,5-diisopropyl salicylato aluminum (III) the charge carriers are formed by dissociation of the compound and their concentration is proportional to the square root of the solute concentration. In hydrocarbon systems that consist of micelle forming compounds (sodium AOT/lecithin) a linear dependence of charge carrier concentration on solute concentration is observed in the concentration regime where micelles are present. The conduction mechanisms are studied by numerical solution of a Poisson-Nernst-Planck system that describes the charge transport. We follow two different approaches to extract the degree of micelle dissociation from the impedance data. Firstly, by computing the response of a linear approximation of the Poisson-Nernst-Planck model, and secondly by computing the fully nonlinear response from direct numerical simulations using finite elements. For high and moderate frequencies both approaches agree very well with the experimental data. For small frequencies the response becomes nonlinear and the concept of impedance fails. Furthermore, the numerically computed values for the degree of dissociation are of the same order of magnitude as the values obtained with classical formulas, but still differ by a factor of about 1/3. The direct numerical simulation allows new insight into the conduction mechanisms for different frequency regimes. © 2012 Elsevier Inc.

- G Richardson, CP Please and G Denuault (2012). Multiscale modelling and analysis of lithium-ion battery charge and discharge. Journal of Engineering Mathematics
**72**(1): 41-72. (doi:10.1007/s10665-011-9461-9) (Abstract)A microscopic model of a lithium battery is developed, which accounts for lithium diffusion within particles, transfer of lithium from particles to the electrolyte and transport within the electrolyte assuming a dilute electrolyte and Butler-Volmer reaction kinetics. Exploiting the small size of the particles relative to the electrode dimensions, a homogenised model (in agreement with existing theories) is systematically derived and studied. Details of how the various averaged quantities relate to the underlying geometry and assumptions are given. The novel feature of the homogenisation process is that it allows the coefficients in the electrode-scale model to be derived in terms of the microscopic features of the electrode (e. g. particle size and shape) and can thus be used as a systematic way of investigating the effects of changes in particle design. Asymptotic methods are utilised to further simplify the model so that one-dimensional behaviour can be described with relatively simpler expressions. It is found that for low discharge currents, the battery acts almost uniformly while above a critical current, regions of the battery become depleted of lithium ions and have greatly reduced reaction rates leading to spatially nonuniform use of the electrode. The asymptotic approximations are valid for electrode materials where the OCV is a strong function of intercalated lithium concentration, such as Li C , but not for materials with a flat discharge curve, such as LiFePO . © 2011 Springer Science+Business Media B.V.

- JM Foster, CP Please, AD Fitt and G Richardson (2012). The reversing of interfaces in slow diffusion processes with strong absorption. SIAM Journal on Applied Mathematics
**72**(1): 144-162. (doi:10.1137/100798089) (Abstract)This paper considers a family of one-dimensional nonlinear diffusion equations with absorption. In particular, the solutions that have interfaces that change their direction of propagation are examined. Although this phenomenon of reversing interfaces has been seen numerically, and some special exact solutions have been obtained, there was previously no analytical insight into how this occurs in the general case. The approach taken here is to seek self-similar solutions local to the interface and local to the reversing time. The analysis is split into two parts, one for the solution prior to the reversing time and the other for the solution after the reversing time. In each case the governing PDE is reduced to an ODE by introducing a self-similar coordinate system. These ODEs do not readily admit any nontrivial exact solutions and so the asymptotic behavior of solutions is studied. By doing this the adjustable parameters, or degrees of freedom, which may be used in a numerical shooting scheme are determined. A numerical algorithm is then proposed to furnish solutions to the ODEs and hence the PDE in the limit of interest. As examples of physical problems in which a PDE of this type may be used as a model the authors study the spreading of a viscous film under gravity and subject to evaporation, the dispersion of a population, and a nonlinear heat conduction problem. The numerical algorithm is demonstrated using these examples. Results are also given on the possible existence of self-similar solutions and types of reversing behavior that can be exhibited by PDEs in the family of interest. © 2012 Society for Industrial and Applied Mathematics.

- JM Foster, CP Please and AD Fitt (2012). The slow spreading of several viscous films over a deep viscous pool. Physics of Fluids
**24**(6): 063601. (doi:10.1063/1.4726080) (Abstract)In this study, a previously derived two-dimensional model is used to describe the slow spreading of viscous films on the surface of a quiescent deep viscous pool due to gravity. It is assumed that the densities and viscosities of the fluids in the films and pool are comparable, but may be different. It is also assumed that surface tension effects are negligible. The fluid in the films and in the pool are both modelled using the Stokes flow equations. By exploiting the slenderness of the spreading films, asymptotic techniques are used to analyse the flow. It is shown that the dominant forces controlling the spreading are gravity and the tangential stress induced in the films by the underlying pool. As a consequence, the rate of spreading of the films is independent of their viscosity. For the case special of a symmetric configuration of films on the surface of the pool, the flow is studied by assuming that the solution becomes self-similar and hence the problem is recast in a self-similar coordinate system. Stokeslet analysis is then used to derive a singular integral equation for the stresses on the interfaces between the films and the pool. The form of this integral equation depends on the configuration of spreading films that are to be considered. A number of different cases are then studied, namely, a single film, two films, and an infinite periodic array of films. Finally, some results are derived that apply to a general symmetric configuration of films. It is shown that the profile of a spreading film close to its front (where the film thickness becomes zero) is proportional to x1/4. It is also shown that fronts move, and hence, the distance between adjacent fronts increases proportional to t . © 2012 American Institute of Physics.

- D Friedrich, T Melvin and CP Please (2012). Design of novel microfluidic concentration gradient generators suitable for linear and exponential concentration ranges. Chemical Engineering Journal
**193-194**: 296-303. (doi:10.1016/j.cej.2012.04.041) (Abstract)A novel microfluidic concentration gradient generator is designed where secondary flow, induced via a surface groove, is used to yield a concentration gradient across the output of the microfluidic device. The concentration gradient generator design consists of a single microfluidic channel with two inputs and a single obliquely angled surface groove within the base of the channel to induce the secondary flow and create the concentration gradient. The design allows a concentration gradient to be chosen, either linear or exponential, at the exit of the microfluidic channel with the shape and dimensions of the surface groove within the channel obtained by numerical optimisation. The designed device has a small footprint, suitable for integration within lab-on-a-chip structures for the delivery of a series of an agent to either (i) a single channel and with a concentration gradient or to (ii) a series of reactors with concentrations across a defined range, for bioscience or pharmaceutical screening applications or for chemical reactions. © 2012 Elsevier B.V.

- D Friedrich, CP Please and T Melvin (2012). The design of microfluidic affinity chromatography systems for the separation of bioanalytes. Journal of Chromatography B: Analytical Technologies in the Biomedical and Life Sciences : .
- T Hoffmann, MA Porter and R Lambiotte (2012). Generalized master equations for non-Poisson dynamics on networks. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**86**(4): 046102. (doi:10.1103/PhysRevE.86.046102) (Abstract)The traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static weighted network. This implicitly assumes that the edges are governed by Poisson processes, which is not typically the case in empirical temporal networks. Accordingly, we examine the effects of non-Poisson inter-event statistics on the dynamics of edges, and we apply the concept of a generalized master equation to the study of continuous-time random walks on networks. We show that this equation reduces to the standard rate equations when the underlying process is Poissonian and that its stationary solution is determined by an effective transition matrix whose leading eigenvector is easy to calculate. We conduct numerical simulations and also derive analytical results for the stationary solution under the assumption that all edges have the same waiting-time distribution. We discuss the implications of our work for dynamical processes on temporal networks and for the construction of network diagnostics that take into account their nontrivial stochastic nature. © 2012 American Physical Society.

- M Taroni and D Vella (2012). Multiple equilibria in a simple elastocapillary system. Journal of Fluid Mechanics
**712**: 273-294. (doi:10.1017/jfm.2012.418) (Abstract)We consider the elastocapillary interaction of a liquid drop placed between two elastic beams, which are both clamped at one end to a rigid substrate. This is a simple model system relevant to the problem of surface-tension-induced collapse of flexible micro-channels that has been observed in the manufacture of microelectromechanical systems (MEMS). We determine the conditions under which the beams remain separated, touch at a point, or stick along a portion of their length. Surprisingly, we show that in many circumstances multiple equilibrium states are possible. We develop a lubrication-type model for the flow of liquid out of equilibrium and thereby investigate the stability of the multiple equilibria. We demonstrate that for given material properties two stable equilibria may exist, and show via numerical solutions of the dynamic model that it is the initial state of the system that determines which stable equilibrium is ultimately reached. © 2012 Cambridge University Press.

- D Vella, A Ajdari, A Vaziri and A Boudaoud (2012). Indentation of ellipsoidal and cylindrical elastic shells. Physical Review Letters
**109**(14): 144302. (doi:10.1103/PhysRevLett.109.144302) (Abstract)Thin shells are found in nature at scales ranging from viruses to hens' eggs; the stiffness of such shells is essential for their function. We present the results of numerical simulations and theoretical analyses for the indentation of ellipsoidal and cylindrical elastic shells, considering both pressurized and unpressurized shells. We provide a theoretical foundation for the experimental findings of Lazarus etal. [following paper, Phys. Rev. Lett. 109, 144301 (2012)PRLTAO0031-9007] and for previous work inferring the turgor pressure of bacteria from measurements of their indentation stiffness; we also identify a new regime at large indentation. We show that the indentation stiffness of convex shells is dominated by either the mean or Gaussian curvature of the shell depending on the pressurization and indentation depth. Our results reveal how geometry rules the rigidity of shells. © 2012 American Physical Society.

- CE Morgan, CJW Breward, IM Griffiths, PD Howell, J Penfold, RK Thomas, JRP Webster, I Tucker and JT Petkov (2012). Kinetics of surfactant desorption at an air-solution interface. Langmuir
**28**(50): 17339-17348. (doi:10.1021/la304091g) (Abstract)The kinetics of re-equilibration of the anionic surfactant sodium dodecylbenzene sulfonate at the air-solution interface have been studied using neutron reflectivity. The experimental arrangement incorporates a novel flow cell in which the subphase can be exchanged (diluted) using a laminar flow while the surface region remains unaltered. The rate of the re-equilibration is relatively slow and occurs over many tens of minutes, which is comparable with the dilution time scale of approximately 10-30 min. A detailed mathematical model, in which the rate of the desorption is determined by transport through a near-surface diffusion layer into a diluted bulk solution below, is developed and provides a good description of the time-dependent adsorption data. A key parameter of the model is the ratio of the depth of the diffusion layer, H , to the depth of the fluid, H, and we find that this is related to the reduced Péclet number, Pe*, for the system, via H/H = C/Pe*. Although from a highly idealized experimental arrangement, the results provide an important insight into the "rinse mechanism", which is applicable to a wide variety of domestic and industrial circumstances. © 2012 American Chemical Society.

- RJ Shipley, MJ Ellis and SL Waters (2012). Fluid and mass transport modelling to drive the design of cell-packed hollow fibre bioreactors. Journal of Tissue Engineering and Regenerative Medicine
**6**: 390-390. - AL Stewart and PJ Dellar (2012). Cross-equatorial channel flow with zero potential vorticity under the complete Coriolis force. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
**77**(5): 626-651. (doi:10.1093/imamat/hxs045) (Abstract)We investigate the cross-equatorial flow of deep ocean currents through a channel along the sea floor using the 1-layer or equivalent-barotropic shallow water equations. We restrict our attention to flows with zero potential vorticity, motivated by measurements of the deep Atlantic ocean, and focus on the role of the so-called non-traditional components of the Coriolis force due to the locally horizontal component of the Earth's rotation vector. These components are typically neglected in theoretical studies of ocean dynamics. We first obtain steady asymptotic solutions in a straight-walled channel by assuming that the channel half-length is much larger than the intrinsic lengthscale, the equatorial Rossby deformation radius. The leading-order solution describes a current that switches from the western to the eastern side of the channel as it crosses from the southern to the northern hemisphere. Including the non-traditional component of the Coriolis force substantially increases the cross-equatorial transport as long as the flow is everywhere northwards, but substantially decreases the transport if part of the flow retroflects and returns to the southern hemisphere. We compare our steady asymptotic solutions with time-dependent numerical solutions of the non-traditional shallow water equations. We impose the a priori assumption of zero potential vorticity by writing the fluid's absolute angular momentum as the gradient of a scalar potential. The time-average of our numerical solutions converges to our steady asymptotic solutions in the limit of large channel length, with an error of only around 1 even when our asymptotic parameter ε is equal to 1. However, our solutions diverge when the layer depth is sufficiently small that portions of the flow become super-critical. The resulting formation of steady shocks prevents the time-dependent solution from approaching the asymptotic solution. © 2012 The authors.

- RD O'Dea, JM Osborne, HM Byrne, AJ El Haj and SL Waters (2012). The interplay between tissue growth and scaffold degradation in engineered tissue constructs. Journal of Mathematical Biology
**67**: 1-27. (doi:10.1007/s00285-012-0587-9) (Abstract)In vitro tissue engineering is emerging as a potential tool to meet the high demand for replacement tissue, caused by the increased incidence of tissue degeneration and damage. A key challenge in this field is ensuring that the mechanical properties of the engineered tissue are appropriate for the in vivo environment. Achieving this goal will require detailed understanding of the interplay between cell proliferation, extracellular matrix (ECM) deposition and scaffold degradation. In this paper, we use a mathematical model (based upon a multiphase continuum framework) to investigate the interplay between tissue growth and scaffold degradation during tissue construct evolution in vitro. Our model accommodates a cell population and culture medium, modelled as viscous fluids, together with a porous scaffold and ECM deposited by the cells, represented as rigid porous materials. We focus on tissue growth within a perfusion bioreactor system, and investigate how the predicted tissue composition is altered under the influence of (1) differential interactions between cells and the supporting scaffold and their associated ECM, (2) scaffold degradation, and (3) mechanotransduction-regulated cell proliferation and ECM deposition. Numerical simulation of the model equations reveals that scaffold heterogeneity typical of that obtained from {Mathematical expression}CT scans of tissue engineering scaffolds can lead to significant variation in the flow-induced mechanical stimuli experienced by cells seeded in the scaffold. This leads to strong heterogeneity in the deposition of ECM. Furthermore, preferential adherence of cells to the ECM in favour of the artificial scaffold appears to have no significant influence on the eventual construct composition; adherence of cells to these supporting structures does, however, lead to cell and ECM distributions which mimic and exaggerate the heterogeneity of the underlying scaffold. Such phenomena have important ramifications for the mechanical integrity of engineered tissue constructs and their suitability for implantation in vivo. © 2012 Springer-Verlag.

- C Luo, A Majumdar and R Erban (2012). Multistability in planar liquid crystal wells.. Phys Rev E Stat Nonlin Soft Matter Phys
**85**(6 Pt 1): 061702. (Abstract)A planar bistable liquid crystal device, reported in Tsakonas et al. [Appl. Phys. Lett. 90, 111913 (2007)], is modeled within the Landau-de Gennes theory for nematic liquid crystals. This planar device consists of an array of square micrometer-sized wells. We obtain six different classes of equilibrium profiles and these profiles are classified as diagonal or rotated solutions. In the strong anchoring case, we propose a Dirichlet boundary condition that mimics the experimentally imposed tangent boundary conditions. In the weak anchoring case, we present a suitable surface energy and study the multiplicity of solutions as a function of the anchoring strength. We find that diagonal solutions exist for all values of the anchoring strength W ≥ 0, while rotated solutions only exist for W ≥ W_{c}>0, where W_{c} is a critical anchoring strength that has been computed numerically. We propose a dynamic model for the switching mechanisms based on only dielectric effects. For sufficiently strong external electric fields, we numerically demonstrate diagonal-to-rotated and rotated-to-diagonal switching by allowing for variable anchoring strength across the domain boundary.

- CA Yates, RE Baker, R Erban and PK Maini (2012). Going from microscopic to macroscopic on nonuniform growing domains.. Phys Rev E Stat Nonlin Soft Matter Phys
**86**(2 Pt 1): 021921. (Abstract)Throughout development, chemical cues are employed to guide the functional specification of underlying tissues while the spatiotemporal distributions of such chemicals can be influenced by the growth of the tissue itself. These chemicals, termed morphogens, are often modeled using partial differential equations (PDEs). The connection between discrete stochastic and deterministic continuum models of particle migration on growing domains was elucidated by Baker, Yates, and Erban [Bull. Math. Biol. 72, 719 (2010)] in which the migration of individual particles was modeled as an on-lattice position-jump process. We build on this work by incorporating a more physically reasonable description of domain growth. Instead of allowing underlying lattice elements to instantaneously double in size and divide, we allow incremental element growth and splitting upon reaching a predefined threshold size. Such a description of domain growth necessitates a nonuniform partition of the domain. We first demonstrate that an individual-based stochastic model for particle diffusion on such a nonuniform domain partition is equivalent to a PDE model of the same phenomenon on a nongrowing domain, providing the transition rates (which we derive) are chosen correctly and we partition the domain in the correct manner. We extend this analysis to the case where the domain is allowed to change in size, altering the transition rates as necessary. Through application of the master equation formalism we derive a PDE for particle density on this growing domain and corroborate our findings with numerical simulations.

- CA Yates, RE Baker, PK Maini and R Erban (2012). Going from microscopic to macroscopic on nonuniform growing domains. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**86**(2): 021921. (doi:10.1103/PhysRevE.86.021921) (Abstract)Throughout development, chemical cues are employed to guide the functional specification of underlying tissues while the spatiotemporal distributions of such chemicals can be influenced by the growth of the tissue itself. These chemicals, termed morphogens, are often modeled using partial differential equations (PDEs). The connection between discrete stochastic and deterministic continuum models of particle migration on growing domains was elucidated by Baker, Yates, and Erban in which the migration of individual particles was modeled as an on-lattice position-jump process. We build on this work by incorporating a more physically reasonable description of domain growth. Instead of allowing underlying lattice elements to instantaneously double in size and divide, we allow incremental element growth and splitting upon reaching a predefined threshold size. Such a description of domain growth necessitates a nonuniform partition of the domain. We first demonstrate that an individual-based stochastic model for particle diffusion on such a nonuniform domain partition is equivalent to a PDE model of the same phenomenon on a nongrowing domain, providing the transition rates (which we derive) are chosen correctly and we partition the domain in the correct manner. We extend this analysis to the case where the domain is allowed to change in size, altering the transition rates as necessary. Through application of the master equation formalism we derive a PDE for particle density on this growing domain and corroborate our findings with numerical simulations. © 2012 American Physical Society.

- S Stoye, M Stamp Dawkins and MA Porter (2012). Synchronized lying in cattle in relation to time of day. Livestock Science
**149**(1-2): 70-73. (doi:10.1016/j.livsci.2012.06.028) (Abstract)Postural synchrony, in which cattle lie down or stand up at the same time as other members of their herd, occurs both in animals housed indoors when enough resources are available and in those out at pasture, but the mechanisms by which such synchrony is achieved are poorly understood. We report a study of 6 groups of young cattle (Bos taurus) at pasture in which our aim was to study postural synchrony at different times of day and in relation to the postures of neighbouring cattle. All of the observed groups exhibited a high degree of synchrony in lying/standing, as 70% of animals in a group exhibited the same posture over 93% of the time. Time of day had a significant effect (P≈0.0046): cattle were least synchronized in the middle of the day and most synchronized in the morning and evening. With the increasing use of synchrony of lying as a measure of welfare in cattle, such temporal effects need to be taken into account. Cattle were more synchronized with the posture of a near neighbour than they were with that of a randomly chosen member of the herd (P≈0.016), suggesting that cattle were actively synchronizing their postures with that of their neighbours. These results indicate that a full understanding of the mechanisms of postural synchronization in cattle herds will need to incorporate both collective (allelomimetic) and concurrent (individual) responses. © 2012 Elsevier B.V.

- VS Zubkov, CJW Breward and EA Gaffney (2012). Coupling Fluid and Solute Dynamics Within the Ocular Surface Tear Film: A Modelling Study of Black Line Osmolarity. Bulletin of Mathematical Biology
**74**(9): 2062-2093. (doi:10.1007/s11538-012-9746-9) (Abstract)We present a mathematical model describing the spatial distribution of tear film osmolarity across the ocular surface of a human eye during one blink cycle, incorporating detailed fluid and solute dynamics. Based on the lubrication approximation, our model comprises three coupled equations tracking the depth of the aqueous layer of the tear film, the concentration of the polar lipid, and the concentration of physiological salts contained in the aqueous layer. Diffusive boundary layers in the salt concentration occur at the thinnest regions of the tear film, the black lines. Thus, despite large Peclet numbers, diffusion ameliorates osmolarity around the black lines, but nonetheless is insufficient to eliminate the build-up of solute in these regions. More generally, a heterogeneous distribution of solute concentration is predicted across the ocular surface, indicating that measurements of lower meniscus osmolarity are not globally representative, especially in the presence of dry eye. Vertical saccadic eyelid motion can reduce osmolarity at the lower black line, raising the prospect that select eyeball motions more generally can assist in alleviating tear film hyperosmolarity. Finally, our results indicate that measured evaporative rates will induce excessive hyperosmolarity at the black lines, even for the healthy eye. This suggests that further evaporative retardation at the black lines, for instance due to the cellular glycocalyx at the ocular surface or increasing concentrations of mucus, will be important for controlling hyperosmolarity as the black line thins. © 2012 Society for Mathematical Biology.

- DJ Fenn, MA Porter, NF Johnson, NS Jones, PJ Mucha, M Mcdonald and S Williams (2012). Dynamical clustering of exchange rates. Quantitative Finance
**12**(10): 1493-1520. (doi:10.1080/14697688.2012.668288) (Abstract)We use techniques from network science to study correlations in the foreign exchange (FX) market during the period 1991-2008. We consider an FX market network in which each node represents an exchange rate and each weighted edge represents a time-dependent correlation between the rates. To provide insights into the clustering of the exchange-rate time series, we investigate dynamic communities in the network. We show that there is a relationship between an exchange rate's functional role within the market and its position within its community and use a node-centric community analysis to track the temporal dynamics of such roles. This reveals which exchange rates dominate the market at particular times and also identifies exchange rates that experienced significant changes in market role. We also use the community dynamics to uncover major structural changes that occurred in the FX market. Our techniques are general and will be similarly useful for investigating correlations in other markets. © 2012 Copyright Taylor and Francis Group, LLC.

- J-P Onnela, S Reid, NS Jones, DJ Fenn, MA Porter, MD Fricker and PJ Mucha (2012). Taxonomies of networks from community structure. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**86**(3): 036104. (doi:10.1103/PhysRevE.86.036104) (Abstract)The study of networks has become a substantial interdisciplinary endeavor that encompasses myriad disciplines in the natural, social, and information sciences. Here we introduce a framework for constructing taxonomies of networks based on their structural similarities. These networks can arise from any of numerous sources: They can be empirical or synthetic, they can arise from multiple realizations of a single process (either empirical or synthetic), they can represent entirely different systems in different disciplines, etc. Because mesoscopic properties of networks are hypothesized to be important for network function, we base our comparisons on summaries of network community structures. Although we use a specific method for uncovering network communities, much of the introduced framework is independent of that choice. After introducing the framework, we apply it to construct a taxonomy for 746 networks and demonstrate that our approach usefully identifies similar networks. We also construct taxonomies within individual categories of networks, and we thereby expose nontrivial structure. For example, we create taxonomies for similarity networks constructed from both political voting data and financial data. We also construct network taxonomies to compare the social structures of 100 Facebook networks and the growth structures produced by different types of fungi. © 2012 American Physical Society.

- CD Martin and MA Porter (2012). The extraordinary SVD. American Mathematical Monthly
**119**(10): 838-851. (doi:10.4169/amer.math.monthly.119.10.838) (Abstract)The singular value decomposition (SVD) is a popular matrix factorization that has been used widely in applications ever since an efficient algorithm for its computation was developed in the 1970s. In recent years, the SVD has become even more prominent due to a surge in applications and increased computational memory and speed. To illustrate the vitality of the SVD in data analysis, we highlight three of its lesser-known yet fascinating applications. The SVD can be used to characterize political positions of congressmen, measure the growth rate of crystals in igneous rock, and examine entanglement in quantum computation. We also discuss higher-dimensional generalizations of the SVD, which have become increasingly crucial with the newfound wealth of multidimensional data, and have launched new research initiatives in both theoretical and applied mathematics. With its bountiful theory and applications, the SVD is truly extraordinary. © THE MATHEMATICAL ASSOCIATION OF AMERICA.

- S Stoye, MA Porter and M Stamp Dawkins (2012). Synchronized lying in cattle in relation to time of day. Livestock Science : .
- ACF Lewis, CM Deane, NS Jones and MA Porter (2012). What Evidence Is There for the Homology of Protein-Protein Interactions?. Plos Computational Biology
**8**(9): e1002645. (doi:10.1371/journal.pcbi.1002645) (Abstract)The notion that sequence homology implies functional similarity underlies much of computational biology. In the case of protein-protein interactions, an interaction can be inferred between two proteins on the basis that sequence-similar proteins have been observed to interact. The use of transferred interactions is common, but the legitimacy of such inferred interactions is not clear. Here we investigate transferred interactions and whether data incompleteness explains the lack of evidence found for them. Using definitions of homology associated with functional annotation transfer, we estimate that conservation rates of interactions are low even after taking interactome incompleteness into account. For example, at a blastp E-value threshold of 10, we estimate the conservation rate to be about 11% between S. cerevisiae and H. sapiens. Our method also produces estimates of interactome sizes (which are similar to those previously proposed). Using our estimates of interaction conservation we estimate the rate at which protein-protein interactions are lost across species. To our knowledge, this is the first such study based on large-scale data. Previous work has suggested that interactions transferred within species are more reliable than interactions transferred across species. By controlling for factors that are specific to within-species interaction prediction, we propose that the transfer of interactions within species might be less reliable than transfers between species. Protein-protein interactions appear to be very rarely conserved unless very high sequence similarity is observed. Consequently, inferred interactions should be used with care. © 2012 Lewis et al.

- DE Moulton, A Goriely and R Chirat (2012). Mechanical growth and morphogenesis of seashells. Journal of Theoretical Biology
**311**: 69-79. (doi:10.1016/j.jtbi.2012.07.009) (Abstract)Seashells grow through the local deposition of mass along the aperture. Many mathematical descriptions of the shapes of shells have been provided over the years, and the basic logarithmic coiling seen in mollusks can be simulated with few parameters. However, the developmental mechanisms underlying shell coiling are largely not understood and the ubiquitous presence of ornamentation such as ribs, tubercles, or spines presents yet another level of difficulty. Here we develop a general model for shell growth based entirely on the local geometry and mechanics of the aperture and mantle. This local description enables us to efficiently describe both arbitrary growth velocities and the evolution of the shell aperture itself. We demonstrate how most shells can be simulated within this framework. We then turn to the mechanics underlying the shell morphogenesis, and develop models for the evolution of the aperture. We demonstrate that the elastic response of the mantle during shell deposition provides a natural mechanism for the formation of three-dimensional ornamentation in shells. © 2012 Elsevier Ltd.

- VS Zubkov, CJW Breward and EA Gaffney (2012). Coupling Fluid and Solute Dynamics Within the Ocular Surface Tear Film: A Modelling Study of Black Line Osmolarity. Bulletin of Mathematical Biology : 1-32.
- MJ Mcguinness, B Scheu and AC Fowler (2012). Explosive fragmentation criteria and velocities for vesicular magma. Journal of Volcanology and Geothermal Research
**237-238**: 81-96. (doi:10.1016/j.jvolgeores.2012.05.019) (Abstract)We present a new criterion for primary fragmentation of natural rock samples in shock-tube experiments, and new formulae for the effective strength of natural rock samples, for the size of primary fragments, and for fragmentation velocities. These formulae and the fragmentation criterion are given in terms of the physical properties of the rock and in terms of experimental parameters. The formulae and criterion are derived from numerical solutions and asymptotic analytic solutions to a novel recently published mathematical model for the explosive fragmentation of vesicular magma in shock-tube experiments. This model is singularly successful in accounting for the length-scales observed in these experiments. The criterion and formulae provide good matches to data from shock-tube experiments on natural samples. © 2012 Elsevier B.V.

- H Cooray, P Cicuta and D Vella (2012). The capillary interaction between two vertical cylinders. Journal of Physics Condensed Matter
**24**(28): 284104. (doi:10.1088/0953-8984/24/28/284104) (Abstract)Particles floating at the surface of a liquid generally deform the liquid surface. Minimizing the energetic cost of these deformations results in an inter-particle force which is usually attractive and causes floating particles to aggregate and form surface clusters. Here we present a numerical method for determining the three-dimensional meniscus around a pair of vertical circular cylinders. This involves the numerical solution of the fully nonlinear Laplace-Young equation using a mesh-free finite difference method. Inter-particle force-separation curves for pairs of vertical cylinders are then calculated for different radii and contact angles. These results are compared with previously published asymptotic and experimental results. For large inter-particle separations and conditions such that the meniscus slope remains small everywhere, good agreement is found between all three approaches (numerical, asymptotic and experimental). This is as expected since the asymptotic results were derived using the linearized Laplace-Young equation. For steeper menisci and smaller inter-particle separations, however, the numerical simulation resolves discrepancies between existing asymptotic and experimental results, demonstrating that this discrepancy was due to the nonlinearity of the Laplace-Young equation. © 2012 IOP Publishing Ltd.

- M Taroni, CJW Breward, PD Howell and JM Oliver (2012). Boundary conditions for free surface inlet and outlet problems. Journal of Fluid Mechanics
**708**: 100-110. (doi:10.1017/jfm.2012.275) (Abstract)We investigate and compare the boundary conditions that are to be applied to free-surface problems involving inlet and outlets of Newtonian fluid, typically found in coating processes. The flux of fluid is a priori known at an inlet, but unknown at an outlet, where it is governed by the local behaviour near the film-forming meniscus. In the limit of vanishing capillary number Ca it is well known that the flux scales with Ca , but this classical result is non-uniform as the contact angle approaches π. By examining this limit we find a solution that is uniformly valid for all contact angles. Furthermore, by considering the far-field behaviour of the free surface we show that there exists a critical capillary number above which the problem at an inlet becomes over-determined. The implications of this result for the modelling of coating flows are discussed. © 2012 Cambridge University Press.

- MR Moore, SD Howison, JR Ockendon and JM Oliver (2012). Three-dimensional oblique water-entry problems at small deadrise angles. Journal of Fluid Mechanics
**711**: 259-280. (doi:10.1017/jfm.2012.391) (Abstract)This paper extends Wagner theory for the ideal, incompressible normal impact of rigid bodies that are nearly parallel to the surface of a liquid half-space. The impactors considered are three-dimensional and have an oblique impact velocity. A formulation in terms of the displacement potential is used to reveal the relationship between the oblique and corresponding normal impact solutions. In the case of axisymmetric impactors, several geometries are considered in which singularities develop in the boundary of the effective wetted region. We present the corresponding pressure profiles and models for the splash sheets. © 2012 Cambridge University Press.

- MR Moore, SD Howison, JR Ockendon and JM Oliver (2012). A note on oblique water entry. Journal of Engineering Mathematics
**81**: 1-8. (doi:10.1007/s10665-012-9570-0) (Abstract)A minor error in Howison et al. (J. Eng. Math. 48:321-337, 2004) obscured the fact that the points at which the free surface turns over in the solution of the Wagner model for the oblique impact of a two-dimensional body are directly related to the turnover points in the equivalent normal impact problem. This note corrects some of the earlier results given in Howison et al. (J. Eng. Math. 48:321-337, 2004) and discusses the implications for the applicability of the Wagner model. © 2012 Springer Science+Business Media B.V.

- TJW Wagner and D Vella (2012). The sensitivity of graphene "snap-through" to substrate geometry. Applied Physics Letters
**100**(23): 233111. (doi:10.1063/1.4724329) (Abstract)We study theoretically the deposition of few layer graphene sheets onto a grooved substrate incorporating adhesion between substrate and sheet. We develop a model to understand the equilibrium of the sheet allowing for partial conformation of sheet to substrate. This model gives physical insight into recent observations of snap-through from flat to conforming states and emphasizes the crucial role of substrate shape in determining the nature of this transition. Our analytical results are consistent with numerical simulations using a van der Waals-like interaction. Finally, we propose a substrate shape that should exhibit a continuous, rather than snap-through, transition. © 2012 American Institute of Physics.

- A Yavari and A Goriely (2012). Riemann-Cartan Geometry of Nonlinear Dislocation Mechanics. Archive for Rational Mechanics and Analysis
**205**(1): 59-118. (doi:10.1007/s00205-012-0500-0) (Abstract)We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold-where the body is stress free-is a Weitzenböck manifold, that is, a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan's moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields, assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance. © 2012 Springer-Verlag.

- C Luo, A Majumdar and R Erban (2012). Multistability in planar liquid crystal wells. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**85**(6): 061702. (doi:10.1103/PhysRevE.85.061702) (Abstract)A planar bistable liquid crystal device, reported in Tsakonas, is modeled within the Landau-de Gennes theory for nematic liquid crystals. This planar device consists of an array of square micrometer-sized wells. We obtain six different classes of equilibrium profiles and these profiles are classified as diagonal or rotated solutions. In the strong anchoring case, we propose a Dirichlet boundary condition that mimics the experimentally imposed tangent boundary conditions. In the weak anchoring case, we present a suitable surface energy and study the multiplicity of solutions as a function of the anchoring strength. We find that diagonal solutions exist for all values of the anchoring strength W≥0, while rotated solutions only exist for W≥W >0, where W is a critical anchoring strength that has been computed numerically. We propose a dynamic model for the switching mechanisms based on only dielectric effects. For sufficiently strong external electric fields, we numerically demonstrate diagonal-to-rotated and rotated-to-diagonal switching by allowing for variable anchoring strength across the domain boundary. © 2012 American Physical Society.

- NF Wymbs, ST Grafton, DS Bassett, PJ Mucha and MA Porter (2012). Differential Recruitment of the Sensorimotor Putamen and Frontoparietal Cortex during Motor Chunking in Humans. Neuron
**74**(5): 936-946. (doi:10.1016/j.neuron.2012.03.038) (Abstract)Motor chunking facilitates movement production by combining motor elements into integrated units of behavior. Previous research suggests that chunking involves two processes: concatenation, aimed at the formation of motor-motor associations between elements or sets of elements, and segmentation, aimed at the parsing of multiple contiguous elements into shorter action sets. We used fMRI to measure the trial-wise recruitment of brain regions associated with these chunking processes as healthy subjects performed a cued-sequence production task. A dynamic network analysis identified chunking structure for a set of motor sequences acquired during fMRI and collected over 3 days of training. Activity in the bilateral sensorimotor putamen positively correlated with chunk concatenation, whereas a left-hemisphere frontoparietal network was correlated with chunk segmentation. Across subjects, there was an aggregate increase in chunk strength (concatenation) with training, suggesting that subcortical circuits play a direct role in the creation of fluid transitions across chunks. Separable concatenation and segmentation processes underpin motor chunking. Wymbs et al. demonstrate that the basal ganglia support the concatenation of sequential movements, whereas the frontoparietal cortex supports the segmentation of long sequences into smaller chunks. © 2012 Elsevier Inc..

- CB Prior and A Goriely (2012). The Fourier transform of tubular densities. Journal of Physics A: Mathematical and Theoretical
**45**(22): 225208. (doi:10.1088/1751-8113/45/22/225208) (Abstract)We consider the Fourier transform of tubular volume densities, with arbitrary axial geometry and (possibly) twisted internal structure. This density can be used to represent, among others, magnetic flux or the electron density of biopolymer molecules. We consider tubes of both finite radii and unrestricted radius. When there is overlap of the tube structure the net density is calculated using the super-position principle. The Fourier transform of this density is composed of two expressions, one for which the radius of the tube is less than the curvature of the axis and one for which the radius is greater (which must have density overlap). This expression can accommodate an asymmetric density distribution and a tube structure which has non-uniform twisting. In addition we give several simpler expressions for isotropic densities, densities of finite radius, densities which decay at a rate sufficient to minimize local overlap and finally individual surfaces of the tube manifold. These simplified cases can often be expressed as arclength integrals and can be evaluated using a system of first-order ODEs. © 2012 IOP Publishing Ltd.

- AL Traud, PJ Mucha and MA Porter (2012). Social structure of Facebook networks. Physica A: Statistical Mechanics and its Applications
**391**(16): 4165-4180. (doi:10.1016/j.physa.2011.12.021) (Abstract)We study the social structure of Facebook "friendship" networks at one hundred American colleges and universities at a single point in time, and we examine the roles of user attributes-gender, class year, major, high school, and residence-at these institutions. We investigate the influence of common attributes at the dyad level in terms of assortativity coefficients and regression models. We then examine larger-scale groupings by detecting communities algorithmically and comparing them to network partitions based on user characteristics. We thereby examine the relative importance of different characteristics at different institutions, finding for example that common high school is more important to the social organization of large institutions and that the importance of common major varies significantly between institutions. Our calculations illustrate how microscopic and macroscopic perspectives give complementary insights on the social organization at universities and suggest future studies to investigate such phenomena further. © 2012 Elsevier B.V. All rights reserved.

- S Bennett, P Asinari and PJ Dellar (2012). A lattice Boltzmann model for diffusion of binary gas mixtures that includes diffusion slip. International Journal For Numerical Methods in Fluids
**69**(1): 171-189. (doi:10.1002/fld.2549) - CG Bell, W-J Lim, JH Siggers, PD Howell and HA Stone (2012). The long-time chronoamperometric current at an inlaid disk electrode. Journal of Electroanalytical Chemistry
**673**: 48-57. (doi:10.1016/j.jelechem.2012.03.013) (Abstract)Existing analytical solutions for the long-time chronoamperometric current response at an inlaid disk electrode are restricted to diffusion-limited currents due to extreme polarisation or reversible kinetics at the electrode surface. In this article, we derive an approximate analytical solution for the long-time-dependent current when the kinetics of the redox reaction at the electrode surface are quasi-reversible and the diffusion coefficients of the oxidant and reductant are different. We also detail a novel method for calculating the steady-state current. We show that our new method encapsulates and extends the existing solutions, and agrees with numerically simulated currents. © 2012 Elsevier B.V. All rights reserved.

- M Taroni, CJW Breward, PD Howell, JM Oliver and RJS Young (2012). Erratum to: The screen printing of a power-law fluid. Journal of Engineering Mathematics : 1-1.
- AL Traud, PJ Mucha and MA Porter (2012). Social structure of Facebook networks. Physica A: Statistical Mechanics and its Applications : .
- M Taroni, CJW Breward, PD Howell, JM Oliver and RJS Young (2012). The screen printing of a power-law fluid. Journal of Engineering Mathematics
**73**(1): 93-119. (doi:10.1007/s10665-011-9500-6) (Abstract)We present a two-dimensional large-aspect-ratio model for the off-contact screen printing of a power-law fluid. We extend the work of White et al. (J Eng Math 54:49-70, 2005) by explicitly including the fluid/air free surface that is present beneath the screen ahead of the squeegee. In the distinguished parameter limit of greatest interest to industry, the process is quasi-steady on the time-scale of a print and can be analysed in three separate regions using the method of matched asymptotic expansions. This allows us to predict where the fluid transfers through the screen, the point at which it first makes contact with the substrate, and the amount of fluid deposited on the substrate during a print stroke. Finally, we show that using a shear-thinning fluid will decrease the amount of fluid transferred ahead of the squeegee, but increase the amount of fluid deposited on the substrate. © 2011 Springer Science+Business Media B.V.

- RL Machete and IM Moroz (2012). Initial distribution spread: A density forecasting approach. Physica D: Nonlinear Phenomena
**241**(8): 805-815. (doi:10.1016/j.physd.2012.01.007) (Abstract)Ensemble forecasting of nonlinear systems involves the use of a model to run forward a discrete ensemble (or set) of initial states. Data assimilation techniques tend to focus on estimating the true state of the system, even though model error limits the value of such efforts. This paper argues for choosing the initial ensemble in order to optimise forecasting performance rather than estimating the true state of the system. Density forecasting and choosing the initial ensemble are treated as one problem. Forecasting performance can be quantified by some scoring rule. In the case of the logarithmic scoring rule, theoretical arguments and empirical results are presented. It turns out that, if the underlying noise dominates model error, we can diagnose the noise spread. © 2012 Elsevier B.V. All rights reserved.

- MQ De Luna, B Duričković and A Goriely (2012). Non-linear waves in heterogeneous elastic rods via homogenization. International Journal of Non-Linear Mechanics
**47**(2): 197-205. (doi:10.1016/j.ijnonlinmec.2011.05.005) (Abstract)We consider the propagation of a planar loop on a heterogeneous elastic rod with a periodic microstructure consisting of two alternating homogeneous regions with different material properties. The analysis is carried out using a second-order homogenization theory based on a multiple scale asymptotic expansion. © 2011 Elsevier Ltd. All rights reserved.

- IM Griffiths, CJW Breward, SJ Chapman, PD Howell, SL Waters and CD Bain (2012). An asymptotic theory for the re-equilibration of a micellar surfactant solution. SIAM Journal on Applied Mathematics
**72**(1): 201-215. (doi:10.1137/110842089) (Abstract)Micellar surfactant solutions are characterized by a distribution of aggregates made up predominantly of premicellar aggregates (monomers, dimers, trimers, etc.) and a region of proper micelles close to the peak aggregation number, connected by an intermediate region containing a very low concentration of aggregates. Such a distribution gives rise to a distinct two-timescale reequilibration following a system dilution, known as the t1 and t2 processes, whose dynamics may be described by the Becker-Döring equations. We use a continuum version of these equations to develop a reduced asymptotic description that elucidates the behavior during each of these processes.© 2012 Society for Industrial and Applied Mathematics.

- S Howison (2012).
Asymptotic Approximations for Asian, European, and American Options with Discrete Averaging or Discrete Dividend/Coupon Payments. SIAM Journal on Financial Mathematics
**3**(1): 215-241. (doi:10.1137/090771636) - GW Jones and SJ Chapman (2012). Modeling growth in biological materials. SIAM Review
**54**(1): 52-118. (doi:10.1137/080731785) (Abstract)The biomechanical modeling of growing tissues has recently become an area of intense interest. In particular, the interplay between growth patterns and mechanical stress is of great importance, with possible applications to arterial mechanics, embryo morphogenesis, tumor development, and bone remodeling. This review aims to give an overview of the theories that have been used to model these phenomena, categorized according to whether the tissue is considered as a continuum object or a collection of cells. Among the continuum models discussed is the deformation gradient decomposition method, which allows a residual stress field to develop from an incompatible growth field. The cell-based models are further subdivided into cellular automata, center-dynamics, and vertex-dynamics models. Of these the second two are considered in more detail, especially with regard to their treatment of cell-cell interactions and cell division. The review concludes by assessing the prospects for reconciliation between these two fundamentally different approaches to tissue growth, and by identifying possible avenues for further research. © 2012 Society for Industrial and Applied Mathematics.

- A Zazzo, R Bendrey, D Vella, AP Moloney, FJ Monahan and O Schmidt (2012). A refined sampling strategy for intra-tooth stable isotope analysis of mammalian enamel. Geochimica et Cosmochimica Acta : .
- JP Gleeson, S Melnik, JA Ward, MA Porter and PJ Mucha (2012). Accuracy of mean-field theory for dynamics on real-world networks. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**85**(2): 026106. (doi:10.1103/PhysRevE.85.026106) (Abstract)Mean-field analysis is an important tool for understanding dynamics on complex networks. However, surprisingly little attention has been paid to the question of whether mean-field predictions are accurate, and this is particularly true for real-world networks with clustering and modular structure. In this paper, we compare mean-field predictions to numerical simulation results for dynamical processes running on 21 real-world networks and demonstrate that the accuracy of such theory depends not only on the mean degree of the networks but also on the mean first-neighbor degree. We show that mean-field theory can give (unexpectedly) accurate results for certain dynamics on disassortative real-world networks even when the mean degree is as low as 4. © 2012 American Physical Society.

- MPH Stumpf and MA Porter (2012). Critical truths about power laws. Science
**335**(6069): 665-666. (doi:10.1126/science.1216142) (Abstract)Most reported power laws lack statistical support and mechanistic backing.

- AL Stewart and PJ Dellar (2012). Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves. Journal of Fluid Mechanics
**690**: 16-50. (doi:10.1017/jfm.2011.364) - RL Machete and IM Moroz (2012). Initial distribution spread: A density forecasting approach. Physica D: Nonlinear Phenomena : .
- X Luo, I Hoteit and IM Moroz (2012). On a nonlinear Kalman filter with simplified divided difference approximation. Physica D: Nonlinear Phenomena
**241**(6): 671-680. (doi:10.1016/j.physd.2011.12.003) (Abstract)We present a new ensemble-based approach that handles nonlinearity based on a simplified divided difference approximation through Stirling's interpolation formula, which is hence called the simplified divided difference filter (sDDF). The sDDF uses Stirling's interpolation formula to evaluate the statistics of the background ensemble during the prediction step, while at the filtering step the sDDF employs the formulae in an ensemble square root filter (EnSRF) to update the background to the analysis. In this sense, the sDDF is a hybrid of Stirling's interpolation formula and the EnSRF method, while the computational cost of the sDDF is less than that of the EnSRF. Numerical comparison between the sDDF and the EnSRF, with the ensemble transform Kalman filter (ETKF) as the representative, is conducted. The experiment results suggest that the sDDF outperforms the ETKF with a relatively large ensemble size, and thus is a good candidate for data assimilation in systems with moderate dimensions. © 2011 Elsevier B.V. All rights reserved.

- A Majumdar, C Prior and A Goriely (2012). Stability Estimates for a Twisted Rod Under Terminal Loads: A Three-dimensional Study. Journal of Elasticity : 1-19.
- M Bruna and SJ Chapman (2012). Excluded-volume effects in the diffusion of hard spheres. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**85**(1): 011103. (doi:10.1103/PhysRevE.85.011103) (Abstract)Excluded-volume effects can play an important role in determining transport properties in diffusion of particles. Here, the diffusion of finite-sized hard-core interacting particles in two or three dimensions is considered systematically using the method of matched asymptotic expansions. The result is a nonlinear diffusion equation for the one-particle distribution function, with excluded-volume effects enhancing the overall collective diffusion rate. An expression for the effective (collective) diffusion coefficient is obtained. Stochastic simulations of the full particle system are shown to compare well with the solution of this equation for two examples. © 2012 American Physical Society.

- KE Murphy, CL Hall, PK Maini, SW Mccue and DLS Mcelwain (2012). A Fibrocontractive Mechanochemical Model of Dermal Wound Closure Incorporating Realistic Growth Factor Kinetics. Bulletin of Mathematical Biology : 1-28.
- VM Burlakov, A Goriely, N Emptage and PC Bressloff (2012). Synaptic bistability due to nucleation and evaporation of receptor clusters. Physical Review Letters
**108**(2): 028101. (doi:10.1103/PhysRevLett.108.028101) (Abstract)We introduce a bistability mechanism for long-term synaptic plasticity based on switching between two metastable states that contain significantly different numbers of synaptic receptors. One state is characterized by a two-dimensional gas of mobile interacting receptors and is stabilized against clustering by a high nucleation barrier. The other state contains a receptor gas in equilibrium with a large cluster of immobile receptors, which is stabilized by the turnover rate of receptors into and out of the synapse. Transitions between the two states can be initiated by either an increase (potentiation) or a decrease (depotentiation) of the net receptor flux into the synapse. This changes the saturation level of the receptor gas and triggers nucleation or evaporation of receptor clusters. © 2012 American Physical Society.

- E Aumaitre, S Wongsuwarn, P Cicuta, D Rossetti, ND Hedges, AR Cox and D Vella (2012). A viscoelastic regime in dilute hydrophobin monolayers. Soft Matter
**8**(4): 1175-1183. (doi:10.1039/c1sm06139a) (Abstract)We show unexpected viscoelastic response in dilute conditions for hydrophobin layers subjected to compression in a Langmuir trough. The surface pressure, surface dilatational and shear viscoelasticity, and layer thickness, are probed by combining four different experimental techniques: Wilhelmy plate tensiometry, Surface Quasi-Elastic Light Scattering (SQELS), Surface Shear Rheology and Ellipsometry. At a surface pressure around 6-15 mN m , the isotherm measured by a Wilhelmy plate displays erratically a kink or small plateau feature. The build up to this point has been probed with SQELS. SQELS data itself require careful interpretation because the Wilhelmy plate measurements cannot be taken as a valid independent measurement of surface pressure for solid-like films. The SQELS raw signal shows a decrease of the capillary wave frequency, while the surface pressure measured by a Wilhelmy plate is still approximately zero. This could be attributed to either an increase of the layer surface pressure not seen by the Wilhelmy plate, and/or to the presence of significant dilatational response; analysis of the SQELS signals indicates a small finite dilatational modulus in the dilute hydrophobin layer. Shear rheology measurements indicate that within this region, the layer acquires some rigidity. Ellipsometry measurements interestingly show that the layer already has a finite thickness in these dilute conditions, and that the thickness increases linearly through the compression until the layer collapses. Combining these elements allows us to develop a clear picture of the build up of surface tension and viscoelasticity within the hydrophobin layer: the layer becomes viscoelastic at pressures which are still negligible. This is consistent with previous reports in the literature of co-existence of condensed and dilute protein domains, if the domain lengthscale is smaller than the area probed by the techniques employed here. © 2012 The Royal Society of Chemistry.

- G Klingbeil, R Erban, M Giles and PK Maini (2012). Fat versus thin threading approach on GPUs: Application to stochastic simulation of chemical reactions. Ieee Transactions on Parallel and Distributed Systems
**23**(2): 280-287. (doi:10.1109/TPDS.2011.157) (Abstract)We explore two different threading approaches on a graphics processing unit (GPU) exploiting two different characteristics of the current GPU architecture. The fat thread approach tries to minimize data access time by relying on shared memory and registers potentially sacrificing parallelism. The thin thread approach maximizes parallelism and tries to hide access latencies. We apply these two approaches to the parallel stochastic simulation of chemical reaction systems using the stochastic simulation algorithm (SSA) by Gillespie [14]. In these cases, the proposed thin thread approach shows comparable performance while eliminating the limitation of the reaction system's size. © 2006 IEEE.

- AG Fletcher, CJW Breward and S Jonathan Chapman (2012). Mathematical modeling of monoclonal conversion in the colonic crypt. Journal of Theoretical Biology
**300**: 118-133. (doi:10.1016/j.jtbi.2012.01.021) (Abstract)A novel spatial multiscale model of a colonic crypt is described, which couples the cell cycle (including cell division) with the mechanics of cell movement. The model is used to investigate the process of monoclonal conversion under two hypotheses concerning stem cell behavior. Under the first hypothesis, 'stem-ness' is an intrinsic cell property, and the stem cell population is maintained through asymmetric division. Under the second hypothesis, the proliferative behavior of each cell is governed by its microenvironment through a biochemical signalling cue, and all cell division is symmetric. Under each hypothesis, the model is used to run virtual experiments, in which a harmless labeling mutation is bestowed upon a single cell in the crypt and the mutant clonal population is tracked over time to check if and when the crypt becomes monoclonal. It is shown that under the first hypothesis, a stable structured cell population is not possible without some form of population-dependent feedback; in contrast, under the second hypothesis, a stable crypt architecture arises naturally. Through comparison with an existing spatial crypt model and a non-spatial stochastic population model, it is shown that the spatial structure of the crypt has a significant effect on the time scale over which a crypt becomes monoclonal. © 2012 Elsevier Ltd.

- KE Murphy, SW Mccue, S Mcelwain, CL Hall and PK Maini (2012). A Fibrocontractive Mechanochemical Model of Dermal Wound Closure Incorporating Realistic Growth Factor Kinetics. Bulletin of Mathematical Biology
**74**(5): 1143-1170. (doi:10.1007/s11538-011-9712-y) (Abstract)Fibroblasts and their activated phenotype, myofibroblasts, are the primary cell types involved in the contraction associated with dermal wound healing. Recent experimental evidence indicates that the transformation from fibroblasts to myofibroblasts involves two distinct processes: The cells are stimulated to change phenotype by the combined actions of transforming growth factor β (TGFβ) and mechanical tension. This observation indicates a need for a detailed exploration of the effect of the strong interactions between the mechanical changes and growth factors in dermal wound healing. We review the experimental findings in detail and develop a model of dermal wound healing that incorporates these phenomena. Our model includes the interactions between TGFβ and collagenase, providing a more biologically realistic form for the growth factor kinetics than those included in previous mechanochemical descriptions. A comparison is made between the model predictions and experimental data on human dermal wound healing and all the essential features are well matched. © 2012 Society for Mathematical Biology.

- S-J Dunn, DJ Gavaghan, JM Osborne, AG Fletcher and SJ Chapman (2012). Modelling the role of the basement membrane beneath a growing epithelial monolayer. Journal of Theoretical Biology
**298**: 82-91. (doi:10.1016/j.jtbi.2011.12.013) (Abstract)The role of the basement membrane is vital in maintaining the integrity and structure of an epithelial layer, acting as both a mechanical support and forming the physical interface between epithelial cells and the surrounding connective tissue. The function of this membrane is explored here in the context of a growing epithelial monolayer, defined such that the epithelial cells divide and migrate along a deformable substrate. A discrete, off-lattice cell-centre modelling approach is undertaken, which permits definition of a basement membrane component, separating the epithelial cells from the tissue stroma whilst responding to forces from both that arise due to cell division, migration and apoptosis. This model is applicable to a range of biological epithelia, including the self-renewing interfollicular epidermis, the olfactory epithelium and the intestinal crypts of Lieberkühn, to inform response and recovery of such tissues following injury. Model simulations show that homeostasis of the growing monolayer can be achieved and sustained, and the necessary balance of interactive cell forces, cell migration and cell death is presented. This work is proposed as a novel extension to the body of discrete models of biological epithelia, permitting investigation of the growth and migration of epithelial cells in a deformable environment. © 2011 Elsevier Ltd.

- RJ Shipley and SL Waters (2012). Fluid and mass transport modelling to drive the design of cell-packed hollow fibre bioreactors for tissue engineering applications. Mathematical Medicine and Biology
**29**(4): 329-359. (doi:10.1093/imammb/dqr025) (Abstract)A model for fluid and mass transport in a single module of a tissue engineering hollow fibre bioreactor (HFB) is developed. Cells are seeded in alginate throughout the extra-capillary space (ECS), and fluid is pumped through a central lumen to feed the cells and remove waste products. Fluid transport is described using Navier-Stokes or Darcy equations as appropriate; this is overlaid with models of mass transport in the form of advection-diffusion-reaction equations that describe the distribution and uptake/production of nutrients/waste products. The small aspect ratio of a module is exploited and the option of opening an ECS port is explored. By proceeding analytically, operating equations are determined that enable a tissue engineer to prescribe the geometry and operation of the HFB by ensuring the nutrient and waste product concentrations are consistent with a functional cell population. Finally, results for chondrocyte and cardiomyocyte cell populations are presented, typifying two extremes of oxygen uptake rates. © The Author 2011. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

- AL Hazel, M Heil, SL Waters and JM Oliver (2012). On the liquid lining in fluid-conveying curved tubes. Journal of Fluid Mechanics
**705**: 213-233. (doi:10.1017/jfm.2011.346) (Abstract)We consider axially uniform, two-phase flow through a rigid curved tube in which a fluid (air) core is surrounded by a film of a second, immiscible fluid (water): a simplified model for flow in a conducting airway of the lung. Jensen (1997) showed that, in the absence of a core flow, surface tension drives the system towards a configuration in which the film thickness tends to zero on the inner wall of the bend. In the present work, we demonstrate that the presence of a core flow, driven by a steady axial pressure gradient, allows the existence of steady states in which the film thickness remains finite, a consequence of the fact that the tangential stresses at the interface, imposed by secondary flows in the core, can oppose the surface-tension-driven flow. For sufficiently strong surface tension, the steady configurations are symmetric about the plane containing the tube's centreline, but as the surface tension decreases the symmetry is lost through a pitchfork bifurcation, which is closely followed by a limit point on the symmetric solution branch. This solution structure is found both in simulations of the Navier-Stokes equations and a thin-film model appropriate for weakly curved tubes. Analysis of the thin-film model reveals that the bifurcation structure arises from a perturbation of the translational degeneracy of the interface location in a straight tube. © 2011 Cambridge University Press.

- M Robertson-Tessi, A El-Kareh and A Goriely (2012). A mathematical model of tumor-immune interactions. Journal of Theoretical Biology
**294**: 56-73. (doi:10.1016/j.jtbi.2011.10.027) - MB Flegg, SJ Chapman and R Erban (2012). The two-regime method for optimizing stochastic reaction-diffusion simulations. Journal of the Royal Society Interface
**9**(70): 859-868. (doi:10.1098/rsif.2011.0574) (Abstract)Spatial organization and noise play an important role in molecular systems biology. In recent years, a number of software packages have been developed for stochastic spatio-temporal simulation, ranging from detailed molecular-based approaches to less detailed compartment-based simulations. Compartment-based approaches yield quick and accurate mesoscopic results, but lack the level of detail that is characteristic of the computationally intensive molecularbased models. Often microscopic detail is only required in a small region (e.g. close to the cell membrane). Currently, the best way to achieve microscopic detail is to use a resource-intensive simulation over the whole domain.We develop the two-regime method (TRM) in which a molecular-based algorithm is used where desired and a compartment-based approach is used elsewhere. We present easy-to-implement coupling conditions which ensure that the TRM results have the same accuracy as a detailed molecular-based model in the whole simulation domain. Therefore, the TRM combines strengths of previously developed stochastic reaction-diffusion software to efficiently explore the behaviour of biological models. Illustrative examples and the mathematical justification of the TRM are also presented. © 2011 The Royal Society.

- KT Macon, PJ Mucha and MA Porter (2012). Community structure in the United Nations General Assembly. Physica A: Statistical Mechanics and its Applications
**391**(1-2): 343-361. (doi:10.1016/j.physa.2011.06.030) (Abstract)We study the community structure of networks representing voting on resolutions in the United Nations General Assembly. We construct networks from the voting records of the separate annual sessions between 1946 and 2008 in three different ways: (1) by considering voting similarities as weighted unipartite networks; (2) by considering voting similarities as weighted, signed unipartite networks; and (3) by examining signed bipartite networks in which countries are connected to resolutions. For each formulation, we detect communities by optimizing network modularity using an appropriate null model. We compare and contrast the results that we obtain for these three different network representations. We thereby illustrate the need to consider multiple resolution parameters and explore the effectiveness of each network representation for identifying voting groups amidst the large amount of agreement typical in General Assembly votes. © 2011 Elsevier B.V. All rights reserved.

- D Vella, A Ajdari, A Vaziri and A Boudaoud (2012). The indentation of pressurized elastic shells: From polymeric capsules to yeast cells. Journal of the Royal Society Interface
**9**(68): 448-455. (doi:10.1098/rsif.2011.0352) (Abstract)Pressurized elastic capsules arise at scales ranging from the 10 m diameter pressure vessels used to store propane at oil refineries to the microscopic polymeric capsules that may be used in drug delivery. Nature also makes extensive use of pressurized elastic capsules: plant cells, bacteria and fungi have stiff walls, which are subject to an internal turgor pressure. Here, we present theoretical, numerical and experimental investigations of the indentation of a linearly elastic shell subject to a constant internal pressure. We show that, unlike unpressurized shells, the relationship between force and displacement demonstrates two linear regimes. We determine analytical expressions for the effective stiffness in each of these regimes in terms of the material properties of the shell and the pressure difference. As a consequence, a single indentation experiment over a range of displacements may be used as a simple assay to determine both the internal pressure and elastic properties of capsules. Our results are relevant for determining the internal pressure in bacterial, fungal or plant cells. As an illustration of this, we apply our results to recent measurements of the stiffness of baker's yeast and infer from these experiments that the internal osmotic pressure of yeast cells may be regulated in response to changes in the osmotic pressure of the external medium. © 2011 The Royal Society.

- R Erban and J Haskovec (2012). From individual to collective behaviour of coupled velocity jump processes: A locust example. Kinetic and Related Models
**5**(4): 817-842. (doi:10.3934/krm.2012.5.817) (Abstract)A class of stochastic individual-based models, written in terms of coupled velocity jump processes, is presented and analysed. This modelling approach incorporates recent experimental findings on the behaviour of locusts. It exhibits nontrivial dynamics with a pitchfork bifurcation and recovers the observed group directional switching. Estimates of the expected switching times, in terms of the number of individuals and values of the model coefi-cients, are obtained using the corresponding Fokker-Planck equation. In the limit of large populations, a system of two kinetic equations (with nonlocal and nonlinear right hand side) is derived and analyzed. The existence of its solutions is proven and the system's long-time behaviour is investigated. Finally, a first step towards the mean field limit of topological interactions is made by studying the efiect of shrinking the interaction radius in the individual-based model. © American Institute of Mathematical Sciences.

### 2011

- H Hansen-Goos and JS Wettlaufer (2011).
A fundamental measure theory for the sticky hard sphere fluid. The Journal of Chemical Physics
**134**(1): 014506-014506. (doi:10.1063/1.3528226) - JA Neufeld and JS Wettlaufer (2011).
Shear flow, phase change and matched asymptotic expansions: Pattern formation in mushy layers. Physica D: Nonlinear Phenomena
**240**(2): 140-149. (doi:10.1016/j.physd.2010.10.007) - IJ Hewitt, NJ Balmforth and JN Mcelwaine (2011). Continual skipping on water. Journal of Fluid Mechanics
**669**: 328-353. (doi:10.1017/S0022112010005057) (Abstract)Experiments are conducted to study the planing and skipping of a rectangular paddle on the surface of a shallow stream. The paddle is allowed to move freely up and down by attaching it to a pivoted arm. A steady planing state, in which the lift force from the water balances the weight on the paddle, is found to be stable for small stream velocities but to become unstable above a certain threshold velocity which depends upon the weight and the angle of attack. Above this threshold, the paddle oscillates in the water and can take off into a continual bouncing, or skipping, motion, with a well-defined amplitude and frequency. The transition is sometimes bistable so that both a steady planing state and a regular skipping state are possible for the same experimental parameters. Shallow-water theory is used to construct simple models that explain the qualitative features of the planing and skipping states in the experiments. It is found that a simple parameterisation of the lift force on the paddle proportional to the depth of entry is not sufficient to explain the observations, and it is concluded that the rise of water ahead of the paddle, in particular the way this varies over time, is responsible for causing the planing state to become unstable and for enabling a continual skipping state. © 2011 Cambridge University Press.

- IJ Hewitt (2011). Modelling distributed and channelized subglacial drainage: The spacing of channels. Journal of Glaciology
**57**(202): 302-314. (Abstract)Models are proposed for channelized and distributed flow of meltwater at the base of an ice sheet. The volumes of both channel and distributed systems evolve according to a competition between processes that open drainage space (e.g. sliding over bedrock,melting of the ice) and processes that close it (e.g. viscous creep of the ice due to a positive effective pressure). Channels are generally predicted to have lower water pressure and therefore capture water from the surrounding regions of distributed flow. There is a natural length scale associated with the distributed system that determines the width of the bed from which water can be drawn into a channel. It is suggested that this determines the spacing between major channels and that this may be reflected in the spacing of eskers. A more permeable distributed system results in more widely spaced, and therefore larger, channels. Calculations of the flow into the head of a channel reveal that there is a critical discharge necessary for it to form, and provide a criterion for where channels can exist.

- IJ Hewitt, NJ Balmforth and H Scolan (2011). Flow-destabilized seiches in a reservoir with a movable dam. Journal of Fluid Mechanics
**678**: 294-316. (doi:10.1017/jfm.2011.113) (Abstract)Using a combination of theoretical modelling and experiments, seiches in a reservoir are shown to become linearly unstable due to the coupling with flow under a dam that opens and closes in response to the upstream water pressure. The phenomenon is related to the mechanism commonly attributed to generate sound in musical instruments like the clarinet. Shallow water theory is used to model waves in the reservoir, and these are coupled, by an outflow condition, to a nonlinear oscillator equation for the dam opening. In general, several modes of oscillation are predicted to be unstable, and the frequency of the most unstable mode compares well with the dominant frequencies observed in the experiments. The experiments also show a systematic variation of the amplitude and spatial structure of the oscillations with the weight of the dam, reflecting the nonlinear coupling between the unstable modes of the system. © 2011 Cambridge University Press.

- JM Foster, CP Please and AD Fitt (2011). The slow spreading of a viscous fluid film over a deep viscous pool. Journal of Engineering Mathematics
**71**(4): 393-408. (doi:10.1007/s10665-011-9460-x) (Abstract)A model is introduced for the spreading of an isolated viscous foam film over a deep viscous fluid pool. The effects of the bubbles rising in the foam are neglected, but the O(1) alteration to the density and viscosity of the foam due to the bubbles are accounted for. It is assumed that the foam phase is well modelled by the Stokes-flow equations. By exploiting the slenderness of the spreading layer, asymptotic techniques are used to analyse the flow in the foam. It is shown that in this regime the dominant horizontal force balance is between the hydrostatic pressure and the tangential stress induced in the layer by the underlying pool. Boundary-integral techniques are used to determine the form of the flow in the pool and from this an expression for the evolution of the spreading foam is given analytically. In this parameter regime it is found that this expression does not depend upon the viscosity of the foam. © 2011 Springer Science+Business Media B.V.

- CJ Catt, CP Please, W Schuurman, PR van Weeren, WJA Dhert, J Malda and BG Sengers (2011). Mathematical modelling of tissue formation in chondrocyte filter cultures. European Cells and Materials
**22**: 377-392. (Abstract)In the field of cartilage tissue engineering, filter cultures are a frequently used three-dimensional differentiation model. However, understanding of the governing processes of in vitro growth and development of tissue in these models is limited. Therefore, this study aimed to further characterise these processes by means of an approach combining both experimental and applied mathematical methods. A mathematical model was constructed, consisting of partial differential equations predicting the distribution of cells and glycosaminoglycans (GAGs), as well as the overall thickness of the tissue. Experimental data was collected to allow comparison with the predictions of the simulation and refinement of the initial models. Healthy mature equine chondrocytes were expanded and subsequently seeded on collagen-coated filters and cultured for up to 7 weeks. Resulting samples were characterised biochemically, as well as histologically. The simulations showed a good representation of the experimentally obtained cell and matrix distribution within the cultures. The mathematical results indicate that the experimental GAG and cell distribution is critically dependent on the rate at which the cell differentiation process takes place, which has important implications for interpreting experimental results. This study demonstrates that large regions of the tissue are inactive in terms of proliferation and growth of the layer. In particular, this would imply that higher seeding densities will not significantly affect the growth rate. A simple mathematical model was developed to predict the observed experimental data and enable interpretation of the principal underlying mechanisms controlling growth-related changes in tissue composition.

- RJ Shipley, SL Waters and MJ Ellis (2011). Fluid and mass transport modelling to drive the design of cell-packed hollow fibre bioreactors. European Cells and Materials
**22**(SUPPL.3): 72-72. - CL Hall (2011). Asymptotic analysis of a pile-up of regular edge dislocation walls. Materials Science and Engineering A
**530**(1): 144-148. (doi:10.1016/j.msea.2011.09.065) (Abstract)The idealised problem of a pile-up of regular dislocation walls (that is, of planes each containing an infinite number of parallel, identical and equally spaced dislocations) was presented by Roy et al. [A. Roy, R.H.J. Peerlings, M.G.D. Geers, Y. Kasyanyuk, Materials Science and Engineering A 486 (2008) 653-661] as a prototype for understanding the importance of discrete dislocation interactions in dislocation-based plasticity models. They noted that analytic solutions for the dislocation wall density are available for a pile-up of regular screw dislocation walls, but that numerical methods seem to be necessary for investigating regular edge dislocation walls. In this paper, we use the techniques of discrete-to-continuum asymptotic analysis to obtain a detailed description of a pile-up of regular edge dislocation walls. To leading order, we find that the dislocation wall density is governed by a simple differential equation and that boundary layers are present at both ends of the pile-up. © 2011 Elsevier B.V.

- X Luo, I Hoteit and IM Moroz (2011). On a nonlinear Kalman filter with simplified divided difference approximation. Physica D: Nonlinear Phenomena : .
- SA Myers, PJ Mucha and MA Porter (2011). Mathematical genealogy and department prestige. Chaos
**21**(4): 041104. (doi:10.1063/1.3668043) - PJ Dellar (2011). An exact energy conservation property of the quantum lattice Boltzmann algorithm. Physics Letters A
**376**(1): 6-13. (doi:10.1016/j.physleta.2011.10.014) - PH Trinh, SJ Chapman and J-M Vanden-Broeck (2011). Do waveless ships exist? Results for single-cornered hulls. Journal of Fluid Mechanics
**685**: 413-439. (doi:10.1017/jfm.2011.325) - AL Stewart and PJ Dellar (2011). Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves. Journal of Fluid Mechanics : 1-35.
- B Davidovitch, RD Schroll, D Vella, M Adda-Bedia and EA Cerda (2011). Prototypical model for tensional wrinkling in thin sheets. Proceedings of the National Academy of Sciences of the United States of America
**108**(45): 18227-18232. (doi:10.1073/pnas.1108553108) (Abstract)The buckling and wrinkling of thin films has recently seen a surge of interest among physicists, biologists, mathematicians, and engineers. This activity has been triggered by the growing interest in developing technologies at ever-decreasing scales and the resulting necessity to control the mechanics of tiny structures, as well as by the realization that morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, often emerge from mechanical instabilities of cell sheets. Although the most basic buckling instability of uniaxially compressed plates was understood by Euler more than two centuries ago, recent experiments on nanometrically thin (ultrathin) films have shown significant deviations from predictions of standard buckling theory. Motivated by this puzzle, we introduce here a theoretical model that allows for a systematic analysis of wrinkling in sheets far from their instability threshold. We focus on the simplest extension of Euler buckling that exhibits wrinkles of finite length - a sheet under axisymmetric tensile loads. The first study of this geometry, which is attributed to Lamé, allows us to construct a phase diagram that demonstrates the dramatic variation of wrinkling patterns from near-threshold to far-from-threshold conditions. Theoretical arguments and comparison to experiments show that the thinner the sheet is, the smaller is the compressive load above which the far-from-threshold regime emerges. This observation emphasizes the relevance of our analysis for nanomechanics applications.

- JA Ward, AC Fowler and SBG O'Brien (2011). Acid polishing of lead glass. Journal of Mathematics in Industry
**1**(1): 1-19. (doi:10.1186/2190-5983-1-1) (Abstract)Purpose: The polishing of cut lead glass crystal is effected through the dowsing of the glass in a mixture of two separate acids, which between them etch the surface and as a result cause it to be become smooth. In order to characterise the resultant polishing the rate of surface etching must be known, but when this involves multicomponent surface reactions it becomes unclear what this rate actually is. Methods: We develop a differential equation based discrete model to determine the effective etching rate by means of an atomic scale model of the etching process. Results: We calculate the etching rate numerically and provide an approximate asymptotic estimate. Conclusions: The natural extension of this work would be to develop a continuum advection-diffusion model. © 2011 Springer-Verlag Berlin Heidelberg.

- AC Fowler (2011). Weertman, Lliboutry and the development of sliding theory. Journal of Glaciology
**56**(200): 965-972. (Abstract)This paper reviews the seminal contributions (and tussles) of Weertman and Lliboutry to the theory of glacier sliding in the 1950s and 1960s, and summarizes later developments up to the present day.

- C Mayer, AC Fowler, A Lambrecht and K Scharrer (2011). A surge of North Gasherbrum Glacier, Karakoram, China. Journal of Glaciology
**57**(205): 904-916. (doi:10.3189/002214311798043834) (Abstract)Between 2003 and 2007, North Gasherbrum Glacier on the northeastern slope of the Karakoram mountains in Asia underwent a dramatic acceleration, during which a velocity wave propagated down the glacier. There was a significant transfer of ice from up-glacier downstream, which resulted in a strong surface elevation increase over the lower tongue, but only a moderate advance of the glacier snout. We interpret this behaviour as that of a glacier surge, and we explain the observations by means of a simple version of the Kamb drainage-switching theory.

- AC Fowler, JA Ward and SBG O'Brien (2011). A simple model for multicomponent etching. Journal of Colloid and Interface Science
**354**(1): 421-423. (doi:10.1016/j.jcis.2010.10.056) (Abstract)We consider the situation where a multicomponent solid is etched using one or more acids. Of fundamental interest is the rate of surface etching but when this involves multicomponent surface reactions, it becomes unclear how the overall rate can be estimated. In this paper, we sketch a simple model designed to determine the effective etching rate by means of an atomic scale model of the etching process. © 2010 Elsevier Inc.

- I Drobnjak and AC Fowler (2011). A Model of Oscillatory Blood Cell Counts in Chronic Myelogenous Leukaemia. Bulletin of Mathematical Biology
**73**(12): 2983-3007. (doi:10.1007/s11538-011-9656-2) (Abstract)In certain blood diseases, oscillations are found in blood cell counts. Particularly, such oscillations are sometimes found in chronic myelogenous leukaemia, and then occur in all the derived blood cell types: red blood cells, white blood cells, and platelets. It has been suggested that such oscillations arise because of an instability in the pluri-potential stem cell population, associated with its regulatory control system. In this paper, we consider how such oscillations can arise in a model of competition between normal (S) and genetically altered abnormal (A) stem cells, as the latter population grows at the expense of the former. We use an analytic model of long period oscillations to describe regions of oscillatory behaviour in the S-A phase plane, and give parametric criteria to describe when such oscillations will occur. We also describe a mechanism which can explain dynamically how the transformation from chronic phase to acute phase and blast crisis can occur. © 2011 Society for Mathematical Biology.

- M Chapwanya, CD Clark and AC Fowler (2011). Numerical computations of a theoretical model of ribbed moraine formation. Earth Surface Processes and Landforms
**36**(8): 1105-1112. (doi:10.1002/esp.2138) - MR Nelson, D Howard, OE Jensen, JR King, FRAJ Rose and SL Waters (2011). Growth-induced buckling of an epithelial layer. Biomechanics and Modeling in Mechanobiology
**10**(6): 883-900. (doi:10.1007/s10237-010-0280-0) - A Münch, CP Please and B Wagner (2011). Spin coating of an evaporating polymer solution. Physics of Fluids
**23**(10): 102101. (doi:10.1063/1.3643692) (Abstract)We consider a mathematical model of spin coating of a single polymer blended in a solvent. The model describes the one-dimensional development of a thin layer of the mixture as the layer thins due to flow created by a balance of viscous forces and centrifugal forces and evaporation of the solvent. In the model both the diffusivity of the solvent in the polymer and the viscosity of the mixture are very rapidly varying functions of the solvent mass fraction. Guided by numerical solutions an asymptotic analysis reveals a number of different possible behaviours of the thinning layer dependent on the nondimensional parameters describing the system. The main practical interest is in controlling the appearance and development of a "skin" on the polymer where the solvent concentration reduces rapidly on the outer surface leaving the bulk of the layer still with high concentrations of solvent. In practice, a fast and uniform drying of the film is required. The critical parameters controlling this behaviour are found to be the ratio of the diffusion to advection time scales ε, the ratio of the evaporation to advection time scales δ and the ratio of the diffusivity of the pure polymer and the initial mixture exp(-1/γ). In particular, our analysis shows that for very small evaporation with δ

- K Kostourou, D Peschka, A Muench, B Wagner, S Herminghaus and R Seemann (2011). Interface morphologies in liquid/liquid dewetting. Chemical Engineering and Processing
**50**(5-6): 531-536. (doi:10.1016/j.cep.2010.10.006) - A Münch and B Wagner (2011). Impact of slippage on the morphology and stability of a dewetting rim. Journal of Physics Condensed Matter
**23**(18): 184101. (doi:10.1088/0953-8984/23/18/184101) (Abstract)In this study lubrication theory is used to describe the stability and morphology of the rim that forms as a thin polymer film dewets from a hydrophobized silicon wafer. Thin film equations are derived from the governing hydrodynamic equations for the polymer to enable the systematic mathematical and numerical analysis of the properties of the solutions for different regimes of slippage and for a range of timescales. Dewetting rates and the cross sectional profiles of the evolving rims are derived for these models and compared to experimental results. Experiments also show that the rim is typically unstable in the spanwise direction and develops thicker and thinner parts that may grow into 'fingers'. Linear stability analysis as well as nonlinear numerical solutions are presented to investigate shape and growth rate of the rim instability. It is demonstrated that the difference in morphology and the rate at which the instability develops can be directly attributed to the magnitude of slippage. Finally, a derivation is given for the dominant wavelength of the bulges along the unstable rim. © 2011 IOP Publishing Ltd.

- D Vella, A Ajdari, A Vaziri and A Boudaoud (2011). Wrinkling of pressurized elastic shells. Physical Review Letters
**107**(17): 174301. (doi:10.1103/PhysRevLett.107.174301) (Abstract)We study the formation of localized structures formed by the point loading of an internally pressurized elastic shell. While unpressurized shells (such as a ping-pong ball) buckle into polygonal structures, we show that pressurized shells are subject to a wrinkling instability. We study wrinkling in depth, presenting scaling laws for the critical indentation at which wrinkling occurs and the number of wrinkles formed in terms of the internal pressurization and material properties of the shell. These results are validated by numerical simulations. We show that the evolution of the wrinkle length with increasing indentation can be understood for highly pressurized shells from membrane theory. These results suggest that the position and number of wrinkles may be used in combination to give simple methods for the estimation of the mechanical properties of highly pressurized shells. © 2011 American Physical Society.

- PJ Dellar (2011). An exact energy conservation property of the quantum lattice Boltzmann algorithm. Physics Letters, Section A: General, Atomic and Solid State Physics : .
- IM Moroz (2011). Template analysis of a nonlinear dynamo. Proceedings of the Royal Society of London. Mathematical, Physical and Engineering Sciences
**468**: 288-302. (doi:10.1098/rspa.2011.0216) - IM Moroz (2011). Unstable periodic orbits in a four-dimensional Faraday disk dynamo. Geophysical and Astrophysical Fluid Dynamics
**105**(nos 2-3): 273-286. - J Mcmahon, A Goriely and M Tabor (2011). Nonlinear morphoelastic plates II: Exodus to buckled states. Mathematics and Mechanics of Solids
**16**(8): 833-871. (doi:10.1177/1081286510387234) - J Mcmahon, A Goriely and M Tabor (2011). Nonlinear morphoelastic plates I: Genesis of residual stress. Mathematics and Mechanics of Solids
**16**(8): 812-832. (doi:10.1177/1081286510387233) - LA Mihai and A Goriely (2011). Positive or negative Poynting effect? the role of adscititious inequalities in hyperelastic materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**467**(2136): 3633-3646. (doi:10.1098/rspa.2011.0281) (Abstract)Motivated by recent experiments on biopolymer gels whereby the reverse of the usual (positive) Poynting effect was observed, we investigate the effect of the so-called 'adscititious inequalities' on the behaviour of hyperelastic materials subject to shear. We first demonstrate that for homogeneous isotropic materials subject to pure shear, the resulting deformation consists of a triaxial stretch combined with a simple shear in the direction of the shear force if and only if the Baker-Ericksen inequalities hold. Then for a cube deformed under pure shear, the positive Poynting effect occurs if the 'sheared faces spread apart', whereas the negative Poynting effect is obtained if the 'sheared faces draw together'. Similarly, under simple shear deformation, the positive Poynting effect is obtained if the 'sheared faces tend to spread apart', whereas the negative Poynting effect occurs if the 'sheared faces tend to draw together'. When the Poynting effect occurs under simple shear, it is reasonable to assume that the same sign Poynting effect is btained also under pure shear. Since the observation of the negative Poynting effect in semiflexible biopolymers implies that the (stronger) empirical inequalities may not hold, we conclude that these inequalities must not be imposed when such materials are described. © 2011 The Royal Society.

- S Neukirch, J Frelat, A Goriely and C Maurini (2011). Vibrations of post-buckled rods: The singular inextensible limit. Journal of Sound and Vibration : .
- M Taroni, CJW Breward, PD Howell, JM Oliver and RJS Young (2011). The screen printing of a power-law fluid. Journal of Engineering Mathematics : 1-27.
- PJ Dellar (2011). An interpretation and derivation of the lattice Boltzmann method using Strang splitting. Computers and Mathematics with Applications : .
- MQ de Luna, B Duričković and A Goriely (2011). Non-linear waves in heterogeneous elastic rods via homogenization. International Journal of Non-Linear Mechanics : .
- M Epstein and A Goriely (2011). Self-diffusion in remodeling and growth. Zeitschrift fur Angewandte Mathematik und Physik : 1-17.
- HF Winstanley, M Chapwanya, MJ Mcguinness and AC Fowler (2011). A polymer-solvent model of biofilm growth. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**467**(2129): 1449-1467. (doi:10.1098/rspa.2010.0327) (Abstract)We provide and analyse a model for the growth of bacterial biofilms based on the concept of an extracellular polymeric substance as a polymer solution, whose viscoelastic rheology is described by the classical Flory-Huggins theory. We show that one-dimensional solutions exist, which take the form at large times of travelling waves, and we characterize their form and speed in terms of the describing parameters of the problem. Numerical solutions of the time-dependent problem converge to the travelling wave solutions. © 2010 The Royal Society.

- H Dankowicz, A Hosoi and A Goriely (2011). Special issue on non-linear mechanics of biological structures. International Journal of Non-Linear Mechanics
**46**(4): 555-556. (doi:10.1016/j.ijnonlinmec.2011.04.001) - DE Moulton and A Goriely (2011). Anticavitation and differential growth in elastic shells. Journal of Elasticity
**102**(2): 117-132. (doi:10.1007/s10659-010-9266-5) (Abstract)Elastic anticavitation is the phenomenon of a void in an elastic solid collapsing on itself. Under the action of mechanical loading alone typical materials do not admit anticavitation. We study the possibility of anticavitation as a consequence of an imposed differential growth. Working in the geometry of a spherical shell, we seek radial growth functions which cause the shell to deform to a solid sphere. It is shown, surprisingly, that most material models do not admit full anticavitation, even when infinite growth or resorption is imposed at the inner surface of the shell. However, void collapse can occur in a limiting sense when radial and circumferential growth are properly balanced. Growth functions which diverge or vanish at a point arise naturally in a cumulative growth process. © 2010 Springer Science+Business Media B.V.

- DE Moulton and A Goriely (2011). Possible role of differential growth in airway wall remodeling in asthma. Journal of Applied Physiology
**110**(4): 1003-1012. (doi:10.1152/japplphysiol.00991.2010) (Abstract)Possible role of differential growth in airway wall remodeling in asthma. J Appl Physiol 110: 1003-1012, 2011. First published January 20, 2011; doi:10.1152/japplphysiol.00991.2010.- Airway remodeling in patients with chronic asthma is characterized by a thickening of the airway walls. It has been demonstrated in previous theoretical models that this change in thickness can have an important mechanical effect on the properties of the wall, in particular on the phenomenon of mucosal folding induced by smooth muscle contraction. In this paper, we present a model for mucosal folding of the airway in the context of growth. The airway is modeled as a bilayered cylindrical tube, with both geometric and material nonlinearities accounted for via the theory of finite elasticity. Growth is incorporated into the model through the theory of morphoelasticity. We explore a range of growth possibilities, allowing for anisotropic growth as well as different growth rates in each layer. Such nonuniform growth, referred to as differential growth, can change the properties of the material beyond geometrical changes through the generation of residual stresses. We demonstrate that differential growth can have a dramatic impact on mucosal folding, in particular on the critical pressure needed to induce folding, the buckling pattern, as well as airway narrowing. We conclude that growth may be an important component in airway remodeling. Copyright © 2011 the American Physiological Society.

- R De Pascalis, M Destrade and A Goriely (2011). Nonlinear correction to the Euler buckling formula for compressed cylinders with guided-guided end conditions. Journal of Elasticity
**102**(2): 191-200. (doi:10.1007/s10659-010-9265-6) (Abstract)Euler's celebrated buckling formula gives the critical load N for the buckling of a slender cylindrical column with radius B and length L as N/(πB)=(E/4)(B/L) where E is Young's modulus. Its derivation relies on the assumptions that linear elasticity applies to this problem, and that the slenderness (B/L) is an infinitesimal quantity. Here we ask the following question: What is the first non-linear correction in the right hand-side of this equation when terms up to (B/L) are kept? To answer this question, we specialize the exact solution of incremental non-linear elasticity for the homogeneous compression of a thick compressible cylinder with lubricated ends to the theory of third-order elasticity. In particular, we highlight the way second- and third-order constants-including Poisson's ratio-all appear in the coefficient of (B/L). © 2010 Springer Science+Business Media B.V.

- D Ambrosi, GA Ateshian, EM Arruda, JE Olberding, K Garikipati, SC Cowin, J Dumais, A Goriely, GA Holzapfel, JD Humphrey, R Kemkemer, E Kuhl and LA Taber (2011). Perspectives on biological growth and remodeling. Journal of the Mechanics and Physics of Solids
**59**(4): 863-883. (doi:10.1016/j.jmps.2010.12.011) (Abstract)The continuum mechanical treatment of biological growth and remodeling has attracted considerable attention over the past fifteen years. Many aspects of these problems are now well-understood, yet there remain areas in need of significant development from the standpoint of experiments, theory, and computation. In this perspective paper we review the state of the field and highlight open questions, challenges, and avenues for further development. © 2010 Elsevier Ltd. All rights reserved.

- A Goriely and M Tabor (2011). Spontaneous rotational inversion in phycomyces. Physical Review Letters
**106**(13): 138103. (doi:10.1103/PhysRevLett.106.138103) (Abstract)The filamentary fungus Phycomyces blakesleeanus undergoes a series of remarkable transitions during aerial growth. During what is known as the stagea IV growth phase, the fungus extends while rotating in a counterclockwise manner when viewed from above (stagea IVa) and then, while continuing to grow, spontaneously reverses to a clockwise rotation (stagea IVb). This phase lasts for 24-48Ah and is sometimes followed by yet another reversal (stageAIVc) before the overall growth ends. Here, we propose a continuum mechanical model of this entire process using nonlinear, anisotropic, elasticity and show how helical anisotropy associated with the cell wall structure can induce spontaneous rotation and, under appropriate circumstances, the observed reversal of rotational handedness. © 2011 American Physical Society.

- M Destrade, A Goriely and G Saccomandi (2011). Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK and KP equations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**467**(2131): 1823-1834. (doi:10.1098/rspa.2010.0508) (Abstract)We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then, we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov- Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics. © 2010 The Royal Society.

- DE Moulton and A Goriely (2011). Circumferential buckling instability of a growing cylindrical tube. Journal of the Mechanics and Physics of Solids
**59**(3): 525-537. (doi:10.1016/j.jmps.2011.01.005) (Abstract)A cylindrical elastic tube under uniform radial external pressure will buckle circumferentially to a non-circular cross-section at a critical pressure. The buckling represents an instability of the inner or outer edge of the tube. This is a common phenomenon in biological tissues, where it is referred to as mucosal folding. Here, we investigate this buckling instability in a growing elastic tube. A change in thickness due to growth can have a dramatic impact on circumferential buckling, both in the critical pressure and the buckling pattern. We consider both single- and bi-layer tubes and multiple boundary conditions. We highlight the competition between geometric effects, i.e. the change in tube dimensions, and mechanical effects, i.e. the effect of residual stress, due to differential growth. This competition can lead to non-intuitive results, such as a tube growing to be thinner and yet buckle at a higher pressure. © 2011 Elsevier Ltd. All rights reserved.

- D Vella, JA Neufeld, HE Huppert and JR Lister (2011). Leakage from gravity currents in a porous medium. Part 2. A line sink. Journal of Fluid Mechanics
**666**: 414-427. (doi:10.1017/S002211201000491X) (Abstract)We consider the propagation of a buoyancy-driven gravity current in a porous medium bounded by a horizontal, impermeable boundary. The current is fed by a constant flux injected at a point and leaks through a line sink at a distance from the injection point. This is an idealized model of how a fault in a cap rock might compromise the geological sequestration of carbon dioxide. The temporal evolution of the efficiency of storage, defined as the instantaneous ratio of the rate at which fluid is stored without leaking to the rate at which it is injected, is of particular interest. We show that the 'efficiency of storage' decays like -2/5 for times t that are long compared with the time taken for the current to reach the fault. This algebraic decay is in contrast to the case of leakage through a circular sink (Neufeld et al., J. Fluid Mech., vol. 2010) where the efficiency of storage decays more slowly like 1/lnt. The implications of the predicted decay in the efficiency of storage are discussed in the context of geological sequestration of carbon dioxide. Using parameter values typical of the demonstration project at Sleipner, Norway, we show that the efficiency of storage should remain greater than 90% on a time scale of millennia, provided that there are no significant faults in the cap rock within about 12km of the injection site. © 2011 Cambridge University Press.

- JA Neufeld, D Vella, HE Huppert and JR Lister (2011). Leakage from gravity currents in a porous medium. Part 1. A localized sink. Journal of Fluid Mechanics
**666**: 391-413. (doi:10.1017/S002211201000488X) (Abstract)We consider the buoyancy-driven flow of a fluid injected into a semi-infinite porous medium bounded by a horizontal impermeable barrier through which a single localized sink allows leakage of the injected fluid. Our study is motivated by the geological sequestration of carbon dioxide (CO ), which is less dense than the ambient water, and the possibility that fissures in the bounding 'cap' rock may therefore compromise the long-term storage of CO . A theoretical model is presented in which the leakage through the sink, or fissure, is driven by the hydrostatic pressure at the sink of the injected buoyant fluid. We determine numerical solutions for the evolution of the gravity current in the porous medium and for the quantity of fluid that escapes through the sink as a function of time. A quantity of considerable interest is the efficiency of storage, which we define as the flux of fluid that is stably stored relative to the amount injected. At the later stages in the evolution of the current, the region near the source and sink reaches a quasi-steady state. We find analytical solutions to this asymptotic state which show that the efficiency of storage decreases to zero like 1/lnt, where t is the time since initiation of the current, and predict a dependence on the properties of the sink in agreement with our numerical results. The implications of this result for the geological sequestration of CO are discussed. © 2011 Cambridge University Press.

- E Aumaitre, P Cicuta and D Vella (2011). On the measurement of the surface pressure in Langmuir films with finite shear elasticity. Soft Matter
**7**(6): 2530-2537. (doi:10.1039/c0sm01213k) (Abstract)We investigate the use of the Wilhelmy plate method to measure the surface pressure in a solid-like Langmuir film under compression. Layers of the protein hydrophobin, which exhibits a high shear elastic modulus, are spread and compressed in a Langmuir trough. The resulting isotherms are classified according to the surface pressure and distance between the barriers measured at the onset of buckling. We find that the surface pressure measured in the centre of the layer at the onset of buckling decays with increasing distance between the barriers (which can be tuned by varying the amount of material spread initially). However, unlike the case of particle rafts, the length scale of this decay is not controlled by the width of the trough but rather by the size of the Wilhelmy plate used. We use experiments and a computational model to suggest that this independence of trough width may be attributed to the localised nature of the effect of the trough walls. Our work highlights the potential pitfalls of using the Wilhelmy method to characterize layers with high shear rigidity and may lead to a better understanding of the use of the Wilhlemy plate to measure the surface stress tensor. © 2011 The Royal Society of Chemistry.

- S Gart, S Jung and D Vella (2011). The collective motion of nematodes in a thin liquid layer. Soft Matter
**7**(6): 2444-2448. (doi:10.1039/c0sm01236j) (Abstract)Many organisms live in confined fluidic environments such as the thin liquid layers on the skin of host organisms or in partially-saturated soil. We investigate the collective behaviour of nematodes in a thin liquid layer, which was first described by Gray and Lissmann [J. Exp. Biol., 1964, 41, 135]. We show experimentally that nematodes confined by a thin liquid film come into contact spontaneously. By analysing the time scales of this aggregation, we suggest that the initial aggregation is driven by random collisions between nematodes but the continued collective motion is due to an attractive force between them arising from the surface tension of the layer. We show that for nearby nematodes this surface tension force is typically stronger than the force that may be exerted by the nematodes' muscles. We believe this to be the first demonstration of the "Cheerios effect" acting on a living organism. However, we find that being grouped together does not significantly alter the body stroke and kinematic performance of the nematode: there are no statistically significant changes of the Strouhal number and the ratio of amplitude to wavelength when aggregated. This result implies that nematodes gain neither a mechanical advantage nor disadvantage by being grouped together; the capillary force merely keeps them stuck together after a chance encounter. © 2011 The Royal Society of Chemistry.

- PJ Zimoch, JA Neufeld and D Vella (2011). Leakage from inclined porous reservoirs. Journal of Fluid Mechanics
**673**: 395-405. (doi:10.1017/S0022112011000723) (Abstract)We investigate the effect of localized leakage on the storage of buoyant fluids in inclined porous reservoirs, with application to the geological storage of CO2. We find that once the current has propagated some distance beyond the point of leakage, its profile becomes steady in time, save for the nose, which advances at a constant speed. Crucially, this steady state implies that the efficiency of storage (defined as the instantaneous proportion of the injected fluid that does not leak) tends to a finite value. This contrasts with previous studies of localized leakage in horizontal reservoirs, which found that the efficiency of storage tends to zero at late times. We analyse the steady-state efficiency and the time scales of evolution for a leakage point located either upslope or downslope of the injection point using analytical and numerical methods. These findings are verified by model laboratory experiments. Finally, we consider the implications of our results for the geological storage of CO2 under sloping cap rocks compromised by a fracture or fissure. Copyright © Cambridge University Press 2011.

- TJW Wagner and D Vella (2011). Floating carpets and the delamination of elastic sheets. Physical Review Letters
**107**(4): 044301. (doi:10.1103/PhysRevLett.107.044301) (Abstract)We investigate the deformation of a thin elastic sheet floating on a liquid surface and subject to a uniaxial compression. We show that at a critical compression the sheet delaminates from the liquid over a finite region forming a delamination "blister." This blistering regime adds to the wrinkling and localized folding regimes that have been studied previously. The transition from wrinkled to blistered states occurs when delamination becomes energetically favorable compared with wrinkling. We determine the initial blister size and the evolution of blister size with continuing compression before verifying our theoretical results with experiments at a macroscopic scale. © 2011 American Physical Society.

- CJ Breward, GC Hocking, WL Sweatman and AD Fitt (2011). Deformations during jet-stripping in the galvanizing process. Journal of Engineering Mathematics
**1-3**(70): 297-306. - J Sun, EM Bollt, MA Porter and MS Dawkins (2011). A mathematical model for the dynamics and synchronization of cows. Physica D: Nonlinear Phenomena
**240**(19): 1497-1509. (doi:10.1016/j.physd.2011.06.009) (Abstract)We formulate a mathematical model for the daily activities of a cow (eating, lying down, and standing) in terms of a piecewise linear dynamical system. We analyze the properties of this bovine dynamical system representing the single animal and develop an exact integrative form as a discrete-time mapping. We then couple multiple cow "oscillators" together to study synchrony and cooperation in cattle herds. We comment on the relevant biology and discuss extensions of our model. With this abstract approach, we not only investigate equations with interesting dynamics but also develop biological predictions. In particular, our model illustrates that it is possible for cows to synchronize less when the coupling is increased. © 2011 Elsevier B.V. All rights reserved.

- AL Stewart and PJ Dellar (2011). Cross-equatorial flow through an abyssal channel under the complete Coriolis force: Two-dimensional solutions. Ocean Modelling
**40**(1): 87-104. (doi:10.1016/j.ocemod.2011.07.003) (Abstract)The component of the Coriolis force due to the locally horizontal component of the Earth's rotation vector is commonly neglected, under the so-called traditional approximation. We investigate the role of this " non-traditional" component of the Coriolis force in cross-equatorial flow of abyssal ocean currents. We focus on the Antarctic Bottom Water (AABW), which crosses from the southern to the northern hemisphere through the Ceara abyssal plain in the western Atlantic ocean. The bathymetry in this region resembles a northwestward channel, connecting the Brazil Basin in the south to the Guyana Basin in the north. South of the equator, the AABW leans against the western continental rise, consistent with a northward flow in approximate geostrophic balance. The AABW then crosses to the other side of the abyssal channel as it crosses the equator, and flows into the northern hemisphere leaning towards the east against the Mid-Atlantic Ridge.The non-traditional component of the Coriolis force is strongest close to the equator. The traditional component vanishes at the equator, being proportional to the locally vertical component of the Earth's rotation vector. The weak stratification of the abyssal ocean, and subsequent small internal deformation radius, defines a relatively short characteristic horizontal lengthscale that tends to make non-traditional effects more prominent. Additionally, the steep gradients of the channel bathymetry induce large vertical velocities, which are linked to zonal accelerations by the non-traditional components of the Coriolis force. We therefore expect non-traditional effects to play a substantial role in cross-equatorial transport of the AABW.We present asymptotic steady solutions for non-traditional shallow water flow through an idealised abyssal channel, oriented at an oblique angle to the equator. The current enters from the south, leaning up against the western side of the channel in approximate geostrophic balance, and crosses the channel as it crosses the equator. The " non-traditional" contribution to the planetary angular momentum must be balanced by stronger westward flow in the channel, which leads to an increased transport in a northwestward channel, and a reduced transport in a northeastward channel. Our results suggest that as much as 10-30% of the cross-equatorial flow of the AABW may be attributed to the non-traditional components of the Coriolis force. © 2011 Elsevier Ltd.

- KT Macon, PJ Mucha and MA Porter (2011). Community structure in the United Nations General Assembly. Physica A: Statistical Mechanics and its Applications : .
- SL Cotter, KC Zygalakis, R Erban and IG Kevrekidis (2011). A constrained approach to multiscale stochastic simulation of chemically reacting systems. Journal of Chemical Physics
**135**(9): 094102. (doi:10.1063/1.3624333) (Abstract)Stochastic simulation of coupled chemical reactions is often computationally intensive, especially if a chemical system contains reactions occurring on different time scales. In this paper, we introduce a multiscale methodology suitable to address this problem, assuming that the evolution of the slow species in the system is well approximated by a Langevin process. It is based on the conditional stochastic simulation algorithm (CSSA) which samples from the conditional distribution of the suitably defined fast variables, given values for the slow variables. In the constrained multiscale algorithm (CMA) a single realization of the CSSA is then used for each value of the slow variable to approximate the effective drift and diffusion terms, in a similar manner to the constrained mean-force computations in other applications such as molecular dynamics. We then show how using the ensuing Fokker-Planck equation approximation, we can in turn approximate average switching times in stochastic chemical systems. © 2011 American Institute of Physics.

- SL Cotter, KC Zygalakis, IG Kevrekidis and R Erban (2011). A constrained approach to multiscale stochastic simulation of chemically reacting systems.. J Chem Phys
**135**(9): 094102. (doi:10.1063/1.3624333) (Abstract)Stochastic simulation of coupled chemical reactions is often computationally intensive, especially if a chemical system contains reactions occurring on different time scales. In this paper, we introduce a multiscale methodology suitable to address this problem, assuming that the evolution of the slow species in the system is well approximated by a Langevin process. It is based on the conditional stochastic simulation algorithm (CSSA) which samples from the conditional distribution of the suitably defined fast variables, given values for the slow variables. In the constrained multiscale algorithm (CMA) a single realization of the CSSA is then used for each value of the slow variable to approximate the effective drift and diffusion terms, in a similar manner to the constrained mean-force computations in other applications such as molecular dynamics. We then show how using the ensuing Fokker-Planck equation approximation, we can in turn approximate average switching times in stochastic chemical systems.

- AL Stewart and PJ Dellar (2011). Cross-equatorial flow through an abyssal channel under the complete Coriolis force: Two-dimensional solutions. Ocean Modelling : .
- AL Traud, PJ Mucha, ED Kelsic and MA Porter (2011). Comparing community structure to characteristics in online collegiate social networks. SIAM Review
**53**(3): 526-543. (doi:10.1137/080734315) (Abstract)We study the structure of social networks of students by examining the graphs of Facebook "friendships" at five U.S. universities at a single point in time. We investigate the community structure of each single-institution network and employ visual and quantitative tools, including standardized pair-counting methods, to measure the correlations between the network communities and a set of self-identified user characteristics (residence, class year, major, and high school). We review the basic properties and statistics of the employed pair-counting indices and recall, in simplified notation, a useful formula for the z-score of the Rand coefficient. Our study illustrates how to examine different instances of social networks constructed in similar environments, emphasizes the array of social forces that combine to form "communities," and leads to comparative observations about online social structures, which reflect offline social structures. We calculate the relative contributions of different characteristics to the community structure of individual universities and compare these relative contributions at different universities. For example, we examine the importance of common high school affiliation at large state universities and the varying degrees of influence that common major can have on the social structure at different universities. The heterogeneity of the communities that we observe indicates that university networks typically have multiple organizing factors rather than a single dominant one. © 2011 Society for Industrial and Applied Mathematics.

- DJ Fenn, MA Porter, S Williams, M Mcdonald, NF Johnson and NS Jones (2011). Temporal evolution of financial-market correlations. Physical Review E
**84**(2): 026109. (doi:10.1103/PhysRevE.84.026109) - C Xue, HJ Hwang, KJ Painter and R Erban (2011). Travelling Waves in Hyperbolic Chemotaxis Equations. Bulletin of Mathematical Biology
**73**(8): 1695-1733. (doi:10.1007/s11538-010-9586-4) (Abstract)Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235-248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically. © 2010 Society for Mathematical Biology.

- AL Stewart and PJ Dellar (2011). The role of the complete Coriolis force in cross-equatorial flow of abyssal ocean currents. Ocean Modelling
**38**(3-4): 187-202. (doi:10.1016/j.ocemod.2011.03.001) - G Lemon, OE Jensen, JR King, D Howard, FRAJ Rose, H Yang, SM Ratchev, JI Segal and SL Waters (2011). Growth of the chorioallantoic membrane into a rapid-prototyped model pore system: Experiments and mathematical model. Biomechanics and Modeling in Mechanobiology
**10**(4): 539-558. (doi:10.1007/s10237-010-0254-2) (Abstract)This paper presents a mathematical model to describe the growth of tissue into a rapid-prototyped porous scaffold when it is implanted onto the chorioallantoic membrane (CAM). The scaffold was designed to study the effects of the size and shape of pores on tissue growth into conventional tissue engineering scaffolds, and consists of an array of pores each having a pre-specified shape. The experimental observations revealed that the CAMgrows through each pore as an intact layer of tissue, provided the width of the pore exceeds a threshold value. Based on these results a mathematical model is described to simulate the growth of the membrane, assuming that the growth is a function of the local isotropic membrane tension. The model predictions are compared against measurements of the extent of membrane growth through the pores as a function of time for pores with different dimensions. © Springer-Verlag 2010.

- G Richardson and SJ Chapman (2011). Derivation of the bidomain equat ions for a beating heart with a general micr ostructure. SIAM Journal on Applied Mathematics
**71**(3): 657-675. (doi:10.1137/090777165) (Abstract)A novel multiple scales method is formulated th at can be applied to problems which have an almost periodic microstructure not in Cartesian coordinates but in a general curvilinear coordinate system. The method is applied to a model of the electrical activity of cardiac myocytes and used to derive a version of the bidomain equations describing the macroscopic electrical activity of cardiac tissue. The treatment systematically accounts for the nonuniform orientation of the cells within the tissue and for deformations of the tissue occurring as a result of the heart beat. © 2011 Society for Industrial and Applied Mathematics.

- RJ Whittaker, SL Waters and M Heil (2011). The energetics of flow through a rapidly oscillating tube with slowly varying amplitude. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
**369**(1947): 2989-3006. (doi:10.1098/rsta.2011.0106) (Abstract)Motivated by the problem of self-excited oscillations in fluid-filled collapsible tubes, we examine the flow structure and energy budget of flow through an elastic-walled tube. Specifically, we consider the case in which a background axial flow is perturbed by prescribed small-amplitude high-frequency long-wavelength oscillations of the tube wall, with a slowly growing or decaying amplitude. We use a multiple-scale analysis to show that, at leading order, we recover the constant-amplitude equations derived by Whittaker et al. (Whittaker et al. 2010 J. Fluid Mech. 648, 83-121. (doi:10.1017/S0022112009992904)) with the effects of growth or decay entering only at first order. We also quantify the effects on the flow structure and energy budget. Finally, we discuss how our results are needed to understand and predict an instability that can lead to self-excited oscillations in collapsible-tube systems. © 2011 The Royal Society.

- IM Griffiths, CJW Breward, PD Howell, SL Waters, CD Bain and DM Colegate (2011). On the predictions and limitations of the Becker-Döring model for reaction kinetics in micellar surfactant solutions. Journal of Colloid and Interface Science
**360**(2): 662-671. (doi:10.1016/j.jcis.2011.04.074) (Abstract)We investigate the breakdown of a system of micellar aggregates in a surfactant solution following an order-one dilution. We derive a mathematical model based on the Becker-Döring system of equations, using realistic expressions for the reaction constants fit to results from Molecular Dynamics simulations. We exploit the largeness of typical aggregation numbers to derive a continuum model, substituting a large system of ordinary differential equations for a partial differential equation in two independent variables: time and aggregate size. Numerical solutions demonstrate that re-equilibration occurs in two distinct stages over well-separated timescales, in agreement with experiment and with previous theories. We conclude by exposing a limitation in the Becker-Döring theory for re-equilibration of surfactant solutions. © 2011 Elsevier Inc.

- AL Stewart, PJ Dellar and ER Johnson (2011). Numerical simulation of wave propagation along a discontinuity in depth in a rotating annulus. Computers & Fluids
**46**(1): 442-447. (doi:10.1016/j.compfluid.2010.10.016) - PJ Dellar (2011). Lattice Boltzmann formulation for Braginskii magnetohydrodynamics. Computers & Fluids
**46**(1): 201-205. (doi:10.1016/j.compfluid.2010.12.004) - T Reis and PJ Dellar (2011). A volume-preserving sharpening approach for the propagation of sharp phase boundaries in multiphase lattice Boltzmann simulations. Computers and Fluids
**46**(1): 417-421. (doi:10.1016/j.compfluid.2010.12.005) (Abstract)Lattice Boltzmann models that recover a macroscopic description of multiphase flow of immiscible liquids typically represent the boundaries between phases using a scalar function, the phase field, that varies smoothly over several grid points. Attempts to tune the model parameters to minimise the widths of these interfaces typically lead to the interfaces becoming fixed to the underlying grid instead of advecting with the fluid velocity. This phenomenon, known as lattice pinning, is strikingly similar to that associated with the numerical simulation of conservation laws coupled to stiff algebraic source terms. We present a lattice Boltzmann formulation of the model problem proposed by LeVeque and Yee (1990) [3] to study the latter phenomenon in the context of computational combustion, and offer a volume-conserving extension in multiple space dimensions. Inspired by the random projection method of Bao and Jin (2000) [1] we further generalise this formulation by introducing a uniformly distributed quasi-random variable into the term responsible for the sharpening of phase boundaries. This method is mass conserving, gives correct average propagation speeds over many timesteps, and is shown to significantly delay the onset of pinning as the interface width is reduced. © 2010 Elsevier Ltd.

- KE Murphy, SW Mccue, DL Sean McElwain and CL Hall (2011). A two-compartment mechanochemical model of the roles of transforming growth factor β and tissue tension in dermal wound healing. Journal of Theoretical Biology
**272**(1): 145-159. (doi:10.1016/j.jtbi.2010.12.011) (Abstract)The repair of dermal tissue is a complex process of interconnected phenomena, where cellular, chemical and mechanical aspects all play a role, both in an autocrine and in a paracrine fashion. Recent experimental results have shown that transforming growth factor -β (TGF β) and tissue mechanics play roles in regulating cell proliferation, differentiation and the production of extracellular materials. We have developed a 1D mathematical model that considers the interaction between the cellular, chemical and mechanical phenomena, allowing the combination of TGF β and tissue stress to inform the activation of fibroblasts to myofibroblasts. Additionally, our model incorporates the observed feature of residual stress by considering the changing zero-stress state in the formulation for effective strain. Using this model, we predict that the continued presence of TGF β in dermal wounds will produce contractures due to the persistence of myofibroblasts; in contrast, early elimination of TGF β significantly reduces the myofibroblast numbers resulting in an increase in wound size. Similar results were obtained by varying the rate at which fibroblasts differentiate to myofibroblasts and by changing the myofibroblast apoptotic rate. Taken together, the implication is that elevated levels of myofibroblasts is the key factor behind wounds healing with excessive contraction, suggesting that clinical strategies which aim to reduce the myofibroblast density may reduce the appearance of contractures. © 2010 Elsevier Ltd.

- J Lipková, KC Zygalakis, S Jonathan Chapman and R Erban (2011). Analysis of Brownian dynamics simulations of reversible bimolecular reactions. SIAM Journal on Applied Mathematics
**71**(3): 714-730. (doi:10.1137/100794213) (Abstract)A class of Brownian dynamics algorithms for stochastic reaction-diffusion models which include reversible bimolecular reactions is presented and analyzed. The method is a generalization of the λ-bcȳ model for irreversible bimolecular reactions which was introduced in [R. Erban and S. J. Chapman, Phys. Biol., 6(2009), 046001]. The formulae relating the experimentally measurable quantities (reaction rate constants and diffusion constants) with the algorithm parameters are derived. The probability of geminate recombination is also investigated. © 2011 Society for Industrial and Applied Mathematics.

- DS Bassett, JM Carlson, NF Wymbs, ST Grafton, MA Porter and PJ Mucha (2011). Dynamic reconfiguration of human brain networks during learning. Proceedings of the National Academy of Sciences of the United States of America
**108**(18): 7641-7646. (doi:10.1073/pnas.1018985108) (Abstract)Human learning is a complex phenomenon requiring flexibility to adapt existing brain function and precision in selecting new neurophysiological activities to drive desired behavior. These two attributes - flexibility and selection - must operate over multiple temporal scales as performance of a skill changes from being slow and challenging to being fast and automatic. Such selective adaptability is naturally provided by modular structure, which plays a critical role in evolution, development, and optimal network function. Using functional connectivity measurements of brain activity acquired from initial training through mastery of a simple motor skill, we investigate the role of modularity in human learning by identifying dynamic changes of modular organization spanning multiple temporal scales. Our results indicate that flexibility, which we measure by the allegiance of nodes to modules, in one experimental session predicts the relative amount of learning in a future session. We also develop a general statistical framework for the identification of modular architectures in evolving systems, which is broadly applicable to disciplines where network adaptability is crucial to the understanding of system performance.

- PJ Dellar (2011). Variations on a beta-plane: Derivation of non-traditional beta-plane equations from Hamilton's principle on a sphere. Journal of Fluid Mechanics
**674**: 174-195. (doi:10.1017/S0022112010006464) (Abstract)Starting from Hamilton's principle on a rotating sphere, we derive a series of successively more accurate β-plane approximations. These are Cartesian approximations to motion in spherical geometry that capture the change with latitude of the angle between the rotation vector and the local vertical. Being derived using Hamilton's principle, the different β-plane approximations each conserve energy, angular momentum and potential vorticity. They differ in their treatments of the locally horizontal component of the rotation vector, the component that is usually neglected under the traditional approximation. In particular, we derive an extended set of β-plane equations in which the locally vertical and locally horizontal components of the rotation vector both vary linearly with latitude. This was previously thought to violate conservation of angular momentum and potential vorticity. We show that the difficulty in maintaining these conservation laws arises from the need to express the rotation vector as the curl of a vector potential while approximating the true spherical metric by a flat Cartesian metric. Finally, we derive depth-averaged equations on our extended β-plane with topography, and show that they coincide with the extended non-traditional shallow-water equations previously derived in Cartesian geometry. © 2011 Cambridge University Press.

- D Lapitski and PJ Dellar (2011). Convergence of a three-dimensional quantum lattice Boltzmann scheme towards solutions of the Dirac equation. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
**369**(1944): 2155-2163. (doi:10.1098/rsta.2011.0017) (Abstract)We investigate the convergence properties of a three-dimensional quantum lattice Boltzmann scheme for the Dirac equation. These schemes were constructed as discretizations of the Dirac equation based on operator splitting to separate the streaming along the three coordinate axes, but their output has previously only been compared against solutions of the Schrödinger equation. The Schrödinger equation arises as the nonrelativistic limit of the Dirac equation, describing solutions that vary slowly compared with the Compton frequency. We demonstrate first-order convergence towards solutions of the Dirac equation obtained by an independent numerical method based on fast Fourier transforms and matrix exponentiation. © 2011 The Royal Society.

- PJ Dellar, D Lapitski, S Palpacelli and S Succi (2011). Isotropy of three-dimensional quantum lattice Boltzmann schemes. Physical Review E
**83**(4): 046706. (doi:10.1103/PhysRevE.83.046706) - S Melnik, A Hackett, JP Gleeson, MA Porter and PJ Mucha (2011). The unreasonable effectiveness of tree-based theory for networks with clustering. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**83**(3): 036112. (doi:10.1103/PhysRevE.83.036112) (Abstract)We demonstrate that a tree-based theory for various dynamical processes operating on static, undirected networks yields extremely accurate results for several networks with high levels of clustering. We find that such a theory works well as long as the mean intervertex distance ℓ is sufficiently small-that is, as long as it is close to the value of ℓ in a random network with negligible clustering and the same degree-degree correlations. We support this hypothesis numerically using both real-world networks from various domains and several classes of synthetic clustered networks. We present analytical calculations that further support our claim that tree-based theories can be accurate for clustered networks, provided that the networks are "sufficiently small" worlds. © 2011 American Physical Society.

- SJ Chapman and SE Mcburnie (2011). A unified multiple-scales approach to one-dimensional composite materials and multiphase flow. SIAM Journal on Applied Mathematics
**71**(1): 200-217. (doi:10.1137/100784515) (Abstract)The method of multiple scales is adapted to homogenization problems in which the size of the unit cell is not constant but varies with the solution (corresponding to frequency modulation as well as amplitude modulation). The method is illustrated by two one-dimensional examples taken from solid mechanics and fluid mechanics, respectively, namely, the nonlinear elastic deformation of a composite material and the ab initio derivation of the equations of multiphase slug flow. © 2011 Society for Industrial and Applied Mathematics.

- PJ Dellar (2011). Planar channel flow in Braginskii magnetohydrodynamics. Journal of Fluid Mechanics
**667**: 520-543. (doi:10.1017/S0022112010004507) - MA Porter and K Boerner (2011). Atlas of Science Visualizing What We Know. Science
**331**(6018): 676-677. (doi:10.1126/science.1201504) - SJ Chapman, J Lottes and LN Trefethen (2011). Four bugs on a rectangle. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**467**(2127): 881-896. (doi:10.1098/rspa.2010.0506) (Abstract)The idealized mathematical problem of four bugs in cyclic pursuit starting from a 2-by-1 rectangle is considered, and asymptotic formulas are derived to describe the motion. In contrast to the famous case of four bugs on a square, here the trajectories quickly freeze to essentially one dimension. After the first rotation about the centre point, the scale of the configuration has shrunk by a factor of 10, and this number is then exponentiated four more times with each successive cycle. Relations to Knuth's double-arrow notation and level-index arithmetic are discussed. This journal is © 2011 The Royal Society.

- RJ Shipley, SL Waters, AJ Davidson, JB Chaudhuri, MJ Ellis and K Chan (2011). A strategy to determine operating parameters in tissue engineering hollow fiber bioreactors. Biotechnology and Bioengineering
**108**(6): 1450-1461. (doi:10.1002/bit.23062) (Abstract)The development of tissue engineering hollow fiber bioreactors (HFB) requires the optimal design of the geometry and operation parameters of the system. This article provides a strategy for specifying operating conditions for the system based on mathematical models of oxygen delivery to the cell population. Analytical and numerical solutions of these models are developed based on Michaelis-Menten kinetics. Depending on the minimum oxygen concentration required to culture a functional cell population, together with the oxygen uptake kinetics, the strategy dictates the model needed to describe mass transport so that the operating conditions can be defined. If c ≫K we capture oxygen uptake using zero-order kinetics and proceed analytically. This enables operating equations to be developed that allow the user to choose the medium flow rate, lumen length, and ECS depth to provide a prescribed value of c . When c ≫K , we use numerical techniques to solve full Michaelis-Menten kinetics and present operating data for the bioreactor. The strategy presented utilizes both analytical and numerical approaches and can be applied to any cell type with known oxygen transport properties and uptake kinetics. © 2011 Wiley Periodicals, Inc.

- G Klingbeil, PK Maini, R Erban and M Giles (2011). STOCHSIMGPU: Parallel stochastic simulation for the Systems Biology Toolbox 2 for MATLAB. Bioinformatics
**27**(8): 1170-1171. (doi:10.1093/bioinformatics/btr068) (Abstract)Motivation: The importance of stochasticity in biological systems is becoming increasingly recognized and the computational cost of biologically realistic stochastic simulations urgently requires development of efficient software. We present a new software tool STOCHSIMGPU that exploits graphics processing units (GPUs) for parallel stochastic simulations of biological/chemical reaction systems and show that significant gains in efficiency can be made. It is integrated into MATLAB and works with the Systems Biology Toolbox 2 (SBTOOLBOX2) for MATLAB. Results: The GPU-based parallel implementation of the Gillespie stochastic simulation algorithm (SSA), the logarithmic direct method (LDM) and the next reaction method (NRM) is approximately 85 times faster than the sequential implementation of the NRM on a central processing unit (CPU). Using our software does not require any changes to the user's models, since it acts as a direct replacement of the stochastic simulation software of the SBTOOLBOX2. © The Author 2011. Published by Oxford University Press. All rights reserved.

- SL Waters, J Alastruey, KH Parker, DA Beard, PF Davies, G Jayaraman, OE Jensen, J Lee, NP Smith, AS Popel, TW Secomb, M Siebes, PHM Bovendeerd, SJ Sherwin, RJ Shipley and FN van de Vosse (2011). Theoretical models for coronary vascular biomechanics: Progress & challenges. Progress in Biophysics and Molecular Biology
**104**(1-3): 49-76. (doi:10.1016/j.pbiomolbio.2010.10.001) (Abstract)A key aim of the cardiac Physiome Project is to develop theoretical models to simulate the functional behaviour of the heart under physiological and pathophysiological conditions. Heart function is critically dependent on the delivery of an adequate blood supply to the myocardium via the coronary vasculature. Key to this critical function of the coronary vasculature is system dynamics that emerge via the interactions of the numerous constituent components at a range of spatial and temporal scales. Here, we focus on several components for which theoretical approaches can be applied, including vascular structure and mechanics, blood flow and mass transport, flow regulation, angiogenesis and vascular remodelling, and vascular cellular mechanics. For each component, we summarise the current state of the art in model development, and discuss areas requiring further research. We highlight the major challenges associated with integrating the component models to develop a computational tool that can ultimately be used to simulate the responses of the coronary vascular system to changing demands and to diseases and therapies. © 2010 Elsevier Ltd.

- E Aydemir, CJW Breward and TP Witelski (2011). The Effect of Polar Lipids on Tear Film Dynamics. Bulletin of Mathematical Biology
**73**(6): 1171-1201. (doi:10.1007/s11538-010-9555-y) (Abstract)In this paper, we present a mathematical model describing the effect of polar lipids, excreted by glands in the eyelid and present on the surface of the tear film, on the evolution of a pre-corneal tear film. We aim to explain the interesting experimentally observed phenomenon that the tear film continues to move upward even after the upper eyelid has become stationary. The polar lipid is an insoluble surface species that locally alters the surface tension of the tear film. In the lubrication limit, the model reduces to two coupled non-linear partial differential equations for the film thickness and the concentration of lipid. We solve the system numerically and observe that increasing the concentration of the lipid increases the flow of liquid up the eye. We further exploit the size of the parameters in the problem to explain the initial evolution of the system. © 2010 Society for Mathematical Biology.

### 2010

- H Hansen-Goos and JS Wettlaufer (2010).
Theory of ice premelting in porous media. Physical Review E
**81**(3): 031604. (doi:10.1103/PhysRevE.81.031604) - IJ Hewitt (2010). Modelling melting rates in upwelling mantle. Earth and Planetary Science Letters
**300**(3-4): 264-274. (doi:10.1016/j.epsl.2010.10.010) (Abstract)Upwelling regions of the mantle can undergo partial melting as a result of decompression. Many models for the dynamics of these regions have largely ignored the actual melting process or have prescribed a uniform melting rate proportional to the upwelling velocity. This paper uses a simple model for an upwelling column to calculate the melting rate from conservation principles. The model rock comprises two chemical components, and is assumed to be in thermodynamic equilibrium. For idealized linear phase constraints the melting rate can be calculated analytically, and is found to be proportional to the average upwelling velocity of both the matrix and melt.A secondary aim is to discuss reactive instabilities; the model predicts that the one dimensional state will be linearly stable, whereas previous models have suggested that reactive infiltration instability should occur. This is argued to be a result of the 'background' melting rate which has not usually been fully accounted for, but which has a stabilizing effect. The model here can also be applied to a column in which some melt is already present, and in that case it does exhibit a channeling instability. It is concluded that accounting for melt production consistently in mid-ocean ridge models is important when assessing the likely modes of melt transport. © 2010 Elsevier B.V.

- BG Sengers, CP Please and RM Lewis (2010). Computational modelling of amino acid transfer interactions in the placenta. Experimental Physiology
**95**(7): 829-840. (doi:10.1113/expphysiol.2010.052902) (Abstract)Amino acid transfer from mother to fetus via the placenta plays a critical role in normal development, and restricted transfer is associated with fetal growth restriction. Placental amino acid transfer involves the interaction of 15 or more transporters and 20 amino acids. This complexity means that knowing which transporters are present is not sufficient to predict how they operate together as a system. Therefore, in order to investigate how placental amino acid transfer occurs as a system, an integrated mathematical/computational modelling framework was developed to represent the simultaneous transport of multiple amino acids. The approach was based on a compartmental model, in which separate maternal, syncytiotrophoblast and fetal volumes were distinguished, and transporters were modelled on the maternal- and fetal-facing membranes of the syncytiotrophoblast using Michaelis-Menten-type kinetics. The model was tested in comparison with placental perfusion experiments studying serine-alanine exchange and found to correspond well. The results demonstrated how the different transporters can work together as an integrated system and allowed their relative importance to be assessed. Placental-fetal serine exchange was found to be most sensitive to basal membrane transporter characteristics, but a range of secondary, less intuitive effects were also revealed. While this work only addressed a relatively simple three amino acid system, it demonstrates the feasibility of the approach and could be extended to incorporate additional experimental parameters. Ultimately, this approach will allow physiological simulations of amino acid transfer. This will enhance our understanding of these complex systems and placental function in health and disease. © 2010 The Physiological Society.

- PS Stewart, J Billingham, OE Jensen and SL Waters (2010). Spatially localised growth within global instabilities of flexible channel flows. Iutam Bookseries
**18**: 397-402. (doi:10.1007/978-90-481-3723-7_64) (Abstract)We investigate the localised spatial growth of two-dimensional disturbances governed by the long-wavelength Orr-Sommerfeld operator in a rigid channel at sub-transition Reynolds numbers. We discuss the link between this growth and vorticity waves, a striking feature of the self-excited oscillations that arise when flow is driven through a finite-length channel, one wall of which contains a segment of flexible membrane. © 2010 Springer Science+Business Media B.V.

- CL Hall (2010). Asymptotic expressions for the nearest and furthest dislocations in a pile-up against a grain boundary. Philosophical Magazine
**90**(29): 3879-3890. (doi:10.1080/14786435.2010.499859) (Abstract)In 1965, Armstrong and Head explored the problem of a pile-up of screw dislocations against a grain boundary. They used numerical methods to determine the positions of the dislocations in the pile-up and they were able to fit approximate formulae for the locations of the first and last dislocations. These formulae were used to gain insights into the Hall-Petch relationship. More recently, Voskoboinikov et al. used asymptotic techniques to study the equivalent problem of a pile-up of a large number of screw dislocations against a bimetallic interface. In this paper, we extend the work of Voskoboinikov et al. to construct systematic asymptotic expressions for the formulae proposed by Armstrong and Head. The further extension of these techniques to more general pile-ups is also outlined. As a result of this work, we show that a pile-up against a grain boundary can become equivalent to a pile-up against a locked dislocation in the case where the mismatch across the boundary is small. © 2010 Taylor & Francis.

- WT Lee, AC Fowler, O Power, S Healy and J Browne (2010). Blowing of polycrystalline silicon fuses. Applied Physics Letters
**97**(2): 023502. (doi:10.1063/1.3457469) (Abstract)Polycrystalline silicon fuses are one time programmable memory elements which allow the calibration of integrated circuits at wafer and package level. We present a zero-dimensional lumped parameter model of the programming of fuses made from a combination of tungsten silicide and polycrystalline silicon. The components of the model are an electrical model, a thermal model, and a flow model. The model generates quantitatively accurate results and reproduces trends with applied voltage and fuse size. © 2010 American Institute of Physics.

- AC Fowler (2010). The instability theory of drumlin formation applied to Newtonian viscous ice of finite depth. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**466**(2121): 2673-2694. (doi:10.1098/rspa.2010.0017) (Abstract)The Hindmarsh instability theory of drumlin formation is applied to the study of interfacial instabilities, which may arise when ice flows viscously over deformable sediments. Here, the analytic form of this theory is extended to the case where the ice is Newtonian viscous and of finite depth, and where the basal till can be both sheared by the ice and squeezed by basal effective pressure gradients: previous authors assumed infinitely deep ice, based on the assumption that the developing waveforms had wavelength much less than ice depth. The previous infinite depth theory only allowed transverse instabilities to occur, and these have been associated with the formation of ribbed moraine; one of the purposes of extending the analysis to finite depth is to see whether three-dimensional instabilities, which might be associated with the formation of drumlins or mega-scale glacial lineations, can occur: we find that they do not. A second purpose is to calculate under what circumstances the infinite depth theory provides accurate prediction of bedform development in ice of finite depth d .We find that this is the case if the waveforms have a wavelength less than approximately 1.2d . Finally, the finite depth theory allows us to compute, for the first time, the response of the ice surface to the developing unstable bedforms. We find that this response is rapid, and we give explicit recipes for the surface perturbation transfer functions in terms of the perturbations to the basal stress and the basal topography. © 2010 The Royal Society.

- AC Fowler (2010). The formation of subglacial streams and mega-scale glacial lineations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**466**(2123): 3181-3201. (doi:10.1098/rspa.2010.0009) (Abstract)The instability theory of drumlin formation has been very successful in predicting the existence of ribbed moraine, as well as its amplitude and wavelength. However, the theory as it stands has not yet been shown to have the capability of predicting the existence of three-dimensional bedforms-drumlins-or their more extreme cousins, mega-scale glacial lineations. We extend the instability theory to include a dynamic description of the local subglacial drainage system, and in particular, we show that a uniform water-film flow between ice and deformable subglacial till is unstable, and that as a consequence, lineations will form. Predictions of the transverse wavelengths are consistent with observations. © 2010 The Royal Society.

- AC Fowler (2010). A note on the derivation of the quasi-geostrophic potential vorticity equation. Geophysical and Astrophysical Fluid Dynamics
**105**(2-3): 340-350. (doi:10.1080/03091929.2010.531719) (Abstract)The derivation of the quasi-geostrophic potential vorticity equation of mathematical meteorology is usually done using fairly sophisticated techniques of perturbation theory, but stops short of deriving self-consistently the stratification parameter of the mean atmospheric state. In this note we suggest how this should be done within the confines of the theory, and as a consequence we raise the possibility that the atmosphere could become globally unstable, with dramatic consequences. © 2011 Taylor & Francis.

- AC Fowler, WT Lee, B Scheu and MJ Mcguinness (2010). A theoretical model of the explosive fragmentation of vesicular magma. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**466**(2115): 731-752. (doi:10.1098/rspa.2009.0382) (Abstract)Recent experimental work has shown that, when a vertical column of rock under large pressure is suddenly depressurized, the column can 'explode' in a structured and repeatable way. The observations show that a sequence of horizontal fractures forms from the top down, and the resulting blocks are lifted off and ejected. The blocks can suffer secondary internal fractures. This experiment provides a framework for understanding the way in which catastrophic explosion can occur, and is motivated by the corresponding phenomenon of magmatic explosion during Vulcanian eruptions. We build a theoretical model to describe these results, and show that it is capable of describing both the primary sequence of fracturing and the secondary intrablock fracturing. The model allows us to suggest a practical criterion for when such explosions occur: firstly, the initial confining pressure must exceed the yield stress of the rock, and, secondly, the diffusion of the gas by porous flow must be sufficiently slow that a large excess pore pressure is built up. This will be the case if the rock permeability is small enough. © 2010 The Royal Society.

- AC Fowler, R Toja and C Vázquez (2010). Temperature-dependent shear flow and the absence of thermal runaway in valley glaciers. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**466**(2114): 363-382. (doi:10.1098/rspa.2009.0335) (Abstract)We propose a two-dimensional model of a valley glacier in order to reconsider the question of whether thermal runaway could be a viable mechanism for the onset of creep instability in surging glaciers. We do this by providing an approximate solution for the temperature field based on the idea that shear is concentrated at the glacier bed. With this assumption, we show that a closed-form evolution equation for the glacier profile exists. While this is well known for isoviscous flows, it has not been previously derived for variable viscosity flows. During the process of deriving this equation, we show that thermal runaway does not occur.We provide numerical solutions of the model, and are led to infer that enhanced basal heating owing to refreezing of surface meltwater is an essential constituent in raising the bed temperature to the melting point. © 2009 The Royal Society.

- D Peschka, A Münch and B Niethammer (2010). Self-similar rupture of viscous thin films in the strong-slip regime. Nonlinearity
**23**(2): 409-427. (doi:10.1088/0951-7715/23/2/010) - D Peschka, A Münch and B Niethammer (2010). Thin-film rupture for large slip. Journal of Engineering Mathematics
**66**(1-3): 33-51. (doi:10.1007/s10665-009-9342-7) - RJ Zammett and AC Fowler (2010). The morphology of the martian ICE CAPS: A mathematical model of ICE-Dust kinetics. SIAM Journal on Applied Mathematics
**70**(7): 2409-2433. (doi:10.1137/080734558) (Abstract)Spiral-shaped canyons on the polar ice caps of Mars are striking morphological features whose explanation is still mysterious. We pose a model for the kinetics at the ice-atmosphere interface based on the positive feedback properties of both dust and ice albedo within the north polar ice cap, together with katabatic wind transport of exposed dust. Analysis of this model indicates that traveling waves are possible, but only solitary waves are found in the numerical solution. This suggests that the traveling wave hypothesis for trough formation may not be an appropriate explanation. © 2010 Society for Industrial and Applied Mathematics.

- A Goriely and R Vandiver (2010). On the mechanical stability of growing arteries. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
**75**(4): 549-570. (doi:10.1093/imamat/hxq021) (Abstract)Arteries are modelled, within the framework of non-linear elasticity, as incompressible two-layer cylindrical structures that are residually stressed through differential growth. These structures are loaded by an axial force, internal pressure and have non-linear, anisotropic, hyperelastic response to stresses. Parameters for this model are directly related to experimental observations. The possible role of axial residual stress in regulating stress in arteries and preventing buckling instabilities is investigated. It is shown that axial residual stress lowers the critical internal pressure leading to buckling and that a reduction of axial loading may lead to a buckling instability which may eventually lead to arterial tortusity. © 2010 The Author. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

- MA Beauregard, A Goriely and M Tabor (2010). The nonlinear dynamics of elastic tubes conveying a fluid. International Journal of Solids and Structures
**47**(1): 161-168. (doi:10.1016/j.ijsolstr.2009.09.022) (Abstract)The Kirchhoff equations for elastic tubes are modified to include the effect of fluid flow. Using the techniques of linear and nonlinear analysis specially developed for the Kirchhoff equations, the effect of the fluid flow on the basic twist-to-writhe instability is investigated. The results suggest an intriguing modification of the bifurcation threshold due to the flow. Beyond threshold the buckled tube acquires a slight curvature which modifies the flow rate and results in a correction to nonlinearity of the amplitude equation governing the deformation dynamics. © 2009 Elsevier Ltd. All rights reserved.

- J Mcmahon, A Goriely and M Tabor (2010). Spontaneous cavitation in growing elastic membranes. Mathematics and Mechanics of Solids
**15**(1): 57-77. (doi:10.1177/1081286508092010) (Abstract)One of the possible effects of volumetric growth in elastic materials is the creation of residual stresses. These stresses are known to change many of the classical properties of the material and have been studied extensively in the context of volumetric growth in biomechanics. Here we consider the problem of elastic cavitation in a growing compressible elastic membrane. Growth is taken to be homogeneous but anisotropic, and the membrane is assumed to remain axisymmetric during growth and deformation. We prove that neo-Hookean membranes cannot cavitate, but for Varga elastic materials we find conditions under which the material exhibits spontaneous cavitation in the absence of external loads, in marked distinction from the cavitation problem without growth. © 2010 SAGE Publications.

- A Goriely, M Tabor and A Tongen (2010). A morpho-elastic model of hyphal tip growth in filamentous organisms. Iutam Bookseries
**16**: 245-255. (doi:10.1007/978-90-481-3348-2_21) (Abstract)The growth of filamentous cells is modeled through the use of exact, nonlinear, elasticity theory for shells and membranes. The biomechanical model is able to capture the generic features of growth of a broad array of cells including actinomycetes, fungi, and root hairs. It also provides the means of studying the effects of external surface stresses. The growth mechanism is modeled by a process of incremental elastic growth in which the cell wall responds elastically to the continuous addition of new material. © Springer Science+Business Media B.V. 2010.

- A Goriely, DE Moulton and R Vandiver (2010). Elastic cavitation, tube hollowing, and differential growth in plants and biological tissues. Epl
**91**(1): 18001. (doi:10.1209/0295-5075/91/18001) (Abstract)Elastic cavitation is a well-known physical process by which elastic materials under stress can open cavities. Usually, cavitation is induced by applied loads on the elastic body. However, growing materials may generate stresses in the absence of applied loads and could induce cavity opening. Here, we demonstrate the possibility of spontaneous growth-induced cavitation in elastic materials and consider the implications of this phenomenon to biological tissues and in particular to the problem of schizogenous aerenchyma formation. Copyright © EPLA, 2010.

- D Vella and J Li (2010). The impulsive motion of a small cylinder at an interface. Physics of Fluids
**22**(5): 1-12. (doi:10.1063/1.3427241) (Abstract)We study the unsteady motion caused by an impulse acting at time t=0 on a small cylinder floating horizontally at a liquid-gas interface. This is a model for the impact of a cylinder onto a liquid surface after the initial splash. Following the impulse, the motion of the cylinder is determined by its weight per unit length (pulling it into the bulk liquid) and resistance from the liquid, which acts to keep the cylinder at the interface. The range of cylinder radii r and impact speeds U considered is such that the resistance from the liquid comes from both the interfacial tension and hydrodynamic pressures. We use two theoretical approaches to investigate this problem. In the first, we apply the arbitrary Lagrangian Eulerian (ALE) method developed by Li et al. ["An arbitrary Lagrangian Eulerian method for moving-boundary problems and its application to jumping over water," J. Comput. Phys.208, 289 (2005)] to compute the fluid flow caused by the impulse and the (coupled) motion of the cylinder. We show that at early times the interfacial deformation is given by a family of shapes parametrized by r/t. We also find that for a given density and radius there is a critical impulse speed below which the cylinder is captured by the interface and floats but above which it pierces the interface and sinks. Our second theoretical approach is a simplified one in which we assume that the interface is in equilibrium and derive an ordinary differential equation for the motion of the cylinder. Solving this we again find the existence of a critical impulse speed for sinking giving us some quantitative understanding of the results from the ALE simulations. Finally, we compare our theoretical predictions with the results of experiments for cylinder impacts by Vella and Metcalfe ["Surface tension dominated impact," Phys. Fluids19, 072108 (2007)]. This comparison suggests that the influence of contact line effects, neglected here, may be important in the transition from floating to sinking. © 2010 American Institute of Physics.

- D Vella, M Adda-Bedia and E Cerda (2010). Capillary wrinkling of elastic membranes. Soft Matter
**6**(22): 5778-5782. (doi:10.1039/c0sm00432d) (Abstract)We present a physically-based model for the deformation of a floating elastic membrane caused by the presence of a liquid drop. Starting from the equations of membrane theory modified to account for surface energies, we show that the presence of a liquid drop causes an azimuthal compression over a finite region. This explains the origin of the wrinkling of such membranes observed recently (Huang et al., Science, 2007, 317, 650) and suggests a single parameter that determines the extent of the wrinkled region. While experimental data supports the importance of this single parameter, our theory under-predicts the extent of the wrinkled region observed experimentally. We suggest that this discrepancy is likely to be due to the wrinkling observed here being far from the threshold. © 2010 The Royal Society of Chemistry.

- C Harris, S Howison and R Sircar (2010).
Games with Exhaustible Resources. SIAM Journal on Applied Mathematics
**70**(7): 2556-2581. (doi:10.1137/09076862X) - AL Stewart and PJ Dellar (2010). Two-Layer Shallow Water Equations with Complete Coriolis Force and Topography. Progress in Industrial Mathematics at Ecmi 2008
**15**: 1033-1038. (doi:10.1007/978-3-642-12110-4_164) - X Luo, I Hoteit, IM Moroz and TE Simos (2010). On Ensemble Nonlinear Kalman Filtering with Symmetric Analysis Ensembles. Numerical Analysis and Applied Mathematics, Vols I-III
**1281**: 1088-1092. - L Duan, CL Farmer, IM Moroz and TE Simos (2010). Regularized Particle Filter with Langevin Resampling Step. Numerical Analysis and Applied Mathematics, Vols I-III
**1281**: 1080-1083. - I Hoteit, X Luo, D-T Pham, IM Moroz and TE Simos (2010). Particle Kalman Filtering: A Nonlinear Framework for Ensemble Kalman Filters. Numerical Analysis and Applied Mathematics, Vols I-III
**1281**: 1075-1079. (doi:10.1063/1.3497823) - L Duan, CL Farmer, IM Moroz and TE Simos (2010). Sequential Inverse Problems Bayesian Principles and the Logistic Map Example. Numerical Analysis and Applied Mathematics, Vols I-III
**1281**: 1071-1074. - IM Moroz (2010). Unstable periodic orbits in a four-dimensional Faraday disk dynamo. Geophysical and Astrophysical Fluid Dynamics
**105**(2-3): 273-286. (doi:10.1080/03091929.2010.511206) (Abstract)In this article, we return to a four-dimensional model for a self-exciting Faraday disk dynamo, originally investigated by Hide and Moroz [Hide, R. and Moroz, I.M., Effects due to induced azimuthal eddy currents in the self-exciting Faraday disk homopolar dynamo with a nonlinear series motor: I two special cases. Physica D 1999, 134, 387-301]. We extract unstable periodic orbits from long time series of chaotic trajectories by the method of close returns to a Poincaré section. We then employ recently developed topological tools to characterise the underlying chaotic attractor of the dynamo by identifying its branched manifold or template. © 2011 Taylor & Francis.

- X Luo, I Hoteit and IM Moroz (2010). On ensemble nonlinear Kalman filtering with symmetric analysis ensembles. Aip Conference Proceedings
**1281**: 1088-1092. (doi:10.1063/1.3497831) (Abstract)The ensemble square root filter (EnSRF) [1, 2, 3, 4] is a popular method for data assimilation in high dimensional systems (e.g., geophysics models). Essentially the EnSRF is a Monte Carlo implementation of the conventional Kalman filter (KF) [5, 6]. It is mainly different from the KF at the prediction steps, where it is some ensembles, rather then the means and covariance matrices, of the system state that are propagated forward. In doing this, the EnSRF is computationally more efficient than the KF, since propagating a covariance matrix forward in high dimensional systems is prohibitively expensive. In addition, the EnSRF is also very convenient in implementation. By propagating the ensembles of the system state, the EnSRF can be directly applied to nonlinear systems without any change in comparison to the assimilation procedures in linear systems. However, by adopting the Monte Carlo method, the EnSRF also incurs certain sampling errors. One way to alleviate this problem is to introduce certain symmetry to the ensembles, which can reduce the sampling errors and spurious modes in evaluation of the means and covariances of the ensembles [7]. In this contribution, we present two methods to produce symmetric ensembles. One is based on the unscented transform [8, 9], which leads to the unscented Kalman filter (UKF) [8, 9] and its variant, the ensemble unscented Kalman filter (EnUKF) [7]. The other is based on Stirling's interpolation formula (SIF), which results in the divided difference filter (DDF) [10]. Here we propose a simplified divided difference filter (sDDF) in the context of ensemble filtering. The similarity and difference between the sDDF and the EnUKF will be discussed. Numerical experiments will also be conducted to investigate the performance of the sDDF and the EnUKF, and compare them to a well-established EnSRF, the ensemble transform Kalman filter (ETKF) [2]. © 2010 American Institute of Physics.

- L Duan, CL Farmer and IM Moroz (2010). Regularized particle filter with Langevin resampling step. Aip Conference Proceedings
**1281**: 1080-1083. (doi:10.1063/1.3497827) (Abstract)The solution of an inverse problem involves the estimation of variables and parameters values given by the state-space system. While a general (infinite-dimensional) optimal filter theory [1, 2] exists for nonlinear systems with Gaussian or non-Gaussian noise, applications rely on (finite-dimensional) suboptimal approximations to the optimal filter for practical implementations. The most widely-studied filters of this kind include the Regularized Particle Filter (RPF) [3, 4] and the Ensemble Square Root Filter (EnSRF) [5]. The latter is an ad-hoc approximation to the Bayes Filter, while the former is rigorously formulated, based upon the Glivenko-Cantelli theorem. By introducing a new global resampling step to the RPF, the EnSRF is proved to approximate the RPF in a special case. © 2010 American Institute of Physics.

- I Hoteit, X Luo, D-T Pham and IM Moroz (2010). Particle Kalman filtering: A nonlinear framework for ensemble Kalman filters. Aip Conference Proceedings
**1281**: 1075-1079. (doi:10.1063/1.3497823) (Abstract)Optimal nonlinear filtering consists of sequentially determining the conditional probability distribution functions (pdf) of the system state, given the information of the dynamical and measurement processes and the previous measurements. Once the pdfs are obtained, one can determine different estimates, for instance, the minimum variance estimate, or the maximum a posteriori estimate, of the system state. It can be shown that, many filters, including the Kalman filter (KF) and the particle filter (PF), can be derived based on this sequential Bayesian estimation framework. In this contribution, we present a Gaussian mixture-based framework, called the particle Kalman filter (PKF), and discuss how the different EnKF methods can be derived as simplified variants of the PKF. We also discuss approaches to reducing the computational burden of the PKF in order to make it suitable for complex geosciences applications. We use the strongly nonlinear Lorenz-96 model to illustrate the performance of the PKF. © 2010 American Institute of Physics.

- L Duan, CL Farmer and IM Moroz (2010). Sequential inverse problems Bayesian principles and the logistic map example. Aip Conference Proceedings
**1281**: 1071-1074. (doi:10.1063/1.3497821) (Abstract)Bayesian statistics provides a general framework for solving inverse problems, but is not without interpretation and implementation problems. This paper discusses difficulties arising from the fact that forward models are always in error to some extent. Using a simple example based on the one-dimensional logistic map, we argue that, when implementation problems are minimal, the Bayesian framework is quite adequate. In this paper the Bayesian Filter is shown to be able to recover excellent state estimates in the perfect model scenario (PMS) and to distinguish the PMS from the imperfect model scenario (IMS). Through a quantitative comparison of the way in which the observations are assimilated in both the PMS and the IMS scenarios, we suggest that one can, sometimes, measure the degree of imperfection. © 2010 American Institute of Physics.

- N Boechler, G Theocharis, S Job, C Daraio, PG Kevrekidis and MA Porter (2010). Discrete breathers in one-dimensional diatomic granular crystals. Physical Review Letters
**104**(24): 244302. (doi:10.1103/PhysRevLett.104.244302) (Abstract)We report the experimental observation of modulational instability and discrete breathers in a one-dimensional diatomic granular crystal composed of compressed elastic beads that interact via Hertzian contact. We first characterize their effective linear spectrum both theoretically and experimentally. We then illustrate theoretically and numerically the modulational instability of the lower edge of the optical band. This leads to the dynamical formation of long-lived breather structures, whose families of solutions we compute throughout the linear spectral gap. Finally, we experimentally observe the manifestation of the modulational instability and the resulting generation of localized breathing modes with quantitative characteristics that agree with our numerical results. © 2010 The American Physical Society.

- M Beguerisse Díaz, MA Porter and J-P Onnela (2010). Competition for popularity in bipartite networks. Chaos
**20**(4): 043101. (doi:10.1063/1.3475411) (Abstract)We present a dynamical model for rewiring and attachment in bipartite networks. Edges are placed between nodes that belong to catalogs that can either be fixed in size or growing in size. The model is motivated by an empirical study of data from the video rental service Netflix, which invites its users to give ratings to the videos available in its catalog. We find that the distribution of the number of ratings given by users and that of the number of ratings received by videos both follow a power law with an exponential cutoff. We also examine the activity patterns of Netflix users and find bursts of intense video-rating activity followed by long periods of inactivity. We derive ordinary differential equations to model the acquisition of edges by the nodes over time and obtain the corresponding time-dependent degree distributions. We then compare our results with the Netflix data and find good agreement. We conclude with a discussion of how catalog models can be used to study systems in which agents are forced to choose, rate, or prioritize their interactions from a large set of options. © 2010 American Institute of Physics.

- PJ Mucha and MA Porter (2010). Communities in multislice voting networks. Chaos
**20**(4): 041108. (doi:10.1063/1.3518696) - ES Benilov, SJ Chapman, JB Mcleod, JR Ockendon and VS Zubkov (2010). On liquid films on an inclined plate. Journal of Fluid Mechanics : 1-17.
- PS Stewart, M Heil, SL Waters and OE Jensen (2010). Sloshing and slamming oscillations in a collapsible channel flow. Journal of Fluid Mechanics
**662**: 288-319. (doi:10.1017/S0022112010003277) (Abstract)We consider laminar high-Reynolds-number flow through a finite-length planar channel, where a portion of one wall is replaced by a thin massless elastic membrane that is held under longitudinal tension T and subject to a linear external pressure distribution. The flow is driven by a fixed pressure drop along the full length of the channel. We investigate the global stability of two-dimensional Poiseuille flow using a method of matched local eigenfunction expansions, which is compared to direct numerical simulations. We trace the neutral stability curve of the primary oscillatory instability of the system, illustrating a transition from high-frequency sloshing oscillations at high T to vigorous slamming motion at low T. Small-amplitude sloshing at high T can be captured using a low-order eigenmode truncation involving four surface-based modes in the compliant segment of the channel coupled to Womersley flow in the rigid segments. At lower tensions, we show that hydrodynamic modes increasingly contribute to the global instability, and we demonstrate a change in the mechanism of energy transfer from the mean flow, with viscous effects being destabilizing. Simulations of finite-amplitude oscillations at low T reveal a generic slamming motion, in which the flexible membrane is drawn close to the opposite rigid wall before recovering rapidly. A simple model is used to demonstrate how fluid inertia in the downstream rigid channel segment, coupled to membrane curvature downstream of the moving constriction, together control slamming dynamics. © 2010 Cambridge University Press.

- G Theocharis, N Boechler, S Job, C Daraio, PG Kevrekidis and MA Porter (2010). Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystals. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**82**(5): 056604. (doi:10.1103/PhysRevE.82.056604) (Abstract)We present a systematic study of the existence and stability of discrete breathers that are spatially localized in the bulk of a one-dimensional chain of compressed elastic beads that interact via Hertzian contact. The chain is diatomic, consisting of a periodic arrangement of heavy and light spherical particles. We examine two families of discrete gap breathers: (1) an unstable discrete gap breather that is centered on a heavy particle and characterized by a symmetric spatial energy profile and (2) a potentially stable discrete gap breather that is centered on a light particle and is characterized by an asymmetric spatial energy profile. We investigate their existence, structure, and stability throughout the band gap of the linear spectrum and classify them into four regimes: a regime near the lower optical band edge of the linear spectrum, a moderately discrete regime, a strongly discrete regime that lies deep within the band gap of the linearized version of the system, and a regime near the upper acoustic band edge. We contrast discrete breathers in anharmonic Fermi-Pasta-Ulam (FPU)-type diatomic chains with those in diatomic granular crystals, which have a tensionless interaction potential between adjacent particles, and note that the asymmetric nature of the tensionless interaction potential can lead to hybrid bulk-surface localized solutions. © 2010 The American Physical Society.

- RJ Whittaker, SL Waters, M Heil and OE Jensen (2010). Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**466**(2124): 3635-3657. (doi:10.1098/rspa.2009.0641) (Abstract)We present a theoretical description of flow-induced self-excited oscillations in the Starling resistor - a pre-stretched thin-walled elastic tube that is mounted on two rigid tubes and enclosed in a pressure chamber. Assuming that the flow through the elastic tube is driven by imposing the flow rate at the downstream end, we study the development of small-amplitude long-wavelength high-frequency oscillations, combining the results of two previous studies in which we analysed the fluid and solid mechanics of the problem in isolation. We derive a one-dimensional eigenvalue problem for the frequencies and mode shapes of the oscillations, and determine the slow growth or decay of the normal modes by considering the system's energy budget.We compare the theoretical predictions for the mode shapes, frequencies and growth rates with the results of direct numerical simulations, based on the solution of the three-dimensional Navier-Stokes equations, coupled to the equations of shell theory, and find good agreement between the results. Our results provide the first asymptotic predictions for the onset of self-excited oscillations in three-dimensional collapsible tube flows. © 2010 The Royal Society.

- RJ Whittaker, SL Waters, M Heil and OE Jensen (2010). A rational derivation of a tube law from shell theory. Quarterly Journal of Mechanics and Applied Mathematics
**63**(4): 465-496. (doi:10.1093/qjmam/hbq020) (Abstract)We consider small-amplitude deformations of a long thin-walled elastic tube having an initially axially uniform elliptical cross section. The tube is subject to an axial pre-stress, and the deformations result from an applied transmural pressure. An approximate tube law (linking the transmural pressure with the cross-sectional area and its axial derivatives) is derived from shell theory in the distinguished asymptotic limit in which the tube's behaviour is dominated by the restoring forces from the axial pre-stress and azimuthal bending. This is possible because the deformations of the tube induced by both the transmural pressure and the axial forces can be described, to very good approximation, by a single azimuthal mode of deformation of axially varying amplitude. The resulting tube law is compared with numerical solutions of the full shell equations and good agreement is found (provided the tube is sufficiently long and the wall not too thin so that in-plane shearing is negligible). We discuss the applications of our results to the modelling of flow in collapsible tubes. © The author 2010.

- S Agarwal, CM Deane, MA Porter and NS Jones (2010). Revisiting date and party hubs: novel approaches to role assignment in protein interaction networks.. Plos computational biology
**6**(6): e1000817. (doi:10.1371/journal.pcbi.1000817) (Abstract)The idea of "date" and "party" hubs has been influential in the study of protein-protein interaction networks. Date hubs display low co-expression with their partners, whilst party hubs have high co-expression. It was proposed that party hubs are local coordinators whereas date hubs are global connectors. Here, we show that the reported importance of date hubs to network connectivity can in fact be attributed to a tiny subset of them. Crucially, these few, extremely central, hubs do not display particularly low expression correlation, undermining the idea of a link between this quantity and hub function. The date/party distinction was originally motivated by an approximately bimodal distribution of hub co-expression; we show that this feature is not always robust to methodological changes. Additionally, topological properties of hubs do not in general correlate with co-expression. However, we find significant correlations between interaction centrality and the functional similarity of the interacting proteins. We suggest that thinking in terms of a date/party dichotomy for hubs in protein interaction networks is not meaningful, and it might be more useful to conceive of roles for protein-protein interactions rather than for individual proteins.

- JEF Green, SL Waters, JP Whiteley, L Edelstein-Keshet, KM Shakesheff and HM Byrne (2010). Non-local models for the formation of hepatocyte-stellate cell aggregates. Journal of Theoretical Biology
**267**(1): 106-120. (doi:10.1016/j.jtbi.2010.08.013) - C Xue, HJ Hwang, KJ Painter and R Erban (2010). Travelling Waves in Hyperbolic Chemotaxis Equations. Bulletin of Mathematical Biology : 1-39.
- PD Howell, B Scheid and HA Stone (2010). Newtonian pizza: Spinning a viscous sheet. Journal of Fluid Mechanics
**659**: 1-23. (doi:10.1017/S0022112010001564) (Abstract)We study the axisymmetric stretching of a thin sheet of viscous fluid driven by a centrifugal body force. Time-dependent simulations show that the sheet radius R(t) tends to infinity in finite time. As time t approaches the critical time t, the sheet becomes partitioned into a very thin central region and a relatively thick rim. A net momentum and mass balance in the rim leads to a prediction for the sheet radius near the singularity that agrees with the numerical simulations. By asymptotically matching the dynamics of the sheet with the rim, we find that the thickness h in the central region is described by a similarity solution of the second kind, with h (t t) where the exponent α satisfies a nonlinear eigenvalue problem. Finally, for non-zero surface tension, we find that the exponent increases rapidly to infinity at a critical value of the rotational Bond number B = 1/4. For B > 1/4, surface tension defeats the centrifugal force, causing the sheet to retract rather than to stretch, with the limiting behaviour described by a similarity solution of the first kind. © 2010 Cambridge University Press.

- P Pathmanathan, DJ Gavaghan, JP Whiteley and SJ Chapman (2010). Cardiac electromechanics: The effect of contraction model on the mathematical problem and accuracy of the numerical scheme. Quarterly Journal of Mechanics and Applied Mathematics
**63**(3): 375-399. (doi:10.1093/qjmam/hbq014) (Abstract)Models of cardiac electromechanics usually contain a contraction model determining the active tension induced at the cellular level and the equations of nonlinear elasticity to determine tissue deformation in response to this active tension. All contraction models are dependent on cardiac electrophysiology but can also be dependent on the stretch and stretch rate in the fibre direction. This fundamentally affects the mathematical problem being solved, through classification of the governing partial differential equations, which affects numerical schemes that can be used to solve the governing equations. We categorise contraction models into three types, and for each consider questions such as classification and the most appropriate choice from two numerical methods (the explicit and implicit schemes). In terms of mathematical classification, we consider the question of strong ellipticity of the total strain energy (important for precluding 'unnatural' material behaviour) for stretch-rate-independent contraction models; whereas for stretch-rate-dependent contraction models, we introduce a corresponding third-order problem and explain how certain choices of boundary condition could lead to constraints on allowable initial condition. In terms of suitable numerical methods, we show that an explicit approach (where the contraction model is integrated in the time step prior to the bulk deformation being computed) is (i) appropriate for stretch-independent contraction models; (ii) only conditionally stable, with the stability criterion independent of time step, for contractions models which just depend on stretch (but not stretch rate) and (iii) inappropriate for stretch-rate-dependent models. © The author 2010. Published by Oxford University Press; all rights reserved.

- CL Hall, CS Jonathan and JR Ockendon (2010). Asymptotic analysis of a system of algebraic equations arising in dislocation theory. SIAM Journal on Applied Mathematics
**70**(7): 2729-2749. (doi:10.1137/090778444) (Abstract)The system of algebraic equations given by σ, j≠=i a = 1, i = 1, 2, ⋯ , n, x0 = 0, appears in dislocation theory in models of dislocation pile-ups. Specifically, the case a = 1 corresponds to the simple situation where n dislocations are piled up against a locked dislocation, while the case a = 3 corresponds to n dislocation dipoles piled up against a locked dipole. We present a general analysis of systems of this type for a > 0 and n large. In the asymptotic limit n→∞, it becomes possible to replace the system of discrete equations with a continuum equation for the particle density. For 0 < a < 2, this takes the form of a singular integral equation, while for a > 2 it is a first-order differential equation. The critical case a = 2 requires special treatment, but, up to corrections of logarithmic order, it also leads to a differential equation. The continuum approximation is valid only for i neither too small nor too close to n. The boundary layers at either end of the pile-up are also analyzed, which requires matching between discrete and continuum approximations to the main problem. © 2010 Society for Industrial and Applied Mathematics.

- L Ponson, N Boechler, C Daraio, YM Lai, MA Porter and PG Kevrekidis (2010). Nonlinear waves in disordered diatomic granular chains. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**82**(2): 021301. (doi:10.1103/PhysRevE.82.021301) (Abstract)We investigate the propagation and scattering of highly nonlinear waves in disordered granular chains composed of diatomic (two-mass) units of spheres that interact via Hertzian contact. Using ideas from statistical mechanics, we consider each diatomic unit to be a "spin," so that a granular chain can be viewed as a spin chain composed of units that are each oriented in one of two possible ways. Experiments and numerical simulations both reveal the existence of two different mechanisms of wave propagation: in low-disorder chains, we observe the propagation of a solitary pulse with exponentially decaying amplitude. Beyond a critical level of disorder, the wave amplitude instead decays as a power law, and the wave transmission becomes insensitive to the level of disorder. We characterize the spatiotemporal structure of the wave in both propagation regimes and propose a simple theoretical interpretation for a transition between the two regimes. Our investigation suggests that an elastic spin chain can be used as a model system to investigate the role of heterogeneities in the propagation of highly nonlinear waves. © 2010 The American Physical Society.

- X Luo, I Hoteit and IM Moroz (2010). Reply to "comment on 'Ensemble Kalman filter with the unscented transform'". Physica D: Nonlinear Phenomena
**239**(17): 1662-1664. (doi:10.1016/j.physd.2010.04.011) (Abstract)This is a reply to the comment of Dr. Sakov on the work "Ensemble Kalman filter with the unscented transform" of Luo and Moroz (2009) [2]. © 2010 Elsevier B.V. All rights reserved.

- C Escudero, CA Yates, R Erban, PK Maini, J Buhl, ID Couzin and IG Kevrekidis (2010). Ergodic directional switching in mobile insect groups. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**82**(1): 011926. (doi:10.1103/PhysRevE.82.011926) (Abstract)We obtain a Fokker-Planck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including nonmonotonic behavior of several order and disorder transition indicators in noise intensity. This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the group's direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation. © 2010 The American Physical Society.

- PJ Dellar (2010). Electromagnetic waves in lattice Boltzmann magnetohydrodynamics. Epl
**90**(5): 50002. (doi:10.1209/0295-5075/90/50002) - CG Bell, CJW Breward, PD Howell, J Penfold and RK Thomas (2010). A theoretical analysis of the surface tension profiles of strongly interacting polymer-surfactant systems. Journal of Colloid and Interface Science
**350**(2): 486-493. (doi:10.1016/j.jcis.2010.07.020) (Abstract)Measurements of the surface tension of static polymer-surfactant mixtures as the concentrations of polymer and surfactant are varied produce interesting and surprising profiles. It is well known how the critical points in the surface tension profile of a weakly interacting system relate to the formation of complexes in the bulk. However the critical points in the profiles of strongly interacting systems are much less well understood, and an important open question is what conditions are required for the formation of a peak. Here, using a model for the surface tension developed in a previous article (Bell et al., Langmuir 23 (2007) 6042 [. 11]), we apply asymptotic techniques to show explicitly how the structure of the surface tension profile and the critical concentrations depend on the relative stability of the underlying polymer-surfactant complexes. We derive the interesting result that none of the critical concentrations is at the critical aggregation concentration (CAC). We also identify a criterion for the appearance of a peak in the surface tension profile, expressed in terms of the physical and chemical parameters of the system. © 2010 Elsevier Inc.

- MD Johnston, PK Maini, S Jonathan Chapman, CM Edwards and WF Bodmer (2010). On the proportion of cancer stem cells in a tumour. Journal of Theoretical Biology
**266**(4): 708-711. (doi:10.1016/j.jtbi.2010.07.031) (Abstract)It is now generally accepted that cancers contain a sub-population, the cancer stem cells (CSCs), which initiate and drive a tumour's growth. At least until recently it has been widely assumed that only a small proportion of the cells in a tumour are CSCs. Here we use a mathematical model, supported by experimental evidence, to show that such an assumption is unwarranted. We show that CSCs may comprise any possible proportion of the tumour, and that the higher the proportion the more aggressive the tumour is likely to be. © 2010 Elsevier Ltd.

- ACF Lewis, DM Charlotte, NS Jones and MA Porter (2010). The function of communities in protein interaction networks at multiple scales. Bmc Systems Biology
**4**: 100. (doi:10.1186/1752-0509-4-100) (Abstract)Background: If biology is modular then clusters, or communities, of proteins derived using only protein interaction network structure should define protein modules with similar biological roles. We investigate the link between biological modules and network communities in yeast and its relationship to the scale at which we probe the network.Results: Our results demonstrate that the functional homogeneity of communities depends on the scale selected, and that almost all proteins lie in a functionally homogeneous community at some scale. We judge functional homogeneity using a novel test and three independent characterizations of protein function, and find a high degree of overlap between these measures. We show that a high mean clustering coefficient of a community can be used to identify those that are functionally homogeneous. By tracing the community membership of a protein through multiple scales we demonstrate how our approach could be useful to biologists focusing on a particular protein.Conclusions: We show that there is no one scale of interest in the community structure of the yeast protein interaction network, but we can identify the range of resolution parameters that yield the most functionally coherent communities, and predict which communities are most likely to be functionally homogeneous. © 2010 Lewis et al; licensee BioMed Central Ltd.

- RJ Shipley, SL Waters and MJ Ellis (2010). Definition and validation of operating equations for poly(vinyl alcohol)-poly(lactide-co-glycolide) microfiltration membrane-scaffold bioreactors. Biotechnology and Bioengineering
**107**(2): 382-392. (doi:10.1002/bit.22815) (Abstract)The aim of this work is to provide operating data for biodegradable hollow fiber membrane bioreactors. The physicochemical cell culture environment can be controlled with the permeate flowrate, so this aim necessitates the provision of operating equations that enable end-users to set the pressures and feed flowrates to obtain their desired culture environment. In this paper, theoretical expressions for the pure water retentate and permeate flowrates, derived using lubrication theory, are compared against experimental data for a single fiber poly(vinyl alcohol)-poly(lactide-coglycolide) crossflow module to give values for the membrane permeability and slip. Analysis of the width of the boundary layer region where slip effects are important, together with the sensitivity of the retentate and permeate equations to the slip parameter, show that slip is insignificant for these membranes, which have a mean pore diameter of 1.1 μm. The experimental data is used to determine a membrane permeability, of k=1.86×10m, and to validate the model. It was concluded that the operating equation that relates the permeate to feed ratio, c, lumen inlet flowrate, Q, lumen outlet pressure, P, and ECS outlet pressure, P, is P - P = Q(Ac + B) (1) where A and B are constants that depend on the membrane permeability and geometry (and are given explicitly). Finally, two worked examples are presented to demonstrate how a tissue engineer can use Equation (1) to specify operating conditions for their bioreactor. © 2010 Wiley Periodicals, Inc.

- AL Stewart and PJ Dellar (2010). Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. Journal of Fluid Mechanics
**651**: 387-413. (doi:10.1017/S0022112009993922) - A Di Garbo, MD Johnston, PK Maini and SJ Chapman (2010). Variable renewal rate and growth properties of cell populations in colon crypts. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**81**(6): 061909. (doi:10.1103/PhysRevE.81.061909) (Abstract)A nonlinear mathematical model is used to investigate the time evolution of the cell populations in colon crypts (stem, semidifferentiated and fully differentiated cells). To mimic pathological alteration of the biochemical pathways leading to abnormal proliferative activity of the population of semidifferentiated cells their renewal rate is assumed to be dependent on the population size. Then, the effects of such perturbation on the population dynamics are investigated theoretically. Using both theoretical methods and numerical simulations it is shown that the increase in the renewal rate of semidifferentiated cells strongly impacts the dynamical behavior of the cell populations. © 2010 The American Physical Society.

- PJ Mucha, T Richardson, K Macon, MA Porter and J-P Onnela (2010). Community structure in time-dependent, multiscale, and multiplex networks. Science
**328**(5980): 876-878. (doi:10.1126/science.1184819) (Abstract)Network science is an interdisciplinary endeavor, with methods and applications drawn from across the natural, social, and information sciences. A prominent problem in network science is the algorithmic detection of tightly connected groups of nodes known as communities. We developed a generalized framework of network quality functions that allowed us to study the community structure of arbitrary multislice networks, which are combinations of individual networks coupled through links that connect each node in one network slice to itself in other slices. This framework allows studies of community structure in a general setting encompassing networks that evolve over time, have multiple types of links (multiplexity), and have multiple scales.

- RJ Whittaker, SL Waters, M Heil, J Boyle and OE Jensen (2010). The energetics of flow through a rapidly oscillating tube. Part 2. Application to an elliptical tube. Journal of Fluid Mechanics
**648**: 123-153. (doi:10.1017/S0022112009992916) (Abstract)In Part 1 of this work, we derived general asymptotic results for the three-dimensional flow field and energy fluxes for flow within a tube whose walls perform prescribed small-amplitude periodic oscillations of high frequency and large axial wavelength. In the current paper, we illustrate how these results can be applied to the case of flow through a finite-length axially non-uniform tube of elliptical cross-section-a model of flow in a Starling resistor. The results of numerical simulations for three model problems (an axially uniform tube under pressure-flux and pressure-pressure boundary conditions and an axially non-uniform tube with prescribed flux) with prescribed wall motion are compared with the theoretical predictions made in Part 1, each showing excellent agreement. When upstream and downstream pressures are prescribed, we show how the mean flux adjusts slowly under the action of Reynolds stresses using a multiple-scale analysis. We test the asymptotic expressions obtained for the mean energy transfer E from the flow to the wall over a period of the motion. In particular, the critical point at which E = 0 is predicted accurately: this point corresponds to energetically neutral oscillations, the condition which is relevant to the onset of global instability in the Starling resistor. © 2010 Cambridge University Press.

- RJ Whittaker, SL Waters, OE Jensen, J Boyle and M Heil (2010). The energetics of flow through a rapidly oscillating tube. Part 1. General theory. Journal of Fluid Mechanics
**648**: 83-121. (doi:10.1017/S0022112009992904) (Abstract)We examine the effect of prescribed wall-driven oscillations of a flexible tube of arbitrary cross-section, through which a flow is driven by prescribing either a steady flux at the downstream end or a steady pressure difference between the ends. A large-Womersley-number large-Strouhal-number regime is considered, in which the oscillations of the wall are small in amplitude, but sufficiently rapid to ensure viscous effects are confined to a thin boundary layer. We derive asymptotic expressions for the flow fields and evaluate the energy budget. A general result for the conditions under which there is zero net energy transfer from the flow to the wall is provided. This is presented as a critical inverse Strouhal number (a dimensionless measure of the background flow rate) which is expressed only in terms of the tube geometry, the fluid properties and the profile of the prescribed wall oscillations. Our results identify an essential component of a fundamental mechanism for self-excited oscillations in three-dimensional collapsible tube flows, and enable us to assess how geometric and flow properties affect the stability of the system. © 2010 Cambridge University Press.

- JM Osborne, JP Whiteley, RD O'Dea, HM Byrne and SL Waters (2010). The influence of bioreactor geometry and the mechanical environment on engineered tissues. Journal of Biomechanical Engineering
**132**(5): 051006. (doi:10.1115/1.4001160) (Abstract)A three phase model for the growth of a tissue construct within a perfusion bioreactor is examined. The cell population (and attendant extracellular matrix), culture medium, and porous scaffold are treated as distinct phases. The bioreactor system is represented by a two-dimensional channel containing a cell-seeded rigid porous scaffold (tissue construct), which is perfused with a culture medium. Through the prescription of appropriate functional forms for cell proliferation and extracellular matrix deposition rates, the model is used to compare the influence of cell density-, pressure-, and culture medium shear stress-regulated growth on the composition of the engineered tissue. The governing equations are derived in O'Dea et al. "A Three Phase Model for Tissue Construct Growth in a Perfusion Bioreactor," Math. Med. Biol., in which the long-wavelength limit was exploited to aid analysis; here, finite element methods are used to construct two-dimensional solutions to the governing equations and to investigate thoroughly their behavior. Comparison of the total tissue yield and averaged pressures, velocities, and shear stress demonstrates that quantitative agreement between the two-dimensional and long-wavelength approximation solutions is obtained for channel aspect ratios of order 10 and that much of the qualitative behavior of the model is captured in the long-wavelength limit, even for relatively large channel aspect ratios. However, we demonstrate that in order to capture accurately the effect of mechanotransduction mechanisms on tissue construct growth, spatial effects in at least two dimensions must be included due to the inherent spatial variation of mechanical stimuli relevant to perfusion bioreactors, most notably, fluid shear stress, a feature not captured in the long-wavelength limit. Copyright © 2010 by ASME.

- E Pérez-Pérez, PL Read and IM Moroz (2010). Assessing eddy parameterization schemes in a differentially heated rotating annulus experiment. Ocean Modelling
**32**(3-4): 118-131. (doi:10.1016/j.ocemod.2009.11.003) (Abstract)We investigate three of the most common hypotheses underpinning parameterizations of baroclinic eddy fluxes in the context of the differentially heated rotating annulus experiment. The investigation is carried out over a region of parameter space which embraces the onset of baroclinic instability, the regular wave regime and the onset of irregular flows, the latter of which is arguably most relevant to oceanic conditions. Through diagnostics from a 2D axisymmetric and a 3D eddy-resolving numerical model, it was found that the transport of heat by baroclinic eddies is not strictly an adiabatic process but that diffusive 'ventilation' of the flow in the thermal boundary layers is significant during the nonlinear development of the flow. Total heat transport, however, is conserved overall. Depending on the stages of flow evolution and on the region in parameter space under consideration, either heat, quasi-geostrophic potential vorticity (QGPV) or relative vorticity (QGRV) may become a suitable variable on which to parameterize baroclinic eddy fluxes in a down-gradient manner. These results raise issues for eddy parameterization schemes that rely on these assumptions in ocean and atmosphere models. © 2009 Elsevier Ltd.

- X Luo, IM Moroz and I Hoteit (2010). Scaled unscented transform Gaussian sum filter: Theory and application. Physica D: Nonlinear Phenomena
**239**(10): 684-701. (doi:10.1016/j.physd.2010.01.022) (Abstract)In this work we consider the state estimation problem in nonlinear/non-Gaussian systems. We introduce a framework, called the scaled unscented transform Gaussian sum filter (SUT-GSF), which combines two ideas: the scaled unscented Kalman filter (SUKF) based on the concept of scaled unscented transform (SUT) (Julier and Uhlmann (2004) [16]), and the Gaussian mixture model (GMM). The SUT is used to approximate the mean and covariance of a Gaussian random variable which is transformed by a nonlinear function, while the GMM is adopted to approximate the probability density function (pdf) of a random variable through a set of Gaussian distributions. With these two tools, a framework can be set up to assimilate nonlinear systems in a recursive way. Within this framework, one can treat a nonlinear stochastic system as a mixture model of a set of sub-systems, each of which takes the form of a nonlinear system driven by a known Gaussian random process. Then, for each sub-system, one applies the SUKF to estimate the mean and covariance of the underlying Gaussian random variable transformed by the nonlinear governing equations of the sub-system. Incorporating the estimations of the sub-systems into the GMM gives an explicit (approximate) form of the pdf, which can be regarded as a "complete" solution to the state estimation problem, as all of the statistical information of interest can be obtained from the explicit form of the pdf (Arulampalam et al. (2002) [7]). In applications, a potential problem of a Gaussian sum filter is that the number of Gaussian distributions may increase very rapidly. To this end, we also propose an auxiliary algorithm to conduct pdf re-approximation so that the number of Gaussian distributions can be reduced. With the auxiliary algorithm, in principle the SUT-GSF can achieve almost the same computational speed as the SUKF if the SUT-GSF is implemented in parallel. As an example, we will use the SUT-GSF to assimilate a 40-dimensional system due to Lorenz and Emanuel (1998) [27]. We will present the details of implementing the SUT-GSF and examine the effects of filter parameters on the performance of the SUT-GSF. © 2010 Elsevier B.V. All rights reserved.

- PS Stewart, OE Jensen and SL Waters (2010). Local instabilities of flow in a flexible channel: Asymmetric flutter driven by a weak critical layer. Physics of Fluids
**22**(3): 2-18. (doi:10.1063/1.3337824) (Abstract)We examine the linear stability of two-dimensional Poiseuille flow in a long channel confined by a rigid wall and a massless damped-tensioned membrane. We seek solutions that are periodic in the streamwise spatial direction and time, solving the homogeneous eigenvalue problem using a Chebyshev spectral method and asymptotic analysis. Several modes of instability are identified, including Tollmien-Schlichting (TS) waves and traveling-wave flutter (TWF). The eigenmode for neutrally stable downstream-propagating TWF in the absence of wall damping is shown to have a novel asymptotic structure at high Reynolds numbers, not reported in symmetric flexible-walled channels, involving a weak but destabilizing critical layer at the channel centerline where the wave speed is marginally greater than the maximum Poiseuille flow speed. We also show that TS instabilities along the lower branch of the neutral curve are modified remarkably little by wall compliance, but can be either stabilized or destabilized by wall damping. We discuss the energy budget underlying TWF and briefly describe the structure of other flow-induced surface instabilities. © 2010 American Institute of Physics.

- P Pathmanathan, JP Whiteley, DJ Gavaghan and S Jonathan Chapman (2010). A note on the effect of the choice of weak form on GMRES convergence for incompressible nonlinear elasticity problems. Journal of Applied Mechanics, Transactions Asme
**77**(3): 1-6. (doi:10.1115/1.4000414) (Abstract)The generalized minimal residual (GMRES) method is a common choice for solving the large nonsymmetric linear systems that arise when numerically computing solutions of incompressible nonlinear elasticity problems using the finite element method. Analytic results on the performance of GMRES are available on linear problems such as linear elasticity or Stokes' flow (where the matrices in the corresponding linear systems are symmetric), or on the nonlinear problem of the Navier-Stokes flow (where the matrix is block-symmetric/block-skew-symmetric); however, there has been very little investigation into the GMRES performance in incompressible nonlinear elasticity problems, where the nonlinearity of the incompressibility constraint means the matrix is not block-symmetric/block-skew-symmetric. In this short paper, we identify one feature of the problem formulation, which has a huge impact on unpreconditioned GMRES convergence. We explain that it is important to ensure that the matrices are perturbations of a block-skew-symmetric matrix rather than a perturbation of a block-symmetric matrix. This relates to the choice of sign before the incompressibility constraint integral in the weak formulation (with both choices being mathematically equivalent). The incorrect choice is shown to have a hugely detrimental effect on the total computation time. © 2010 by ASME.

- M Aguareles, SJ Chapman and T Witelski (2010). Motion of spiral waves in the complex Ginzburg-Landau equation. Physica D: Nonlinear Phenomena
**239**(7): 348-365. (doi:10.1016/j.physd.2009.12.003) (Abstract)Solutions of the general cubic complex Ginzburg-Landau equation comprising multiple spiral waves are considered. For parameters close to the vortex limit, and for a system of spiral waves with well-separated centres, laws of motion of the centres are found which vary depending on the order of magnitude of the separation of the centres. In particular, the direction of the interaction changes from along the line of centres to perpendicular to the line of centres as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wavenumber and frequency are determined. These depend on the positions of the centres of the spirals, and so evolve slowly as the spirals move. © 2009 Elsevier B.V.

- S Saavedra, S Powers, PJ Mucha, T Mccotter and MA Porter (2010). Mutually-antagonistic interactions in baseball networks. Physica A: Statistical Mechanics and its Applications
**389**(5): 1131-1141. (doi:10.1016/j.physa.2009.10.038) (Abstract)We formulate the head-to-head matchups between Major League Baseball pitchers and batters from 1954 to 2008 as a bipartite network of mutually-antagonistic interactions. We consider both the full network and single-season networks, which exhibit structural changes over time. We find interesting structure in the networks and examine their sensitivity to baseball's rule changes. We then study a biased random walk on the matchup networks as a simple and transparent way to (1) compare the performance of players who competed under different conditions and (2) include information about which particular players a given player has faced. We find that a player's position in the network does not correlate with his placement in the random walker ranking. However, network position does have a substantial effect on the robustness of ranking placement to changes in head-to-head matchups. © 2009 Elsevier B.V. All rights reserved.

- F Fraternali, MA Porter and C Daraio (2010). Optimal design of composite granular protectors. Mechanics of Advanced Materials and Structures
**17**(1): 1-19. (doi:10.1080/15376490802710779) (Abstract)We employ an evolutionary algorithm to investigate the optimal design of composite protectors using one-dimensional granular chains composed of beads of various sizes, masses, and stiffnesses. We define a fitness function using the maximum force transmitted from the protector to a "wall" that represents the body to be protected and accordingly optimize the topology (arrangement), size, and material of the chain. We obtain optimally randomized granular protectors characterized by high-energy equipartition and the transformation of incident waves into interacting solitary pulses. We consistently observe that the pulses traveling to the wall combine to form an extended (long-wavelength), small-amplitude pulse.

- RJ Shipley and SJ Chapman (2010). Multiscale Modelling of Fluid and Drug Transport in Vascular Tumours. Bulletin of Mathematical Biology
**72**(6): 1464-1491. (doi:10.1007/s11538-010-9504-9) - GW Jones and SJ Chapman (2010). Modelling apical constriction in epithelia using elastic shell theory. Biomechanics and Modeling in Mechanobiology
**9**(3): 247-261. (doi:10.1007/s10237-009-0174-1) (Abstract)Apical constriction is one of the fundamental mechanisms by which embryonic tissue is deformed, giving rise to the shape and form of the fully-developed organism. The mechanism involves a contraction of fibres embedded in the apical side of epithelial tissues, leading to an invagination or folding of the cell sheet. In this article the phenomenon is modelled mechanically by describing the epithelial sheet as an elastic shell, which contains a surface representing the continuous mesh formed from the embedded fibres. Allowing this mesh to contract, an enhanced shell theory is developed in which the stiffness and bending tensors of the shell are modified to include the fibres' stiffness, and in which the active effects of the contraction appear as body forces in the shell equilibrium equations. Numerical examples are presented at the end, including the bending of a plate and a cylindrical shell (modelling neurulation) and the invagination of a spherical shell (modelling simple gastrulation). © Springer-Verlag 2009.

- RE Baker, CA Yates and R Erban (2010). From microscopic to macroscopic descriptions of cell migration on growing domains. Bulletin of Mathematical Biology
**72**(3): 719-762. (doi:10.1007/s11538-009-9467-x) (Abstract)Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is an almost ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last 20 years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs. © 2009 Society for Mathematical Biology.

### 2009

- ES Thomson, LA Wilen and JS Wettlaufer (2009).
Light scattering from an isotropic layer between uniaxial crystals. Journal of Physics: Condensed Matter
**21**(19): 195407-195407. (doi:10.1088/0953-8984/21/19/195407) - I Eisenman and JS Wettlaufer (2009).
Nonlinear threshold behavior during the loss of Arctic sea ice. Proceedings of the National Academy of Sciences
**106**(1): 28-32. (doi:10.1073/pnas.0806887106) - MA Porter (2009). Experimental results related to DNLS equations. Springer Tracts in Modern Physics
**232**: 175-189. (doi:10.1007/978-3-540-89199-4_8) - C Gundlach and C Please (2009). Generic behavior of nonlinear sound waves near the surface of a star: Smooth solutions. Physical Review D - Particles, Fields, Gravitation and Cosmology
**79**(6): 067501. (doi:10.1103/PhysRevD.79.067501) (Abstract)We are interested in the generic behavior of nonlinear sound waves as they approach the surface of a star, here assumed to have the polytropic equation of state P=KρΓ. Restricting to spherical symmetry, and considering only the region near the surface, we generalize the methods of Carrier and Greenspan (1958) for the shallow water equations on a sloping beach to this problem. We give a semiquantitative criterion for a shock to form near the surface during the evolution of generic initial data with support away from the surface. We show that in smooth solutions the velocity and the square of the sound speed remain regular functions of Eulerian radius at the surface. © 2009 The American Physical Society.

- AJ Trewenack, KA Landman and CP Please (2009). A continuum model for the development of tissue-engineered cartilage around a chondrocyte. Mathematical Medicine and Biology
**26**(3): 241-262. (doi:10.1093/imammb/dqp013) (Abstract)The limited ability of cartilage to repair when damaged has led to the investigation of tissue engineering as a method for reconstructing cartilage. We propose a continuum multispecies model for the development of cartilage around a single chondrocyte. As in healthy cartilage, the model predicts a balance between synthesis, transport, binding and decay of matrix components. Two mechanisms are investigated for the transport of soluble matrix components: diffusion and advection, caused by displacement of the scaffold medium. Numerical results indicate that a parameter defined by the ratio of the flux of soluble components out of the chondrocyte and its diffusive flux determines which of these mechanisms is dominant. We investigate the diffusion-dominated and advection-dominated limiting cases using perturbation analysis. Using parameter values from the literature, our modelling results suggest that both diffusion and advection are significant mechanisms in developing cartilage. Moreover, in this parameter regime, results are particularly sensitive to parameter values. These two observations could explain differences observed experimentally between various scaffold media. Modelling results are also used to predict the minimum chondrocyte seeding density required to produce functional cartilage. © The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

- BG Sengers, ROC Oreffo, CP Please and M Taylor (2009). Experimental-computational evaluation of human bone marrow stromal cell spreading on trabecular bone structures. Annals of Biomedical Engineering
**37**(6): 1165-1176. (doi:10.1007/s10439-009-9676-3) (Abstract)The clinical application of macro-porous scaffolds for bone regeneration is significantly affected by the problem of insufficient cell colonization. Given the wide variety of different scaffold structures used for tissue engineering it is essential to derive relationships for cell colonization independent of scaffold architecture. To study cell population spreading on 3D structures decoupled from nutrient limitations, an in vitro culture system was developed consisting of thin slices of human trabecular bone seeded with Human Bone Marrow Stromal Cells, combined with dedicated μCT imaging and computational modeling of cell population spreading. Only the first phase of in vitro scaffold colonization was addressed, in which cells migrate and proliferate up to the stage when the surface of the bone is covered as a monolayer, a critical prerequisite for further tissue formation. The results confirm the model's ability to represent experimentally observed cell population spreading. The key advantage of the computational model was that by incorporating complex 3D structure, cell behavior can be characterized quantitatively in terms of intrinsic migration parameters, which could potentially be used for predictions on different macro-porous scaffolds subject to additional experimental validation. This type of modeling will prove useful in predicting cell colonization and improving strategies for skeletal tissue engineering. © 2009 Biomedical Engineering Society.

- RJ Shipley, GW Jones, RJ Dyson, BG Sengers, CL Bailey, CJ Catt, CP Please and J Malda (2009). Design criteria for a printed tissue engineering construct: A mathematical homogenization approach. Journal of Theoretical Biology
**259**(3): 489-502. (doi:10.1016/j.jtbi.2009.03.037) (Abstract)Cartilage tissue repair procedures currently under development aim to create a construct in which patient-derived cells are seeded and expanded ex vivo before implantation back into the body. The key challenge is producing physiologically realistic constructs that mimic real tissue structure and function. One option with vast potential is to print strands of material in a 3D structure called a scaffold that imitates the real tissue structure; the strands are composed of gel seeded with cells and so provide a template for cartilaginous tissue growth. The scaffold is placed in the construct and pumped with nutrient-rich culture medium to supply nutrients to the cells and remove waste products, thus promoting tissue growth. In this paper we use asymptotic homogenization to determine the effective flow and transport properties of such a printed scaffold system. These properties are used to predict the distribution of nutrient/waste products through the construct, and to specify design criteria for the scaffold that will optimize the growth of functional tissue. © 2009 Elsevier Ltd. All rights reserved.

- MJ Mcguinness, AC Fowler, WT Lee and B Scheu (2009). Exploding rocks. Aip Conference Proceedings
**1168**: 1442-1445. (doi:10.1063/1.3241365) (Abstract)This modelling is motivated by recent experimental work, in which a vertical column of rock under large pressure is suddenly depressurised, so that it explodes, in a sequence of horizontal fractures that forms from the top down. The resulting blocks are lifted off and ejected. This experiment provides a framework for understanding the way in which catastrophic explosion can occur, and is motivated by the corresponding phenomenon of magmatic explosion during Vulcanian eruptions. We summarise a theoretical model built to describe these results, and show that it is capable of describing both the primary sequence of fracturing, and the secondary intra-block fracturing. © 2009 American Institute of Physics.

- AC Fowler (2009). Dynamics of subglacial floods. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**465**(2106): 1809-1828. (doi:10.1098/rspa.2008.0488) (Abstract)The Nye model of jökulhlaups is able to explain both their periodicity and even the detailed shape of the flood hydrograph. In this paper, we show how the model can be used to predict the shape of the hydrographs from the subglacial lake Grímsvötn beneath Vatnajökull in Iceland, and we comment on three particular issues that have proved contentious in the application of the Nye model: the role of water temperature; the shape of the flood channel; and the value of the bed roughness coefficient. © 2009 The Royal Society.

- W Lee, O Power, A Fowler, S Healey and J Browne (2009). Blowing polysilicon fuses. Aip Conference Proceedings
**1168**: 1461-1464. (doi:10.1063/1.3241370) (Abstract)Polysilicon fuses are micron sized components of silicon chips. Unlike conventional fuses, they are blown as an irreversible programming step. They are used to isolate circuitry, to tune performance or to specialise the device. A fuse must be blown by applying a precise voltage which currently can only be determined by lengthly experimentation. A mathematical model of the blowing process could be used to eliminate the experimental step reducing the development time needed for new devices. We report progress towards a model describing the electronic, thermodynamic and fluid mechanical aspects of fuse blowing. © 2009 American Institute of Physics.

- AC Fowler (2009). Instability modelling of drumlin formation incorporating lee-side cavity growth. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**465**(2109): 2681-2702. (doi:10.1098/rspa.2008.0490) (Abstract)It is proposed that the formation of the subglacial bedforms known as drumlins occurs through an instability associated with the flow of ice over a wet deformable till. We pose a mathematical model that describes this instability, and we solve a simplified version of the model numerically in order to establish the form of finite-amplitude two-dimensional waveforms. A feature of the solutions is that cavities frequently form downstream of the bedforms; we allow the model to cater for this possibility and we provide an efficient numerical method to solve the resulting free boundary problem.

- JH Siggers, SL Waters, JAD Wattis and LJ Cummings (2009). Flow dynamics in a stented ureter. Mathematical Medicine and Biology
**26**(1): 1-24. - JEF Green, SL Waters, HM Byrne and KM Shakesheff (2009). A mathematical model of liver cell aggregation in vitro. Bulletin of Mathematical Biology
**71**(4): 906-930. (doi:10.1007/s11538-008-9387-1) (Abstract)The behavior of mammalian cells within three-dimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multicellular aggregates has been shown to improve cell viability and functionality compared to traditional monolayer culture techniques. We propose a simple mathematical model for the early stages of this aggregation process, when cell clusters form on the surface of the extracellular matrix (ECM) layer on which they are seeded. We focus on interactions between the cells and the viscoelastic ECM substrate. Governing equations for the cells, culture medium, and ECM are derived using the principles of mass and momentum balance. The model is then reduced to a system of four partial differential equations, which are investigated analytically and numerically. The model predicts that provided cells are seeded at a suitable density, aggregates with clearly defined boundaries and a spatially uniform cell density on the interior will form. While the mechanical properties of the ECM do not appear to have a significant effect, strong cell-ECM interactions can inhibit, or possibly prevent, the formation of aggregates. The paper concludes with a discussion of our key findings and suggestions for future work. © 2008 Society for Mathematical Biology.

- O Baumchen, R Fetzer, K Jacobs, A Munch and B Wagner (2009). Comprehensive analysis of dewetting profiles to quantify hydrodynamic slip. Iutam Bookseries
**15**: 51-65. (doi:10.1007/978-90-481-2626-2_4) (Abstract)Hydrodynamic slip of Newtonian liquids is a new phenomenon, the origin of which is not yet clarified. There are various direct and indirect techniques to measure slippage. Here we describe a method to characterize the influence of slippage on the shape of rims surrounding growing holes in thin polymer films. Atomic force microscopy is used to study the shape of the rim; by analyzing its profile and applying an appropriate lubrication model we are able to determine the slip length for polystyrene films. In the experiments we study polymer films below the entanglement length that dewet from hydrophobized (silanized) surfaces. We show that the slip length at the solid/liquid interface increases with increasing viscosity. The correlation between viscosity and slip length is dependent on the type of silanization. This indicates a link between the molecular mechanism of the interaction of polymer chains and silane molecules under flow conditions that we will discuss in detail. © Springer Science + Business Media B.V. 2009.

- JR King, A Münch and BA Wagner (2009). Linear stability analysis of a sharp-interface model for dewetting thin films. Journal of Engineering Mathematics
**63**(2-4): 177-195. (doi:10.1007/s10665-008-9242-2) (Abstract)The stability of the receding front at the growing rim of a thin liquid film dewetting from a substrate is studied. The underlying forces that drive the dewetting motion are given by the intermolecular potential between the liquid film and the substrate. The role of slippage in the emerging instability is studied via a sharp-interface model for the dewetting thin film, which is derived from the lubrication model via matched asymptotic expansions. Using the separation of the time-scale for the slow growth of the rim and the time-scale on which the rim destabilises, the sharp-interface results are compared to earlier results for the lubrication model and good agreement for the unstable modes is obtained. The main advantage of the sharp-interface model is that it allows for the derivation of traveling solutions for the base state and subsequently a systematic linear stability analysis via normal modes. Interestingly, unlike the dispersion relations that are typically encountered for the well-known finger-instability in thin-film flows, where the dependence of the growth rate on the wave number is quadratic, here it is linear. © Springer Science+Business Media B.V. 2008.

- R Vandiver and A Goriely (2009). Morpho-elastodynamics: the long-time dynamics of elastic growth.. Journal of biological dynamics
**3**(2-3): 180-195. (Abstract)As elastic tissues grow and remodel, they generate stresses that influence their mechanical states which influence the way growth proceeds. This complex feedback leads to a dynamic evolution of growth and stresses which can be modelled within the theory of exact nonlinear elasticity. Here, we first review the different theoretical results considering growth laws and then, we present a new approach to look at morphoelasticity as a continuous dynamical process. These evolution laws lead to new dynamical systems that can be studied by the classical methods of dynamical systems theory.

- AC Fowler and IJ Hewitt (2009). Melt channelization in ascending mantle. Journal of Geophysical Research B: Solid Earth
**114**(6): B06210. (doi:10.1029/2008JB006185) (Abstract)We study a model for channels of magma flow within mantle undergoing decompression melting. Cylindrical conduits in a viscous, porous, compacting matrix are considered, and it is found that the dynamics of the conduit walls are governed by the competition between melting (caused by decompression) and viscous closure (caused by the reduced pressure in the conduit). There are many similarities with the Röthlisberger channels which transport melt water beneath glaciers. Pressure in these mantle conduits is very nearly magmastatic, and ascent velocities on the order of 100 m a are predicted. Flow from the surrounding porous partially molten matrix into the low-pressure channel is considered and can supply a continual source of melt. The accumulation region is on the order of the compaction length, and the residual matrix is reduced to very low melt fractions, typically <0.5%. Channels form naturally from porous flow in the matrix because of the enhanced melting rate in regions of higher porosity, which have a larger heat flux from below. The vast majority of melt could be expected to flow eventually into one of these channels, which therefore offer a physical grounding for the conceptual near-fractional melting models used to explain field observations. Copyright 2009 by the American Geophysical Union.

- A Goriely, S Neukirch and A Hausrath (2009). Polyhelices through n points. International Journal of Bioinformatics Research and Applications
**5**(2): 118-132. (doi:10.1504/IJBRA.2009.024032) (Abstract)A polyhelix is continuous space curve with continuous Frenet frame that consists of a sequence of connected helical segments. The main result of this paper is that given n points in space, there exist infinitely many polyhelices passing through these points. These curves are by construction continuous with continuous derivatives and are completely specified by 3n numbers, i.e., the initial position, the signed curvature, torsion, and length of each helical segment. Polyhelices can be parametrised by the arc length and easily expressed in terms of product of matrices. Copyright © 2009 Inderscience Enterprises Ltd.

- A Goriely and J Rose (2009). LINEAR AND NONLINEAR FRONT SELECTION FOR REACTION-DIFFUSION EQUATIONS. Biomat 2008 : 29-51. (doi:10.1142/9789814271820_0002)
- R Vandiver and A Goriely (2009). Differential growth and residual stress in cylindrical elastic structures. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
**367**(1902): 3607-3630. (doi:10.1098/rsta.2009.0114) (Abstract)Cylindrical forms are among one of Nature's fundamental building blocks. They serve many different purposes, from sustaining body weight to carrying flows. Their mechanical properties are generated through the often complex arrangements of the walls. In particular, in many structures that have elastic responses, such as stems and arteries, the walls are in a state of tension generated by differential growth. Here, the effect of differential growth and residual stress on the overall mechanical response of the cylindrical structure is studied within the framework of morpho-elasticity © 2009 The Royal Society.

- B Duričković, A Goriely and G Saccomandi (2009). Compact waves on planar elastic rods. International Journal of Non-Linear Mechanics
**44**(5): 538-544. (doi:10.1016/j.ijnonlinmec.2008.10.007) (Abstract)Planar Kirchhoff elastic rods with non-linear constitutive relations are shown to admit traveling wave solutions with compact support. The existence of planar compact waves is a general property of all non-linearly elastic intrinsically straight rods, while intrinsically curved rods do not exhibit this type of behavior. © 2008 Elsevier Ltd. All rights reserved.

- AC Hausrath and A Goriely (2009). The Fourier transforms of curves and filaments and their application to low-resolution protein crystallography. Journal of Applied Crystallography
**42**(2): 268-278. (doi:10.1107/S0021889809002544) (Abstract)A numerical method for computing the Fourier transform of an arbitrary space curve is described. The method is applicable to all sufficiently smooth curves and relies on the local geometric parameters describing a curve. The numerical results for a helical curve are compared with the exact analytical theory for the transform of a helix. It is shown that the transform of a filamentary density distribution radially symmetric around a curve is equivalent to the transform of that curve scaled by an appropriately defined weight function. These filamentary density distributions in conjunction with the numerical transform evaluation method can be used for simulating low-resolution diffraction data for protein crystals. Crystallographic structure factors obtained from a filament model representing a simple three-helix-bundle protein are compared with those calculated from conventional coordinate models. At low resolution, the filamentary representation provides an excellent approximation of the structure factors obtained from the standard coordinate model, but requires far fewer independent parameters. © International Union of Crystallography 2009.

- JA Neufeld, D Vella and HE Huppert (2009). The effect of a fissure on storage in a porous medium. Journal of Fluid Mechanics
**639**: 239-259. (doi:10.1017/S0022112009991030) (Abstract)We consider the two-dimensional buoyancy driven flow of a fluid injected into a saturated semi-infinite porous medium bounded by a horizontal barrier in which a single line sink, representing a fissure some distance from the point of injection, allows leakage of buoyant fluid. Our studies are motivated by the geological sequestration of carbon dioxide (CO) and the possibility that fissures in the cap rock may compromise the safe long-term storage of CO. A theoretical model is presented that accounts for leakage through the fissure using two parameters, which characterize leakage driven both by the hydrostatic pressure within the overriding fluid and by the buoyancy of the fluid within the fissure. We determine numerical solutions for the evolution of both the gravity current within the porous medium and the volume of fluid that has escaped through the fissure as a function of time. A quantity of considerable practical interest is the efficiency of storage, which we define as the amount of fluid remaining in the porous medium relative to the amount injected. This efficiency scales like t at late times, indicating that the efficiency of storage ultimately tends to zero. We confirm the results of our model by comparison with an analogue laboratory experiment and discuss the implications of our two-dimensional model of leakage from a fissure for the geological sequestration of CO. © 2009 Cambridge University Press.

- D Vella, A Boudaoud and M Adda-Bedia (2009). Statics and inertial dynamics of a ruck in a rug. Physical Review Letters
**103**(17): 174301. (doi:10.1103/PhysRevLett.103.174301) (Abstract)We consider the familiar problem of a bump, or ruck, in a rug. Under lateral compression, a rug bends to form a ruck-a localized region in which it is no longer in contact with the floor. We show that when the external force that created the ruck is removed, the ruck flattens out unless the initial compression is greater than a critical value, which we determine. We also study the inertial motion of a ruck that is generated when one end of the rug is moved rapidly. We show that the equations of motion admit a traveling ruck solution for which a linear combination of the tension and kinetic energy is determined by the ruck size. We confirm these findings experimentally. We end by discussing the potential implications of our work for the analogous propagation of localized slip pulses in the sliding of two bodies in contact. © 2009 The American Physical Society.

- D Vella, J Bico, A Boudaoud, B Roman and PM Reis (2009). The macroscopic delamination of thin films from elastic substrates. Proceedings of the National Academy of Sciences of the United States of America
**106**(27): 10901-10906. (doi:10.1073/pnas.0902160106) - P Cicuta and D Vella (2009). Granular character of particle rafts. Physical Review Letters
**102**(13): 138302. (doi:10.1103/PhysRevLett.102.138302) (Abstract)We consider a monolayer of particles floating at a horizontal liquid-gas interface-a particle raft. Upon compressing the monolayer in a Langmuir trough, the particles at first pack but ultimately the monolayer buckles out of the plane. We measure the stress profile within the raft at the onset of buckling and show for the first time that such systems exhibit a Janssen effect: the stress decays exponentially away from the compressing barriers over a length scale that depends on the width of the trough. We find quantitative agreement between the rate of decay and the simple theory presented by Janssen and others. This demonstrates that particle rafts have a granular, as well as elastic, character, which is neglected by current models. Finally, we suggest that our experimental setup may be suitable for exploring granular effects in two dimensions without the complications of gravity and basal friction. © 2009 The American Physical Society.

- SD Howison and A Cartea (2009). Option pricing with Levy-stable processes generated by Levy-stable integrated variance. Quantitative Finance
**9**(4): 397-409. - SD Howison, B Hambly and T Kluge (2009). Modelling spikes and pricing swing options in electricity markets. Quantitative Finance
**9**(8): 937-949. - SD Howison and M Coulon (2009). Stochastic behavior of the electricity bid stack: from fundamental drivers to power prices. J Energy Markets
**2**: . - CJW Breward and CS Bohun (2009). Yolk Dynamics in Amphibian Embryos. Maths in Industry Case Studies (1): 1-21.
- CJW Breward and CS Bohun (2009). Yolk Dynamics in Amphibian Embryos. Maths in Industry Case Studies (1): 1-21.
- O Martínez-Alvarado, L Montabone, SR Lewis, IM Moroz and PL Read (2009). Transient teleconnection event at the onset of a planet-encircling dust storm on Mars. Annales Geophysicae
**27**(9): 3663-3676. (doi:10.5194/angeo-27-3663-2009) (Abstract)We use proper orthogonal decomposition (POD) to study a transient teleconnection event at the onset of the 2001 planet-encircling dust storm on Mars, in terms of empirical orthogonal functions (EOFs). There are several differences between this and previous studies of atmospheric events using EOFs. First, instead of using a single variable such as surface pressure or geopotential height on a given pressure surface, we use a dataset describing the evolution in time of global and fully three-dimensional atmospheric fields such as horizontal velocity and temperature. These fields are produced by assimilating Thermal Emission Spectrometer observations from NASA's Mars Global Surveyor spacecraft into a Mars general circulation model. We use total atmospheric energy (TE) as a physically meaningful quantity which weights the state variables. Second, instead of adopting the EOFs to define teleconnection patterns as planetary-scale correlations that explain a large portion of long time-scale variability, we use EOFs to understand transient processes due to localised heating perturbations that have implications for the atmospheric circulation over distant regions. The localised perturbation is given by anomalous heating due to the enhanced presence of dust around the northern edge of the Hellas Planitia basin on Mars. We show that the localised disturbance is seemingly restricted to a small number (a few tens) of EOFs. These can be classified as low-order, transitional, or high-order EOFs according to the TE amount they explain throughout the event. Despite the global character of the EOFs, they show the capability of accounting for the localised effects of the perturbation via the presence of specific centres of action. We finally discuss possible applications for the study of terrestrial phenomena with similar characteristics.

- C Xue, HG Othmer and R Erban (2009). From individual to collective behavior of unicellular organisms: Recent results and open problems. Aip Conference Proceedings
**1167**: 3-14. (doi:10.1063/1.3246413) (Abstract)The collective movements of unicellular organisms such as bacteria or amoeboid (crawling) cells are often modeled by partial differential equations (PDEs) that describe the time evolution of cell density. In particular, chemotaxis equations have been used to model the movement towards various kinds of extracellular cues. Well-developed analytical and numerical methods for analyzing the time-dependent and time-independent properties of solutions make this approach attractive. However, these models are often based on phenomenological descriptions of cell fluxes with no direct correspondence to individual cell processes such signal transduction and cell movement. This leads to the question of how to justify these macroscopic PDEs from microscopic descriptions of cells, and how to relate the macroscopic quantities in these PDEs to individual-level parameters. Here we summarize recent progress on this question in the context of bacterial and amoeboid chemotaxis, and formulate several open problems. © 2009 American Institute of Physics.

- MA Porter, J-P Onnela and PJ Mucha (2009). Communities in networks. Notices of the American Mathematical Society
**56**(9): 1082-1097. - MA Porter, NJ Zabusky, B Hu and DK Campbell (2009). Fermi, Pasta, Ulam and the birth of experimental mathematics. American Scientist
**97**(3): 214-221. (doi:10.1511/2009.78.214) (Abstract)Enrico Fermi, John Pasta, and Stanislaw Ulam's discovery of numerical experiment of nonlinear problem continues to inspire researchers. The Los Alamos scientists call this problem as MANIAC (MAthematical Numerical Integrator And Computer) and performs brute force numerical computations. Fermi, Pasta, Ulam's (FPU) original investigation can be described as the birth of experimental mathematics, which led to computer based investigations designed to give insights into complex mathematical and physical problems. Studies such as atomic lattices of solid state physics are highly inspired by FPU problem of nonlinear systems. FPU problems have also led to the creation of concepts like chaos, solitons and breather's invention, development, refinement and application to a number of real world systems. It also reveals facts on a remarkably broad range of topics in nonlinear dynamics, statistical mechanics and computational physics.

- DJ Fenn, SD Howison, M Mcdonald, S Williams and NF Johnson (2009). The mirage of triangular arbitrage in the spot foreign exchange market. International Journal of Theoretical and Applied Finance
**12**(8): 1105-1123. (doi:10.1142/S0219024909005609) (Abstract)We investigate triangular arbitrage within the spot foreign exchange market using high-frequency executable prices. We show that triangular arbitrage opportunities do exist, but that most have short durations and small magnitudes. We find intra-day variations in the number and length of arbitrage opportunities, with larger numbers of opportunities with shorter mean durations occurring during more liquid hours. We demonstrate further that the number of arbitrage opportunities has decreased in recent years, implying a corresponding increase in pricing efficiency. Using trading simulations, we show that a trader would need to beat other market participants to an unfeasibly large proportion of arbitrage prices to profit from triangular arbitrage over a prolonged period of time. Our results suggest that the foreign exchange market is internally self-consistent and provide a limited verification of market efficiency. © 2009 World Scientific Publishing Company.

- SJ Chapman (2009). Multiscale mathematical modelling in medicine and biology. 18th World Imacs Congress and Modsim09 International Congress on Modelling and Simulation : 13-22.
- AL Traud, C Frost, PJ Mucha and MA Porter (2009). Visualization of communities in networks. Chaos
**19**(4): 041104. (doi:10.1063/1.3194108) - RD O'Dea, SL Waters and HM Byrne (2009). A multiphase model for tissue construct growth in a perfusion bioreactor. Mathematical Medicine and Biology
**27**(2): 95-127. (doi:10.1093/imammb/dqp003) (Abstract)The growth of a cell population within a rigid porous scaffold in a perfusion bioreactor is studied, using a three-phase continuum model of the type presented by Lemon et al. (2006, Multiphase modelling of tissue growth using the theory of mixtures. J. Math. Biol., 52, 571-594) to represent the cell population (and attendant extracellular matrix), culture medium and porous scaffold. The bioreactor system is modelled as a 2D channel containing the cell-seeded rigid porous scaffold (tissue construct) which is perfused with culture medium. The study concentrates on (i) the cell-cell and cell-scaffold interactions and (ii) the impact of mechanotransduction mechanisms on construct composition. A numerical and analytical analysis of the model equations is presented and, depending upon the relative importance of cell aggregation and repulsion, markedly different cell movement is revealed. Additionally, mechanotransduction effects due to cell density, pressure and shear stress-mediated tissue growth are shown to generate qualitative differences in the composition of the resulting construct. The results of our simulations indicate that this model formulation (in conjunction with appropriate experimental data) has the potential to provide a means of identifying the dominant regulatory stimuli in a cell population. © The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

- IM Griffiths and PD Howell (2009). The surface-tension-driven retraction of a viscida. SIAM Journal on Applied Mathematics
**70**(5): 1453-1487. (doi:10.1137/090746318) (Abstract)We consider the surface-tension-driven evolution of a thin two-dimensional sheet of viscous fluid whose ends are held a fixed distance apart. We find that the evolution is governed by a nonlocal nonlinear partial differential equation, which may be transformed, via a suitable change of time variable, to a simple linear equation. This possesses an interesting dispersion relation which indicates that it is well posed whether solved forwards or backwards in time, enabling us to determine which initial shapes will evolve to a given shape at a later time. We demonstrate that our model may be used to describe the global evolution of a viscida containing small regions of high curvature, and proceed to investigate the evolution of a profile which contains a corner. We show that the corner is not smoothed out but persists for forward and inverse time. The introduction of a pressure differential across the free surfaces is shown to provide a method of controlling the shape evolution. © by SIAM.

- LJ Cummings, NBE Sawyer, SP Morgan, FRAJ Rose and SL Waters (2009). Tracking large solid constructs suspended in a rotating bioreactor: A combined experimental and theoretical study. Biotechnology and Bioengineering
**104**(6): 1224-1234. (doi:10.1002/bit.22490) (Abstract)We present a combined experimental and theoretical study of the trajectory of a large solid cylindrical disc suspended within a fluid-filled rotating cylindrical vessel. The experimental set-up is relevant to tissue-engineering applications where a disc-shaped porous scaffold is seeded with cells to be cultured, placed within a bioreactor filled with nutrient-rich culture medium, which is then rotated in a vertical plane to keep the growing tissue construct suspended in a state of "free fall." The experimental results are compared with theoretical predictions based on the model of Cummings and Waters (2007), who showed that the suspended disc executes a periodic motion. For anticlockwise vessel rotation three regimes were identified: (i) disc remains suspended at a fixed position on the right-hand side of the bioreactor; (ii) disc executes a periodic oscillatory motion on the right-hand side of the bioreactor; and (iii) disc orbits the bioreactor. All three regimes are captured experimentally, and good agreement between theory and experiment is obtained. For the tissue engineering application, computation of the fluid dynamics allows the nutrient concentration field surrounding a tissue construct (a property that cannot be measured experimentally) to be determined (Cummings and Waters, 2007). The implications for experimental cell-culture protocols are discussed. © 2009 Wiley Periodicals, Inc.

- G Theocharis, PG Kevrekidis, M Kavousanakis, IG Kevrekidis, C Daraio and MA Porter (2009). Localized breathing modes in granular crystals with defects. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**80**(6): 066601. (doi:10.1103/PhysRevE.80.066601) (Abstract)We study localized modes in uniform one-dimensional chains of tightly packed and uniaxially compressed elastic beads in the presence of one or two light-mass impurities. For chains composed of beads of the same type, the intrinsic nonlinearity, which is caused by the Hertzian interaction of the beads, appears not to support localized, breathing modes. Consequently, the inclusion of light-mass impurities is crucial for their appearance. By analyzing the problem's linear limit, we identify the system's eigenfrequencies and the linear defect modes. Using continuation techniques, we find the solutions that bifurcate from their linear counterparts and study their linear stability in detail. We observe that the nonlinearity leads to a frequency dependence in the amplitude of the oscillations, a static mutual displacement of the parts of the chain separated by a defect, and for chains with two defects that are not in contact, it induces symmetry-breaking bifurcations. © 2009 The American Physical Society.

- J Pitt-Francis, P Pathmanathan, MO Bernabeu, R Bordas, J Cooper, JM Osborne, B Rodríguez, JP Whiteley, DJ Gavaghan, AG Fletcher, P Murray, A Walter, SJ Chapman, PK Maini, SL Waters, GR Mirams, A Garny, IMM van Leeuwen and HM Byrne (2009). Chaste: A test-driven approach to software development for biological modelling. Computer Physics Communications
**180**(12): 2452-2471. (doi:10.1016/j.cpc.2009.07.019) (Abstract)Chaste ('Cancer, heart and soft-tissue environment') is a software library and a set of test suites for computational simulations in the domain of biology. Current functionality has arisen from modelling in the fields of cancer, cardiac physiology and soft-tissue mechanics. It is released under the LGPL 2.1 licence. Chaste has been developed using agile programming methods. The project began in 2005 when it was reasoned that the modelling of a variety of physiological phenomena required both a generic mathematical modelling framework, and a generic computational/simulation framework. The Chaste project evolved from the Integrative Biology (IB) e-Science Project, an inter-institutional project aimed at developing a suitable IT infrastructure to support physiome-level computational modelling, with a primary focus on cardiac and cancer modelling. Program summary: Program title: Chaste. Catalogue identifier: AEFD_v1_0. Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEFD_v1_0.html. Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland. Licensing provisions: LGPL 2.1. No. of lines in distributed program, including test data, etc.: 5 407 321. No. of bytes in distributed program, including test data, etc.: 42 004 554. Distribution format: tar.gz. Programming language: C++. Operating system: Unix. Has the code been vectorised or parallelized?: Yes. Parallelized using MPI. RAM:< 90 Megabytes for two of the scenarios described in Section 6 of the manuscript (Monodomain re-entry on a slab or Cylindrical crypt simulation). Up to 16 Gigabytes (distributed across processors) for full resolution bidomain cardiac simulation. Classification: 3. External routines: Boost, CodeSynthesis XSD, CxxTest, HDF5, METIS, MPI, PETSc, Triangle, Xerces. Nature of problem: Chaste may be used for solving coupled ODE and PDE systems arising from modelling biological systems. Use of Chaste in two application areas are described in this paper: cardiac electrophysiology and intestinal crypt dynamics. Solution method: Coupled multi-physics with PDE, ODE and discrete mechanics simulation. Running time: The largest cardiac simulation described in the manuscript takes about 6 hours to run on a single 3 GHz core. See results section (Section 6) of the manuscript for discussion on parallel scaling. © 2009 Elsevier B.V. All rights reserved.

- PS Stewart, OE Jensen and SL Waters (2009). Local and global instabilities of flow in a flexible-walled channel. European Journal of Mechanics, B/Fluids
**28**(4): 541-557. (doi:10.1016/j.euromechflu.2009.03.002) (Abstract)We consider laminar high-Reynolds-number flow through a long finite-length planar channel, where a segment of one wall is replaced by a massless membrane held under longitudinal tension. The flow is driven by a fixed pressure difference across the channel and is described using an integral form of the unsteady boundary-layer equations. The basic flow state, for which the channel has uniform width, exhibits static and oscillatory global instabilities, having distinct modal forms. In contrast, the corresponding local problem (neglecting boundary conditions associated with the rigid parts of the system) is found to be convectively, but not absolutely, unstable to small-amplitude disturbances in the absence of wall damping. We show how amplification of the primary global oscillatory instability can arise entirely from wave reflections with the rigid parts of the system, involving interacting travelling-wave flutter and static-divergence modes that are convectively stable; alteration of the mean flow by oscillations makes the onset of this primary instability subcritical. We also show how distinct mechanisms of energy transfer differentiate the primary global mode from other modes of oscillatory instability. © 2009 Elsevier Masson SAS. All rights reserved.

- X Luo and IM Moroz (2009). Ensemble Kalman filter with the unscented transform. Physica D: Nonlinear Phenomena
**238**(5): 549-562. (doi:10.1016/j.physd.2008.12.003) (Abstract)A modification scheme to the ensemble Kalman filter (EnKF) is introduced based on the concept of the unscented transform [S. Julier, J. Uhlmann, H. Durrant-Whyte, A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE Trans. Automat. Control. 45 (2000) 477-482; S.J. Julier, J.K. Uhlmann, Unscented filtering and nonlinear estimation, Proc. IEEE 92 (2004) 401-422], which therefore will be called the ensemble unscented Kalman filter (EnUKF) in this work. When the error distribution of the analysis is symmetric (not necessarily Gaussian), it can be shown that, compared with the ordinary EnKF, the EnUKF has more accurate estimations of the ensemble mean and covariance of the background by examining the multidimensional Taylor series expansion term by term. This implies that, the EnUKF may have better performance in state estimation than the ordinary EnKF in the sense that the deviations from the true states are smaller. For verification, some numerical experiments are conducted on a 40-dimensional system due to Lorenz and Emanuel [E.N. Lorenz, K.A. Emanuel, Optimal sites for supplementary weather observations: Simulation with a small model, J. Atmos. Sci. 55 (1998) 399-414]. Simulation results support our argument. © 2008 Elsevier B.V. All rights reserved.

- GW Jones, SJ Chapman and DJ Allwright (2009). Asymptotic analysis of a buckling problem for an embedded spherical shell. SIAM Journal on Applied Mathematics
**70**(3): 901-922. (doi:10.1137/080735114) (Abstract)The axisymmetric buckling of a spherical shell embedded in an elastic medium with uniaxial compression at infinity is examined in the limit of small shell thickness ratio. An asymptotic method is developed by considering the paradigm problem of a beam attached to a Winkler substrate of variable stiffness, which in the small aspect ratio limit displays the same behavior as the shell. The asymptotic method is then applied to the Euler-Lagrange equations corresponding to shell buckling. The system is analyzed in two distinguished limits, displaying good agreement with the full numerical results. © 2009 Society for Industrial and Applied Mathematics.

- RJ Dyson, CJW Breward, PD Howell and J Brander (2009). Long-wavelength stability of an unsupported multilayer liquid film falling under gravity. Journal of Engineering Mathematics
**64**(3): 237-250. (doi:10.1007/s10665-009-9278-y) (Abstract)The long-wavelength stability of an unsupported multilayer liquid film falling under the effects of gravity and surface tension is investigated. By considering the Navier-Stokes equations for two fluid layers in the high-Reynolds-number and small-aspect-ratio limits the steady-state solutions are obtained. The stability criterion found by Lin (J Fluid Mech 104:111-118, 1981) for a one-layer fluid curtain is generalised to the two-layer case and the criterion for an n-layer curtain is established. © Springer Science+Business Media B.V. 2009.

- IMM Van Leeuwen, GR Mirams, A Walter, SJ Young, OE Jensen, JR King, HM Byrne, A Fletcher, P Murray, SJ Chapman, PK Maini, SL Waters, J Osborne, J Cooper, J Pitt-Francis, L Momtahan, P Pathmanathan, JP Whiteley, DJ Gavaghan, S Varma and B Doyle (2009). An integrative computational model for intestinal tissue renewal. Cell Proliferation
**42**(5): 617-636. (doi:10.1111/j.1365-2184.2009.00627.x) (Abstract)Objectives: The luminal surface of the gut is lined with a monolayer of epithelial cells that acts as a nutrient absorptive engine and protective barrier. To maintain its integrity and functionality, the epithelium is renewed every few days. Theoretical models are powerful tools that can be used to test hypotheses concerning the regulation of this renewal process, to investigate how its dysfunction can lead to loss of homeostasis and neoplasia, and to identify potential therapeutic interventions. Here we propose a new multiscale model for crypt dynamics that links phenomena occurring at the subcellular, cellular and tissue levels of organisation. Methods: At the subcellular level, deterministic models characterise molecular networks, such as cell-cycle control and Wnt signalling. The output of these models determines the behaviour of each epithelial cell in response to intra-, inter- and extracellular cues. The modular nature of the model enables us to easily modify individual assumptions and analyse their effects on the system as a whole. Results: We perform virtual microdissection and labelling-index experiments, evaluate the impact of various model extensions, obtain new insight into clonal expansion in the crypt, and compare our predictions with recent mitochondrial DNA mutation data. Conclusions: We demonstrate that relaxing the assumption that stem-cell positions are fixed enables clonal expansion and niche succession to occur. We also predict that the presence of extracellular factors near the base of the crypt alone suffices to explain the observed spatial variation in nuclear beta-catenin levels along the crypt axis. © 2009 Blackwell Publishing Ltd.

- O Martinez-Alvarado, L Montabone, SR Lewis, IM Moroz and PL Read (2009). Transient teleconnection event at the onset of a planet-encircling dust storm on Mars. Annales Geophysicae
**27**(9): 3663-3676. - P Pathmanathan, J Cooper, J Osborne, J Pitt-Francis, A Fletcher, P Murray, SJ Chapman, G Mirams and A Walter (2009). A computational study of discrete mechanical tissue models. Physical Biology
**6**(3): 036001. (doi:10.1088/1478-3975/6/3/036001) (Abstract)A computational study of discrete 'cell-centre' approaches to modelling the evolution of a collection of cells is undertaken. The study focuses on the mechanical aspects of the tissue, in order to separate the passive mechanical response of the model from active effects such as cell-growth and cell division. Issues which arise when implementing these models are described, and a series of numerical mechanical experiments is performed. It is shown that discrete tissues modelled this way typically exhibit elastic-plastic behaviour under slow compression, and act as a brittle linear elastic solid under slow tension. Both overlapping spheres and Voronoi-tessellation-based models are examined, and the effect of different cell-cell interaction force laws on the bulk mechanical properties of the tissue is determined. This correspondence allows parameters in the cell-based model to be chosen to be compatible with bulk tissue measurements. © 2009 IOP Publishing Ltd.

- SJ Chapman and G Kozyreff (2009). Exponential asymptotics of localised patterns and snaking bifurcation diagrams. Physica D-NONLINEAR Phenomena
**238**(3): 319-354. (doi:10.1016/j.physd.2008.10.005) - P Pathmanathan, DJ Gavaghan and SJ Chapman (2009). Inverse membrane problems in elasticity. Quarterly Journal of Mechanics and Applied Mathematics
**62**(1): 67-88. (doi:10.1093/qjmam/hbn026) (Abstract)The inverse elasticity problem of determining the undeformed, deflated, configuration of a nonlinear elastic membrane, given the deformed configuration enclosing an incompressible fluid under known pressure, is considered. It is shown that, in practical cases, it is enough to determine only the undeformed metric tensor, and it is also shown how the two- and three-dimensional cases are fundamentally different. For the three-dimensional case, we set up and classify the partial differential equations to be solved, prove existence of an undeformed state given an undeformed metric and study the axisymmetric case in detail. © The author 2009. Published by Oxford University Press; all rights reserved.

- DJ Fenn, MA Porter, NF Johnson, NS Jones, M Mcdonald and S Williams (2009). Dynamic communities in multichannel data: An application to the foreign exchange market during the 2007-2008 credit crisis. Chaos
**19**(3): 033119. (doi:10.1063/1.3184538) (Abstract)We study the cluster dynamics of multichannel (multivariate) time series by representing their correlations as time-dependent networks and investigating the evolution of network communities. We employ a node-centric approach that allows us to track the effects of the community evolution on the functional roles of individual nodes without having to track entire communities. As an example, we consider a foreign exchange market network in which each node represents an exchange rate and each edge represents a time-dependent correlation between the rates. We study the period 2005-2008, which includes the recent credit and liquidity crisis. Using community detection, we find that exchange rates that are strongly attached to their community are persistently grouped with the same set of rates, whereas exchange rates that are important for the transfer of information tend to be positioned on the edges of communities. Our analysis successfully uncovers major trading changes that occurred in the market during the credit crisis. © 2009 American Institute of Physics.

- G Lemon, JR King, D Howard, MJ Tomlinson, LD Buttery, FRAJ Rose and SL Waters (2009). Mathematical modelling of tissue-engineered angiogenesis. Mathematical Biosciences
**221**(2): 101-120. (doi:10.1016/j.mbs.2009.07.003) (Abstract)We present a mathematical model for the vascularisation of a porous scaffold following implantation in vivo. The model is given as a set of coupled non-linear ordinary differential equations (ODEs) which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of scaffold vascularisation changes as a function of the parameter values. For example, it is shown how the loss of seeded cells arising from slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the scaffold with vascular cells. Using certain assumptions it is shown how the system can be simplified to one which is partially tractable and for which some analysis is given. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) assay. © 2009 Elsevier Inc. All rights reserved.

- A Singer, R Erban, IG Kevrekidis and RR Coifman (2009). Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps. Proceedings of the National Academy of Sciences of the United States of America
**106**(38): 16090-16095. (doi:10.1073/pnas.0905547106) (Abstract)Nonlinear independent component analysis is combined with diffusion-map data analysis techniques to detect good observables in high-dimensional dynamic data. These detections are achieved by integrating local principal component analysis of simulation bursts by using eigenvectors of a Markov matrix describing anisotropic diffusion. The widely applicable procedure, a crucial step in model reduction approaches, is illustrated on stochastic chemical reaction network simulations.

- O Martínez-Alvarado, IM Moroz, PL Read, SR Lewis and L Montabone (2009). Low-order dynamical behavior in the martian atmosphere: Diagnosis of general circulation model results. Icarus
**204**(1): 48-62. (doi:10.1016/j.icarus.2009.06.010) (Abstract)The hypothesis of a low dimensional martian climate attractor is investigated by the application of the proper orthogonal decomposition (POD) to a simulation of martian atmospheric circulation using the UK Mars general circulation model (UK-MGCM). In this article we focus on a time series of the interval between autumn and winter in the northern hemisphere, when baroclinic activity is intense. The POD is a statistical technique that allows the attribution of total energy (TE) to particular structures embedded in the UK-MGCM time-evolving circulation. These structures are called empirical orthogonal functions (EOFs). Ordering the EOFs according to their associated energy content, we were able to determine the necessary number to account for a chosen amount of atmospheric TE. We show that for Mars a large fraction of TE is explained by just a few EOFs (with 90% TE in 23 EOFs), which apparently support the initial hypothesis. We also show that the resulting EOFs represent classical types of atmospheric motion, such as thermal tides and transient waves. Thus, POD is shown to be an efficient method for the identification of different classes of atmospheric modes. It also provides insight into the non-linear interaction of these modes. © 2009 Elsevier Inc. All rights reserved.

- T Richardson, PJ Mucha and MA Porter (2009). Spectral tripartitioning of networks. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**80**(3): 036111. (doi:10.1103/PhysRevE.80.036111) (Abstract)We formulate a spectral graph-partitioning algorithm that uses the two leading eigenvectors of the matrix corresponding to a selected quality function to split a network into three communities in a single step. In so doing, we extend the recursive bipartitioning methods developed by Newman to allow one to consider the best available two-way and three-way divisions at each recursive step. We illustrate the method using simple "bucket brigade" examples and then apply the algorithm to examine the community structures of the coauthorship graph of network scientists and of U. S. Congressional networks inferred from roll call voting similarities. © 2009 The American Physical Society.

- AC Fowler, O Clary and T Roose (2009). A dynamic model of annual foliage growth and carbon uptake in trees. Journal of the Royal Society Interface
**6**(40): 1087-1096. (doi:10.1098/rsif.2009.0010) (Abstract)The growth of trees and other plants occurs through the interactive combination of photosynthesis and carbon (and other nutrient) assimilation. Photosynthesis enables the production of carbohydrate that can then be used in growing foliage, whereby photosynthesis is enabled. We construct a mathematical model of carbon uptake and storage, which allows the prediction of the growth dynamics of trees. We find that the simplest model allows uncontrolled foliage production through the positive feedback outlined above, but that leaf shading provides an automatic saturation to carbon assimilation, and hence to foliage production. The model explains the necessity for finite leaf area production at outbreak, and it explains why foliage density reaches a constant value during a growing season, while also non-leaf tissue also continues to grow. It also explains why trees will die when their carbon stores are depleted below a certain threshold, because the cost of foliage growth and maintenance exceeds the dynamic supply of carbon by photosynthesis. © 2009 The Royal Society.

- LR Band, DS Riley, PC Matthews, JM Oliver and SL Waters (2009). Annular thin-film flows driven by azimuthal variations in interfacial tension. Quarterly Journal of Mechanics and Applied Mathematics
**62**(4): 403-430. (doi:10.1093/qjmam/hbp015) (Abstract)We consider a thin viscous film that lines a rigid cylindrical tube and surrounds a core of inviscid fluid, and we model the flow that is driven by a prescribed azimuthally varying tension at the core-film interface, with dimensional form σm*-a* cos(nθ) (where constants n ∈ and σ*m, a* ∈ ). Neglecting axial variations, we seek steady two-dimensional solutions with the full symmetries of the evolution equation. For a* = 0 (constant interfacial tension), the fully symmetric steady solution is neutrally stable and there is a continuum of steady solutions, whereas for a* ≠ 0 and n = 2, 3, 4,..., the fully symmetric steady solution is linearly unstable. For n = 2 and n = 3, we analyse the weakly nonlinear stability of the fully symmetric steady solution, assuming that 0 < εa*/σm* ≪ 1(where ε denotes the ratio between the typical film thickness and the tube radius); for n = 3, this analysis leads us to additional linearly unstable steady solutions. Solving the full nonlinear system numerically, we confirm the stability analysis and furthermore find that for a* gt 0 and n = 1, 2, 3, hellip, the film can evolve towards a steady solution featuring a drained region. We investigate the draining dynamics using matched asymptotic methods.

- MA Porter, C Daraio, I Szelengowicz, EB Herbold and PG Kevrekidis (2009). Highly nonlinear solitary waves in heterogeneous periodic granular media. Physica D: Nonlinear Phenomena
**238**(6): 666-676. (doi:10.1016/j.physd.2008.12.010) (Abstract)We use experiments, numerical simulations, and theoretical analysis to investigate the propagation of highly nonlinear solitary waves in periodic arrangements of dimer (two-mass) and trimer (three-mass) cell structures in one-dimensional granular lattices. To vary the composition of the fundamental periodic units in the granular chains, we utilize beads of different materials (stainless steel, brass, glass, nylon, polytetrafluoroethylene, and rubber). This selection allows us to tailor the response of the system based on the masses, Poisson ratios, and elastic moduli of the components. For example, we examine dimer configurations with two types of heavy particles, two types of light particles, and alternating light and heavy particles. Employing a model with Hertzian interactions between adjacent beads, we find good agreement between experiments and numerical simulations. We also find good agreement between these results and a theoretical analysis of the model in the long-wavelength regime that we derive for heterogeneous environments (dimer chains) and general bead interactions. Our analysis encompasses previously-studied examples as special cases and also provides key insights on the influence of heterogeneous lattices on the properties (width and propagation speed) of the nonlinear wave solutions of this system. © 2009 Elsevier B.V. All rights reserved.

- R Erban, SJ Chapman, IG Kevrekidis and T Vejchodský (2009). Analysis of a stochastic chemical system close to a SNIPER bifurcation of its mean-field model. SIAM Journal on Applied Mathematics
**70**(3): 984-1016. (doi:10.1137/080731360) (Abstract)A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs, for example, in the modeling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) are studied. Our approach is based on the chemical Fokker-Planck equation. To gain some insight into the advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, and then the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size. © 2009 Society for Industrial and Applied Mathematics.

- R Carretero-González, D Khatri, C Daraio, MA Porter and PG Kevrekidis (2009). Dissipative solitary waves in granular crystals. Physical Review Letters
**102**(2): 024102. (doi:10.1103/PhysRevLett.102.024102) (Abstract)We provide a quantitative characterization of dissipative effects in one-dimensional granular crystals. We use the propagation of highly nonlinear solitary waves as a diagnostic tool and develop optimization schemes that allow one to compute the relevant exponents and prefactors of the dissipative terms in the equations of motion. We thereby propose a quantitatively accurate extension of the Hertzian model that encompasses dissipative effects via a discrete Laplacian of the velocities. Experiments and computations with steel, brass, and polytetrafluoroethylene reveal a common dissipation exponent with a material-dependent prefactor. © 2009 The American Physical Society.

- R Erban and SJ Chapman (2009). Stochastic modelling of reaction-diffusion processes: Algorithms for bimolecular reactions. Physical Biology
**6**(4): 046001. (doi:10.1088/1478-3975/6/4/046001) (Abstract)Several stochastic simulation algorithms (SSAs) have recently been proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. In both cases, it is shown that the commonly used implementation of bimolecular reactions (i.e. the reactions of the form A + B → C or A + A → C) might lead to incorrect results. Improvements of both SSAs are suggested which overcome the difficulties highlighted. In particular, a formula is presented for the smallest possible compartment size (lattice spacing) which can be correctly implemented in the first model. This implementation uses a new formula for the rate of bimolecular reactions per compartment (lattice site). © 2009 IOP Publishing Ltd.

- PJ Dellar (2009). Moment equations for magnetohydrodynamics. Journal of Statistical Mechanics-THEORY and Experiment
**2009**: P06003. (doi:10.1088/1742-5468/2009/06/P06003) - S Wirkus and MA Porter (2009). Comment on "Bifurcation analysis of parametrically excited bipolar disorder model". Communications in Nonlinear Science and Numerical Simulation
**14**(6): 2844-2844. (doi:10.1016/j.cnsns.2008.08.014) - RE Voskoboinikov, SJ Chapman, JB Mcleod and JR Ockendon (2009). Asymptotics of edge dislocation pile-up against a bimetallic interface. Mathematics and Mechanics of Solids
**14**(1-2): 284-295. (doi:10.1177/1081286508092616) (Abstract)The approach developed in preceding papers is extended to derive the equilibrium positions of n edge dislocations in a linear pile-up stressed by a constant applied loading against an interface in a bimetallic solid. As n → ∞, the dislocations in the inner region are located with sufficient accuracy that the stress distribution at the interface can be evaluated by a simple computational procedure. Such a prediction is impossible using a conventional continuum dislocation theory. © 2009 SAGE Publications.

- LR Band, JAD Wattis, LJ Cummings and SL Waters (2009). Modelling crystal aggregation and deposition in the catheterised lower urinary tract. Journal of Mathematical Biology
**59**(6): 809-840. (doi:10.1007/s00285-009-0253-z) (Abstract)Urethral catheters often become encrusted with crystals of magnesium struvite and calcium phosphate. The encrustation can block the catheter, which can cause urine retention in the bladder and reflux into the kidneys. We develop a mathematical model to investigate crystal deposition on the catheter surface, modelling the bladder as a reservoir of fluid and the urethral catheter as a rigid channel. At a constant rate, fluid containing crystal particles of unit size enters the reservoir, and flows from the reservoir through the channel and out of the system. The crystal particles aggregate, which we model using Becker-Döring coagulation theory, and are advected through the channel, where they continue to aggregate and are deposited on the channel's walls. Inhibitor particles also enter the reservoir, and can bind to the crystals, preventing further aggregation and deposition. The crystal concentrations are spatially homogeneous in the reservoir, whereas the channel concentrations vary spatially as a result of advection, diffusion and deposition. We investigate the effect of inhibitor particles on the amount of deposition. For all parameter values, we find that crystals deposit along the full length of the channel, with maximum deposition close to the channel's entrance. © The Author(s) 2009.

- D Daugherty, T Roque-Urrea, J Urrea-Roque, J Troyer, S Wirkus and MA Porter (2009). Mathematical models of bipolar disorder. Communications in Nonlinear Science and Numerical Simulation
**14**(7): 2897-2908. (doi:10.1016/j.cnsns.2008.10.027) (Abstract)We use limit cycle oscillators to model bipolar II disorder, which is characterized by alternating hypomanic and depressive episodes and afflicts about 1% of the United States adult population. We consider two non-linear oscillator models of a single bipolar patient. In both frameworks, we begin with an untreated individual and examine the mathematical effects and resulting biological consequences of treatment. We also briefly consider the dynamics of interacting bipolar II individuals using weakly-coupled, weakly-damped harmonic oscillators. We discuss how the proposed models can be used as a framework for refined models that incorporate additional biological data. We conclude with a discussion of possible generalizations of our work, as there are several biologically-motivated extensions that can be readily incorporated into the series of models presented here. © 2008 Elsevier B.V. All rights reserved.

- RJ Whittaker, R Booth, SL Waters, R Dyson, H Woollard, C Bailey, L Parsons Chini, S Naire, S Payvandi, Z Rong, LJ Cummings, L Mawasse, JB Chaudhuri, MJ Ellis, V Michael, NJ Kuiper and S Cartmell (2009). Mathematical modelling of fibre-enhanced perfusion inside a tissue-engineering bioreactor. Journal of Theoretical Biology
**256**(4): 533-546. (doi:10.1016/j.jtbi.2008.10.013) (Abstract)We develop a simple mathematical model for forced flow of culture medium through a porous scaffold in a tissue-engineering bioreactor. Porous-walled hollow fibres penetrate the scaffold and act as additional sources of culture medium. The model, based on Darcy's law, is used to examine the nutrient and shear-stress distributions throughout the scaffold. We consider several configurations of fibres and inlet and outlet pipes. Compared with a numerical solution of the full Navier-Stokes equations within the complex scaffold geometry, the modelling approach is cheap, and does not require knowledge of the detailed microstructure of the particular scaffold being used. The potential of this approach is demonstrated through quantification of the effect the additional flow from the fibres has on the nutrient and shear-stress distribution. © 2008 Elsevier Ltd. All rights reserved.

- G Kozyreff, P Assemat and SJ Chapman (2009). Influence of Boundaries on Localized Patterns. Physical Review Letters
**103**(16): 164501. (doi:10.1103/PhysRevLett.103.164501) - CA Yates, R Erban, PK Maini, C Escudero, ID Couzin, J Buhl, IG Kevrekidis and DJT Sumpter (2009). Inherent noise can facilitate coherence in collective swarm motion. Proceedings of the National Academy of Sciences of the United States of America
**106**(14): 5464-5469. (doi:10.1073/pnas.0811195106) (Abstract)Among the most striking aspects of the movement of many animal groups are their sudden coherent changes in direction. Recent observations of locusts and starlings have shown that this directional switching is an intrinsic property of their motion. Similar direction switches are seen in self-propelled particle and other models of group motion. Comprehending the factors that determine such switches is key to understanding the movement of these groups. Here, we adopt a coarse-grained approach to the study of directional switching in a self-propelled particle model assuming an underlying one-dimensional Fokker-Planck equation for the mean velocity of the particles. We continue with this assumption in analyzing experimental data on locusts and use a similar systematic Fokker-Planck equation coefficient estimation approach to extract the relevant information for the assumed Fokker-Planck equation underlying that experimental data. In the experiment itself the motion of groups of 5 to 100 locust nymphs was investigated in a homogeneous laboratory environment, helping us to establish the intrinsic dynamics of locust marching bands. We determine the mean time between direction switches as a function of group density for the experimental data and the self-propelled particle model. This systematic approach allows us to identify key differences between the experimental data and the model, revealing that individual locusts appear to increase the randomness of their movements in response to a loss of alignment by the group. We giveaquantitative descriptionofhow locusts use noisetomaintain swarm alignment. We discuss further how properties of individual animal behavior, inferred by using the Fokker-Planck equation coefficient estimation approach, can be implemented in the selfpropelled particle model to replicate qualitatively the group level dynamics seen in the experimental data.

### 2008

- JEROMEA Neufeld and JS Wettlaufer (2008).
Shear-enhanced convection in a mushy layer. Journal of Fluid Mechanics
**612**: 339-361. (doi:10.1017/S0022112008002991) - BG Sengers, CP Please, M Taylor and ROC Oreffo (2008). A computational model relating 2D cell spreading to 3D scaffold colonization for skeletal tissue regeneration. Calcified Tissue International
**83**(1): 14-14. - M Penny, T Farrell and C Please (2008). A mathematical model for interfacial charge transfer at the semiconductor-dye-electrolyte interface of a dye-sensitised solar cell. Solar Energy Materials and Solar Cells
**92**(1): 11-23. (doi:10.1016/j.solmat.2007.07.013) (Abstract)A mathematical model for the interfacial charge transfer within dye-sensitised solar cells (DSC) is presented for the semiconductor-dye-electrolyte interface. The model explicitly accounts for each reaction at the interface involving dye molecules, electrolyte species and adsorbed electrons associated with the conduction band surface states of the semiconductor. Additionally, the model accounts for photoelectron injection via singlet and triplet excited dye states. The governing equations can be used to describe the total current produced by the DSC under illuminated and non-illuminated conditions, at steady state. Regular perturbation methods are applied to the model equations to obtain closed form analytic approximations, resulting in approximate solutions that negate the need for numerical solution of the model system. All parameter values associated with the model are obtained from the literature and from experimental data. The presented numerical results and analytic approximations compare favourably to experimental data, capturing the interfacial characteristics of current versus voltage curves of the DSC. © 2007 Elsevier B.V. All rights reserved.

- MB Jones, GR Fulford, DLS Mcelwain, CP Please and MJ Collins (2008). Elastohydrodynamics of the eyelid wiper. Bulletin of Mathematical Biology
**70**(2): 323-343. (doi:10.1007/s11538-007-9252-7) (Abstract)This paper presents an elastohydrodynamic model of the human eyelid wiper. Standard lubrication theory is applied to the fluid layer between the eyelid wiper and ocular surface. The role of the lubrication film is to reduce the shear stresses by preventing solid to solid contact between the eyelid wiper and ocular surface. For the lubrication film to be effective, it is required that the orientation of the eyelid wiper changes between the opening and closing phases of a blink. In order to model this, the hydrodynamic model is coupled with an elastic mattress model for the soft tissue of the eyelid wiper and ocular surface. This leads to a one-dimensional non-linear partial differential equation governing the fluid pressure in the lubrication film. In order to solve the differential equation, a loading condition or constraint equation must be specified. The resulting system is then solved numerically. The model allows predictions of the tear film flux from under the upper eyelid, as well as normal and shear stresses acting on the ocular surface. These factors are important in relation to dry eye syndrome, deformation of the cornea and contact lens design. It is found that the pressure and shear stress under the eyelid act across a length of approximately 0.1 mm which is consistent with clinical observations. It order to achieve a flow of tears from under the upper eyelid during a blink, the model requires that the normal force the eyelid applies to the ocular surface during the closing phase of the blink is significantly higher than during the opening phase of the blink. © 2007 Society for Mathematical Biology.

- D Friedrich, T Melvin and C Please (2008). Optimisation of analyte transport in integrated microfluidic affinity sensors for the quantification of low levels of analyte. Sensors and Actuators, B: Chemical
**131**(1): 323-332. (doi:10.1016/j.snb.2007.11.034) (Abstract)New designs for microfluidic channels to be integrated with small-scale affinity sensors for analytical applications are provided. Theoretical approaches demonstrate efficient and uniform mass transfer of the analyte from the bulk flow to small-scale affinity sensors in the base of fluidic channels by (i) active control of the analyte flow speed over the affinity sensor, (ii) non-rectangular channel geometries and (iii) non-uniform distributions of recognition binding sites over the active area of the sensor. The methodology reported provides generic strategies that can be exploited for small-scale sensors in single or multiplex formats. © 2007 Elsevier B.V. All rights reserved.

- MJ Tindall, CP Please and MJ Peddie (2008). Modelling the formation of necrotic regions in avascular tumours. Mathematical Biosciences
**211**(1): 34-55. (doi:10.1016/j.mbs.2007.09.002) (Abstract)The mechanisms underlying the formation of necrotic regions within avascular tumours are not well understood. In this paper, we examine the relative roles of nutrient deprivation and of cell death, from both the proliferating phase of the cell cycle via apoptosis and from the quiescent phase via necrosis, in changing the structure within multicellular tumour spheroids and particularly the accumulation of dead cell material in the centre. A mathematical model is presented and studied that accounts for nutrient diffusion, changes in cell cycling rates, the two different routes to cell death as well as active motion of cells and passive motion of the dead cell material. In studying the accumulation of dead cell matter we do not distinguish between the route by which each was formed. The resulting mathematical model is examined for a number of scenarios. Results show that in many cases the size of the necrotic core is closely correlated with low levels in nutrient concentration. However, in certain cases, particularly where the rate of necrosis is large, the resulting necrotic core can lead to regions of non-negligible nutrient concentration-dependent upon the mode of cell death. © 2007 Elsevier Inc. All rights reserved.

- BD Macarthur, ROC Oreffo and CP Please (2008). Stochasticity and the molecular mechanisms of induced pluripotency. Plos One
**3**(8): e3086. (doi:10.1371/journal.pone.0003086) (Abstract)The generation of induced pluripotent stem cells from adult somatic cells by ectopic expression of key transcription factors holds significant medical promise. However, current techniques for inducing pluripotency rely on viral infection and are therefore not, at present, viable within a clinical setting. Thus, there is now a need to better understand the molecular basis of stem cell pluripotency and lineage specification in order to investigate alternative methods to induce pluripotency for clinical application. However, the complexity of the underlying molecular circuitry makes this a conceptually difficult task. In order to address these issues, we considered a computational model of transcriptional control of cell fate specification. The model comprises two mutually interacting sub-circuits: a central pluripotency circuit consisting of interactions between stem-cell specific transcription factors OCT4, SOX2 and NANOG coupled to a differentiation circuit consisting of interactions between lineage-specifying master genes. The molecular switches which arise from feedback loops within these circuits give rise to a well-defined sequence of successive gene restrictions corresponding to a controlled differentiation cascade in response to environmental stimuli. Furthermore, we found that this differentiation cascade is strongly unidirectional: once silenced, core transcription factors cannot easily be reactivated. In the context of induced pluripotency, this indicates that differentiated cells are robustly resistant to reprogramming to a more primitive state. However, our model suggests that under certain circumstances, amplification of low-level fluctuations in transcriptional status (transcriptional "noise") may be sufficient to trigger reactivation of the core pluripotency switch and reprogramming to a pluripotent state. This interpretation offers an explanation of a number of experimental observations concerning the molecular mechanisms of cellular reprogramming by defined factors and suggests a role for stochasticity in reprogramming of somatic cells to pluripotency. © 2008 MacArthur et al.

- ML Mather, JA Crowe, RJ Thomas, SL Waters and HM Byrne (2008). Mathematical investigation of population variability in neural stem cells. European Cells and Materials
**16**(SUPPL. 3): 81-81. - D Howard, MJ Tomlinson, LD Buttery, KM Shakesheff, FRAJ Rose, G Lemon, JR King, H Yang, R Ronaldo and SL Waters (2008). The use of rapid prototype pore model arrays to assess tissue ingress in the CAM assay. European Cells and Materials
**16**(SUPPL. 3): 62-62. - JI Díaz, AC Fowler, AI Muñoz and E Schiavi (2008). Mathematical analysis of a model of river channel formation. Pure and Applied Geophysics
**165**(8): 1663-1682. (doi:10.1007/s00024-004-0394-3) (Abstract)The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin. © Birkhaueser 2008.

- NBE Sawyer, LK Worrall, JA Crowe, SL Waters, KM Shakesheff, FRAJ Rose and SP Morgan (2008). In situ monitoring of 3D in vitro cell aggregation using an optical imaging system. Biotechnology and Bioengineering
**100**: 159-167. - JW Fenner, DR Hose, P Lawford, KM Mccormack, D Pinney, B Brook, G Clapworthy, PV Coveney, V Feipel, S Van Sint Jan, H Gregersen, P Kohl, SR Thomas, S Waters and M Viceconti (2008). The EuroPhysiome, STEP and a roadmap for the virtual physiological human. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
**366**(1878): 2979-2999. (doi:10.1098/rsta.2008.0089) (Abstract)Biomedical science and its allied disciplines are entering a new era in which computational methods and technologies are poised to play a prevalent role in supporting collaborative investigation of the human body. Within Europe, this has its focus in the virtual physiological human (VPH), which is an evolving entity that has emerged from the EuroPhysiome initiative and the strategy for the EuroPhysiome (STEP) consortium. The VPH is intended to be a solution to common infrastructure needs for physiome projects across the globe, providing a unifying architecture that facilitates integration and prediction, ultimately creating a framework capable of describing Homo sapiens in silico. The routine reliance of the biomedical industry, biomedical research and clinical practice on information technology (IT) highlights the importance of a tailor-made and robust IT infrastructure, but numerous challenges need to be addressed if the VPH is to become a mature technological reality. Appropriate investment will reap considerable rewards, since it is anticipated that the VPH will influence all sectors of society, with implications predominantly for improved healthcare, improved competitiveness in industry and greater understanding of (patho)physiological processes. This paper considers issues pertinent to the development of the VPH, highlighted by the work of the STEP consortium. © 2008 The Royal Society.

- LA Guerreiro-Lucas, SR Pop, MJC Machado, YL Ma, SL Waters, G Richardson, K Saetzler, OE Jensen and CA Mitchell (2008). Experimental and theoretical modelling of blind-ended vessels within a developing angiogenic plexus.. Microvascular Research
**76**(3): 161-168. - M Rauscher, R Blossey, A Münch and B Wagner (2008). Spinodal dewetting of thin films with large interfacial slip: Implications from the dispersion relation. Langmuir
**24**(21): 12290-12294. (doi:10.1021/la802260b) (Abstract)We compare the dispersion relations for spinodally dewetting thin liquid films for increasing magnitude of interfacial slip length in the lubrication limit. While the shape of the dispersion relation, in particular the position of the maximum, are equal for no-slip up to moderate-slip lengths, the position of the maximum shifts to much larger wavelengths for large slip lengths. Here, we discuss the implications of this fact for recently developed methods to assess the disjoining pressure in spinodally unstable thin films by measuring the shape of the roughness power spectrum. For polystyrene (PS) films on octadecyltrichlorosilane (OTS) covered Si wafers (with slip length b ≈ 1 μ), we predict a 20% shift of the position of the maximum of the power spectrum which should be detectable in experiments. © 2008 American Chemical Society.

- K Afanasiev, A Muench and B Wagner (2008). Thin film dynamics on a vertically rotating disk partially immersed in a liquid bath. Applied Mathematical Modelling
**32**(9): 1894-1911. (doi:10.1016/j.apm.2007.06.020) - A Münch and B Wagner (2008). Galerkin method for feedback controlled Rayleigh-Bénard convection. Nonlinearity
**21**(11): 2625-2651. (doi:10.1088/0951-7715/21/11/008) (Abstract)The problem of feedback controlled Rayleigh-Bénard convection is considered. For this problem with the simple flow structure in the vertical direction, a Galerkin method that uses only a few basis functions in this direction is presented. This approximation yields considerable simplification of the problem, explicitly incorporates the non-classical boundary conditions at the horizontal boundaries of the fluid layer resulting from feedback control and reduces the dimension of the original problem by one. This method is in spirit very similar to lubrication theory, where the simple laminar flow in the vertical direction is integrated out across the height of the fluid layer. Using a minimal set of appropriate basis functions to capture the nonlinear behaviour of the flow, we investigate the effects of feedback control on amplitude, wavelength and selection of patterns via weakly nonlinear analysis and numerical simulations of the resulting dimension-reduced problems in two and three dimensions. In the second part of this study we discuss the derivation of the appropriate basis functions and prove convergence of the Galerkin scheme. © 2008 IOP Publishing Ltd and London Mathematical Society.

- MD Korzec, B Wagner, PL Evans and A Munch (2008). Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard-type equations. SIAM Journal on Applied Mathematics
**69**(2): 348-374. (doi:10.1137/070710949) (Abstract)New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard-type equation that arises as a model for epitaxially growing nanostructures, such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially nonmonotone solutions in the limit of small driving force strength, which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert W function. Using phase-space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Milliard equation, also known as the convective Cahn-Milliard equation. © 2008 Society for Industrial and Applied Mathematics.

- S Neukirch and A Goriely (2008). La chiralité des protéines fibreuses : une affaire d’élasticité. Reflets de la physique (9): 11-13. (doi:10.1051/refdp:2008008)
- M Robinson, AC Fowler, SBG O'Brien and AJ Alexander (2008). Waves in guinness. Physics of Fluids
**20**(6): 067101. (doi:10.1063/1.2929369) (Abstract)We describe a simple model of a bubbly two-phase flow which is able to explain why waves propagate downward when a pint of Guinness is poured, and also how the waves are generated. Our theory involves a physically based regularization of the basic equations of the two-phase flow, using interphasic pressure difference and virtual mass terms, together with bulk or eddy viscosity terms. We show that waves can occur through an instability analogous to that which forms roll waves in inclined fluid flows, and we provide a description of the form of these waves, and compare them to observations. Our theory provides a platform for the description of waves in more general bubbly two-phase flows, and the way in which the flow breaks down to form slug flow. © 2008 American Institute of Physics.

- IJ Hewitt and AC Fowler (2008). Partial melting in an upwelling mantle column. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**464**(2097): 2467-2491. (doi:10.1098/rspa.2008.0045) (Abstract)Decompression melting of hot upwelling rock in the mantle creates a region of partial melt comprising a porous solid matrix through which magma rises buoyantly. Magma transport and the compensating matrix deformation are commonly described by two-phase compaction models, but melt production is less often incorporated. Melting is driven by the necessity to maintain thermodynamic equilibrium between mineral grains in the partial melt; the position and amount of partial melting that occur are thus thermodynamically determined. We present a consistent model for the ascent of a one-dimensional column of rock and provide solutions that reveal where and how much partial melting occurs, the positions of the boundaries of the partial melt being determined by conserving energy across them. Thermodynamic equilibrium of the boundary between partial melt and the solid lithosphere requires a boundary condition on the effective pressure (solid pressure minus melt pressure), which suggests that large effective stresses, and hence fracture, are likely to occur near the base of the lithosphere. Matrix compaction, melt separation and temperature in the partially molten region are all dependent on the effective pressure, a fact that can lead to interesting oscillatory boundary-layer structures. © 2008 The Royal Society.

- IJ Hewitt and AC Fowler (2008). Seasonal waves on glaciers. Hydrological Processes
**22**(19): 3919-3930. (doi:10.1002/hyp.7029) (Abstract)Seasonal waves accompanying annual changes in the sliding velocity of ice travel down glacier at speeds much faster than the ice itself. A simple explanation for these waves in terms of the passage of a pressure wave through the subglacial drainage system is given. Drainage by both distributed and localized systems is explored, with the sliding velocity governed by a dependence on the effective pressure. Waves are caused by drainage through a slow distributed system, but may be damped if this is well connected to an efficient channelized system. A possible connection between these waves and high velocity spring events is discussed. Copyright © 2008 John Wiley & Sons, Ltd.

- A Goriely and M Tabor (2008). Mathematical modeling of hyphal tip growth. Fungal Biology Reviews
**22**(2): 77-83. (doi:10.1016/j.fbr.2008.05.001) (Abstract)The mathematical modelling of growing filamentous cells has been approached in a variety of ways ranging from simple geometric to biomechanically based models using exact, nonlinear, elasticity theory for shells and membranes in which a growth mechanism is included, and alternative approaches using visco-plasticity theory. We describe how the nonlinear elastic model is able to capture essential biomechanical mechanical features of the growth of a broad array of filamentous cells including fungi, actinomycetes, pollen tubes, and root hairs. A comparison between this approach and visco-plasticity based models is made. © 2008 The British Mycological Society.

- A Goriely, R Vandiver and M Destrade (2008). Nonlinear Euler buckling. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
**464**(2099): 3003-3019. (doi:10.1098/rspa.2008.0184) (Abstract)The buckling of hyperelastic incompressible cylindrical tubes of arbitrary length and thickness under compressive axial load is considered within the framework of nonlinear elasticity. Analytical and numerical methods for bifurcation are developed using the exact solution of Wilkes for the linearized problem within the Stroh formalism. Using these methods, the range of validity of the Euler buckling formula and its first nonlinear corrections are obtained for third-order elasticity. The values of the geometric parameters (tube thickness and slenderness) where a transition between buckling and barrelling is observed are also identified. © 2008 The Royal Society.

- S Neukirch, A Goriely and AC Hausrath (2008). Elastic coiled-coils act as energy buffers in the ATP synthase. International Journal of Non-Linear Mechanics
**43**(10): 1064-1073. (doi:10.1016/j.ijnonlinmec.2008.06.008) (Abstract)In the ATP synthase, transmission of energy from the membrane-embedded F0 sector to the catalytic F1 sector is accomplished by two stalks composed of coiled-coils. The great efficiency of the enzyme, despite the presence of a symmetry mismatch between the F1 and F0 sectors, suggests the involvement of elastic elements that store energy during the catalytic cycle. Here, the stalk subunits γ and b are investigated as the source of this elastic compliance using a continuum mechanical model of coiled-coils and energy arguments. The analysis shows that the compliance of both subunits is required for efficient energy transmission between F0 and F1. In addition, the predicted mechanical properties of coiled-coils in the ATP synthase suggest mechanisms whereby regulatory subunits influence the enzyme activity. © 2008 Elsevier Ltd. All rights reserved.

- A Goriely, A Hausrath and S Neukirch (2008). THE DIFFERENTIAL GEOMETRY OF PROTEINS AND ITS APPLICATIONS TO STRUCTURE DETERMINATION. Biomat 2007 : 89-118. (doi:10.1142/9789812812339_0004)
- S Neukirch, A Goriely and AC Hausrath (2008). Chirality of coiled coils: Elasticity matters. Physical Review Letters
**100**(3): 038105. (doi:10.1103/PhysRevLett.100.038105) (Abstract)Coiled coils are important protein-protein interaction motifs with high specificity that are used to assemble macromolecular complexes. Their simple geometric organization, consisting of α helices wrapped around each other, confers remarkable mechanical properties. A geometrical and mechanical continuous model taking into account sequence effects and based on the superhelical winding of the constituent helices is introduced, and a continuous family of solutions in which the oligomerization interactions are satisfied is derived. From these solutions, geometric and structural properties, such as the chirality and pitch of the coiled coil and the location of residues, are obtained. The theoretical predictions are compared to x-ray data from the leucine zipper motif. © 2008 The American Physical Society.

- R Vandiver and A Goriely (2008). Tissue tension and axial growth of cylindrical structures in plants and elastic tissues. Epl
**84**(5): 58004. (doi:10.1209/0295-5075/84/58004) - J Chopin, D Vella and A Boudaoud (2008). The liquid blister test. Proceedings of the Royal Society A-MATHEMATICAL Physical and Engineering Sciences
**464**(2099): 2887-2906. (doi:10.1098/rspa.2008.0095) - D Vella and JS Wettlaufer (2008). Explaining the patterns formed by ice floe interactions. Journal of Geophysical Research-OCEANS
**113**(C11): C11011. (doi:10.1029/2008JC004781) - M Adda-Bedia, E Katzav and D Vella (2008). Solution of the Percus-Yevick equation for hard hyperspheres in even dimensions. Journal of Chemical Physics
**129**(14): 144506. (doi:10.1063/1.2991338) (Abstract)We solve the Percus-Yevick equation in even dimensions by reducing it to a set of simple integrodifferential equations. This work generalizes an approach we developed previously for hard disks. We numerically obtain both the pair correlation function and the virial coefficients for a fluid of hyperspheres in dimensions d = 4, 6, and 8, and find good agreement with the available exact results and Monte Carlo simulations. This paper confirms the alternating character of the virial series for d≥6 and provides the first evidence for an alternating character for d = 4. Moreover, we show that this sign alternation is due to the existence of a branch point on the negative real axis. It is this branch point that determines the radius of convergence of the virial series, whose value we determine explicitly for d = 4, 6, 8. Our results complement, and are consistent with, a recent study in odd dimensions [R. D. Rohrmann, J. Chem. Phys. 129, 014510 (2008)]. © 2008 American Institute of Physics.

- D Vella (2008). Floating objects with finite resistance to bending. Langmuir
**24**(16): 8701-8706. (doi:10.1021/la800245k) (Abstract)We consider the equilibrium flotation of a thin, flexible cylinder at the interface between a liquid and a gas. In particular, we determine the maximum load that such a cylinder can support without sinking. We find that as the length of such a cylinder increases the maximum load at first increases. However, the maximum load reaches a plateau when the length of the cylinder is comparable to the elastocapillary length, which is determined by a balance between the bending of the cylinder and surface tension. We then consider the implications of our analysis for the walking on water of both arthropods and man-made robots. In particular, we show that the legs of water striders are typically slightly shorter than this 'optimal' length, suggesting that elastocapillary effects may act as a selection pressure. © 2008 American Chemical Society.

- M Adda-Bedia, E Katzav and D Vella (2008). Solution of the Percus-Yevick equation for hard disks. Journal of Chemical Physics
**128**(18): 184508. (doi:10.1063/1.2919123) (Abstract)The authors solve the Percus-Yevick equation in two dimensions by reducing it to a set of simple integral equations. They numerically obtain both the pair correlation function and the equation of state for a hard disk fluid and find good agreement with available Monte Carlo results. The present method of resolution may be generalized to any even dimension. © 2008 American Institute of Physics.

- T Vejchodsky and R Erban (2008). DETERMINISTIC AND STOCHASTIC MODELS OF DYNAMICS OF CHEMICAL SYSTEMS. Programs and Algorithms of Numerical Mathematics 14 : 220-225.
- IM Moroz (2008). Template analysis of the hide, skeldon, acheson dynamo. Lecture Notes in Earth Sciences
**112**: 293-309. (doi:10.1007/978-3-540-78938-3_13) (Abstract)Self-exciting dynamos are nonlinear electro-mechanical engineering devices, or naturally-occurring magnetohydrodynamic fluid systems that convert mechanical energy into magnetic energy without the help of permanent magnets. Hide et al. [1] introduced a nonlinear system of three coupled ordinary differential equations to model a self-exciting Faraday disk homopolar dynamo. Since only a small selection of possible behaviours, including two examples of chaotic behaviour, was investigated by them, Moroz [2]performed a more extensive analysis of the dynamo model, including producing bifurcation transition diagrams and generating unstable periodic orbits for the two chaotic examples. We now extend that analysis and use ideas from topology [3] and results from a corresponding analysis of the Lorenz attractor to identify a possible template for the HSA dynamo. ©2008 Springer-Verlag Berlin Heidelberg.

- ML Mather, JA Crowe, SP Morgan, LJ White, KM Shakesheff, SM Howdle, RJ Thomas, DJ Williams, HM Byrne and SL Waters (2008). Remedi: A research consortium applying engineering strategies to establish regenerative medicine as a new industry. Ifmbe Proceedings
**22**: 2209-2212. (doi:10.1007/978-3-540-89208-3_528) (Abstract)Regenerative medicine is a growing field that aims to develop therapies that restore, maintain or improve tissue function. Although a number of regenerative therapies are biologically effective the vast majority of them are lacking in commercial viability. In order to translate the laboratory based biological success of regenerative medicines to large scale production capable, economic and scaleable approaches to manufacturing need to be implemented. To this end the remedi consortium are applying engineering strategies to assist in establishing regenerative medicine on a commercial footing. This paper details the application of quality engineering approaches to two stages in the manufacture of regenerative therapies: tissue scaffold fabrication and cell culture. The utility of the engineering strategies implemented is demonstrated. © 2009 Springer Berlin Heidelberg.

- CM Edwards, SD Howison, H Ockendon and JR Ockendon (2008). Non-classical shallow water flows. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
**73**(1): 137-157. (doi:10.1093/imamat/hxm064) (Abstract)This paper deals with violent discontinuities in shallow water flows with large Froude number F. On a horizontal base, the paradigm problem is that of the impact of two fluid layers in situations where the flow can be modelled as two smooth regions joined by a singularity in the flow field. Within the framework of shallow water theory, we show that, over a certain time-scale, this discontinuity may be described by a delta shock, which is a weak solution of the underlying conservation laws in which the depth and mass and momentum fluxes have both delta function and step function components. We also make some conjectures about how this model evolves from the traditional model for jet impacts in which a spout is emitted. For flows on a sloping base, we show that for flow with an aspect ratio of O( F ) on a base with an O(1) or larger slope, the governing equations admit a new type of discontinuous solution that is also modelled as a delta shock. The physical manifestation of this discontinuity is a small 'tube' of fluid bounding the flow. The delta-shock conditions for this flow are derived and solved for a point source on an inclined plane. This latter delta-shock framework also sheds light on the evolution of the layer impact on a horizontal base. © The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

- RE Voskoboinikov, SJ Chapman and JR Ockendon (2008). Interconnection of continuum and discrete models of dislocation pile-ups. Progress in Industrial Mathematics at Ecmi 2006
**12**: 392-396. - P Pathmanathan, DJ Gavaghan, JP Whiteley, SJ Chapman and JM Brady (2008). Predicting tumor location by modeling the deformation of the breast. Ieee Transactions on Biomedical Engineering
**55**(10): 2471-2480. (doi:10.1109/TBME.2008.925714) (Abstract)Breast cancer is one of the biggest killers in the western world, and early diagnosis is essential for improved prognosis. The shape of the breast varies hugely between the scenarios of magnetic resonance (MR) imaging (patient lies prone, breast hanging down under gravity), X-ray mammography (breast strongly compressed) and ultrasound or biopsy/surgery (patient lies supine), rendering image fusion an extremely difficult task. This paper is concerned with the use of the finite-element method and nonlinear elasticity to build a 3-D, patient-specific, anatomically accurate model of the breast. The model is constructed from MR images and can be deformed to simulate breast shape and predict tumor location during mammography or biopsy/surgery. Two extensions of the standard elasticity problem need to be solved: an inverse elasticity problem (arising from the fact that only a deformed, stressed, state is known initially), and the contact problem of modeling compression. The model is used for craniocaudal mediolateral oblique mammographic image matching, and a number of numerical experiments are performed. © 2006 IEEE.

- PJ Dellar (2008). Two routes from the Boltzmann equation to compressible flow of polyatomic gases. Progress in Computational Fluid Dynamics
**8**(1-4): 84-96. (doi:10.1504/PCFD.2008.018081) - Y Zhang, AJ Friend, AL Traud, PJ Mucha, MA Porter and JH Fowler (2008). Community structure in Congressional cosponsorship networks. Physica A: Statistical Mechanics and its Applications
**387**(7): 1705-1712. (doi:10.1016/j.physa.2007.11.004) (Abstract)We study the United States Congress by constructing networks between Members of Congress based on the legislation that they cosponsor. Using the concept of modularity, we identify the community structure of Congressmen, who are connected via sponsorship/cosponsorship of the same legislation. This analysis yields an explicit and conceptually clear measure of political polarization, demonstrating a sharp increase in partisan polarization which preceded and then culminated in the 104th Congress (1995-1996), when Republicans took control of both chambers of Congress. Although polarization has since waned in the U.S. Senate, it remains at historically high levels in the House of Representatives. © 2007 Elsevier Ltd. All rights reserved.

- T Roose and AC Fowler (2008). Network development in biological gels: Role in lymphatic vessel development. Bulletin of Mathematical Biology
**70**(6): 1772-1789. (doi:10.1007/s11538-008-9324-3) (Abstract)In this paper, we present a model that explains the prepatterning of lymphatic vessel morphology in collagen gels. This model is derived using the theory of two phase rubber material due to Flory and coworkers and it consists of two coupled fourth order partial differential equations describing the evolution of the collagen volume fraction, and the evolution of the proton concentration in a collagen implant; as described in experiments of Boardman and Swartz (Circ. Res. 92, 801-808, 2003). Using linear stability analysis, we find that above a critical level of proton concentration, spatial patterns form due to small perturbations in the initially uniform steady state. Using a long wavelength reduction, we can reduce the two coupled partial differential equations to one fourth order equation that is very similar to the Cahn-Hilliard equation; however, it has more complex nonlinearities and degeneracies. We present the results of numerical simulations and discuss the biological implications of our model. © 2008 Society for Mathematical Biology.

- M Heil and SL Waters (2008). How rapidly oscillating collapsible tubes extract energy from a viscous mean flow. Journal of Fluid Mechanics
**601**: 199-227. (doi:10.1017/S0022112008000463) - SJ Chapman, RJ Shipley and R Jawad (2008). Multiscale modeling of fluid transport in tumors. Bulletin of Mathematical Biology
**70**(8): 2334-2357. (doi:10.1007/s11538-008-9349-7) (Abstract)A model for fluid flow through the leaky neovasculature and porous interstitium of a solid tumor is developed. A network of isolated capillaries is analyzed in the limit of small capillary radius, and analytical expressions for the hydraulic conductivities and fractional leakage coefficients derived. This model is then homogenized to give a continuum description in terms of the vascular density. The resulting equations comprise a double porous medium with coupled Darcy flow through the interstitium and vasculature. © 2008 Society for Mathematical Biology.

- GW Jones, SJ Chapman and DJ Allwright (2008). Axisymmetric buckling of a spherical shell embedded in an elastic medium under uniaxial stress at infinity. Quarterly Journal of Mechanics and Applied Mathematics
**61**(4): 475-495. (doi:10.1093/qjmam/hbn018) (Abstract)The problem of a thin spherical linearly elastic shell perfectly bonded to an infinite linearly elastic medium is considered. A constant axisymmetric stress field is applied at infinity in the matrix, and the displacement and stress fields in the shell and matrix are evaluated by means of harmonic potential functions. In order to examine the stability of this solution, the buckling problem of a shell which experiences this deformation is considered. Using Koiter's nonlinear shallow shell theory, restricting buckling patterns to those which are axisymmetric and using the Rayleigh-Ritz method by expanding the buckling patterns in an infinite series of Legendre functions, an eigenvalue problem for the coefficients in the infinite series is determined. This system is truncated and solved numerically in order to analyse the behaviour of the shell as it undergoes buckling and to identify the critical buckling stress in two cases, namely, where the shell is hollow and the stress at infinity is either uniaxial or radial. © The author 2008. Published by Oxford University Press; all rights reserved.

- BS Brook and SL Waters (2008). Mathematical challenges in integrative physiology. Journal of Mathematical Biology
**56**(6): 893-896. (doi:10.1007/s00285-008-0159-1) - SL Walters, LJ Cummings, JM Oliver, JAD Wattis, K Heaton, R Bayston, JH Siggers, M Bishop and DM Grant (2008). Ureteric stents: Investigating flow and encrustation. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine
**222**(4): 551-561. (doi:10.1243/09544119JEIM317) (Abstract)Blockages of the ureter, e.g. due to calculi (kidney stones), can result in an increase in renal pelvic pressure. This may be relieved by inserting a stent (essentially a permeable hollow tube). However, a number of complications are associated with stent use. Stents can result in reflux (backflow of urine along the ureter), which will promote recurrent urinary infection and possible renal parenchymal damage. Furthermore, long-term stent use is associated with infection and precipitation of salts from the urine, which can lead to a build-up of crystalline deposits on the stent surface, making stent removal difficult and painful. This paper examines factors governing urine flow in a stented ureter, the implications for reflux, and the processes by which the stent surface encrusts, in particular focusing on the influence of bacterial infection. An interdisciplinary approach is adopted, involving a combination of theoretical investigations and novel experiments. © IMechE 2008.

- M Aguareles, SJ Chapman and T Witelski (2008). Interaction of Spiral Waves in the Complex Ginzburg-Landau Equation. Physical Review Letters
**101**(22): 224101. (doi:10.1103/PhysRevLett.101.224101) - S Beheshti, KJH Law, PG Kevrekidis and MA Porter (2008). Averaging of nonlinearity management with dissipation. Physical Review A
**78**(2): 025805. (doi:10.1103/PhysRevA.78.025805) - JH Siggers and SL Waters (2008). Unsteady flows in pipes with finite curvature. Journal of Fluid Mechanics
**600**: 133-165. (doi:10.1017/S002211200800030X) (Abstract)Motivated by the study of blood flow in a curved artery, we consider fluid flow through a curved pipe of uniform curvature, δ, driven by a prescribed oscillatory axial pressure gradient. The curved pipe has finite (as opposed to asymptotically small) curvature, and we determine the effects of both the centrifugal and Coriolis forces on the flow. In addition to δ, the flow is parameterized by the Dean number, D, the Womersley number, α, and a secondary streaming Reynolds number, R. Asymptotic solutions are developed for the case when δ ≪1, α ≪1 and the magnitude of the axial pressure gradient is small, using regular perturbation techniques. For intermediate values of the governing parameters, a pseudospectral code is used to obtain numerical solutions. For flows driven by a sinusoidal pressure gradient (D = 0), we identify three distinct classes of stable solutions: 2π-periodic symmetric, 2π-periodic asymmetric, and asymmetric solutions that are either quasi-periodic, or periodic with period 2π k for k ε ℕ. The transition between solutions is dependent on the value of α thus pipes with finite curvature may exhibit qualitatively different transitions between the solution classes as the governing parameters are varied from those of curved pipes with asymptotically small curvature. When α >1, matched asymptotic expansions are used to simplify the system, and the resulting equations are solved analytically for R ≪1, α ≪1 and numerically for larger parameter values. We then determine the effect of a non-zero steady component of the pressure gradient (D ≠ 0), and show that, for certain parameter values, when D is above a critical value the periodic asymmetric solutions regain spatial symmetry. Finally, we show that the effects of finite curvature can lead to substantial quantitative differences in the wall shear stress distribution and discuss briefly the physiological implications of the results for blood flow in arteries. © 2008 Cambridge University Press.

- RD O'Dea, HM Byrne and SL Waters (2008). A two-fluid model for tissue growth within a dynamic flow environment. European Journal of Applied Mathematics
**19**(6): 607-634. (doi:10.1017/S0956792508007687) (Abstract)We study the growth of a tissue construct in a perfusion bioreactor, focussing on its response to the mechanical environment. The bioreactor system is modelled as a two-dimensional channel containing a tissue construct through which a flow of culture medium is driven. We employ a multiphase formulation of the type presented by G. Lemon, J. King, H. Byrne, O. Jensen and K. Shakesheff in their study (Multiphase modelling of tissue growth using the theory of mixtures. J. Math. Biol. 52(2), 2006, 571-594) restricted to two interacting fluid phases, representing a cell population (and attendant extracellular matrix) and a culture medium, and employ the simplifying limit of large interphase viscous drag after S. Franks in her study (Mathematical Modelling of Tumour Growth and Stability. Ph.D. Thesis, University of Nottingham, UK, 2002) and S. Franks and J. King in their study (Interactions between a uniformly proliferating tumour and its surrounding: Uniform material properties. Math. Med. Biol. 20, 2003, 47-89). The novel aspects of this study are: (i) the investigation of the effect of an imposed flow on the growth of the tissue construct, and (ii) the inclusion of a mechanotransduction mechanism regulating the response of the cells to the local mechanical environment. Specifically, we consider the response of the cells to their local density and the culture medium pressure. As such, this study forms the first step towards a general multiphase formulation that incorporates the effect of mechanotransduction on the growth and morphology of a tissue construct. The model is analysed using analytic and numerical techniques, the results of which illustrate the potential use of the model to predict the dominant regulatory stimuli in a cell population. Copyright © 2008 Cambridge University Press.

- R Barnett, G Refael, MA Porter and HP Büchler (2008). Vortex lattice locking in rotating two-component Bose-Einstein condensates. New Journal of Physics
**10**: 043030. (doi:10.1088/1367-2630/10/4/043030) (Abstract)The vortex density of a rotating superfluid, divided by its particle mass, dictates the superfluid's angular velocity through the Feynman relation. To find how the Feynman relation applies to superfluid mixtures, we investigate a rotating two-component Bose-Einstein condensate, composed of bosons with different masses. We find that in the case of sufficiently strong interspecies attraction, the vortex lattices of the two condensates lock and rotate at the drive frequency, while the superfluids themselves rotate at two different velocities, whose ratio equals the ratio between the particle masses of the two species. In this paper, we characterize the vortex-locked state, establish its regime of stability, and find that it survives within a disk smaller than a critical radius, beyond which vortices become unbound and the two Bose-gas rings rotate together at the frequency of the external drive. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.

- MA Porter, C Daraio, EB Herbold, I Szelengowicz and PG Kevrekidis (2008). Highly nonlinear solitary waves in periodic dimer granular chains. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**77**(1): 015601. (doi:10.1103/PhysRevE.77.015601) (Abstract)We investigate the propagation of highly nonlinear solitary waves in heterogeneous, periodic granular media using experiments, numerical simulations, and theoretical analysis. We examine periodic arrangements of particles in experiments in which stiffer and heavier beads (stainless steel) are alternated with softer and lighter ones (polytetrafluoroethylene beads). We find good agreement between experiments and numerics in a model with Hertzian interactions between adjacent beads, which in turn agrees very well with a theoretical analysis of the model in the long-wavelength regime that we derive for heterogeneous environments and general bead interactions. Our analysis encompasses previously studied examples as special cases and also provides key insights into the influence of the dimer lattice on the properties (width and propagation speed) of the highly nonlinear wave solutions. © 2008 The American Physical Society.

- IM Moroz (2008). Template analysis of a Faraday disk dynamo. European Physical Journal: Special Topics
**165**(1): 211-220. (doi:10.1140/epjst/e2008-00864-x) (Abstract)In a recent paper Moroz [1] returned to a nonlinear three-dimensional model of dynamo action for a self-exciting Faraday disk dynamo introduced by Hide et al. [2]. Since only two examples of chaotic behaviour were shown in [2], Moroz [1] performed a more extensive analysis of the dynamo model, producing a selection of bifurcation transition diagrams, including those encompassing the two examples of chaotic behaviour in [2]. Unstable periodic orbits were extracted and presented in [1], but no attempt was made to identify the underlying chaotic attractor. We rectify that here. Illustrating the procedure with one of the cases considered in [1], we use some of the unstable periodic orbits to identify a possible template for the chaotic attractor, using ideas from topology [3]. In particular, we investigate how the template is affected by changes in bifurcation parameter. © EDP Sciences and Springer 2008.

- AS Rodrigues, PG Kevrekidis, MA Porter, DJ Frantzeskakis, P Schmelcher and AR Bishop (2008). Matter-wave solitons with a periodic, piecewise-constant scattering length. Physical Review A - Atomic, Molecular, and Optical Physics
**78**(1): 013611. (doi:10.1103/PhysRevA.78.013611) (Abstract)Motivated by recent proposals of "collisionally inhomogeneous" Bose-Einstein condensates (BECs), which have a spatially modulated scattering length, we study the existence and stability properties of bright and dark matter-wave solitons of a BEC characterized by a periodic, piecewise-constant scattering length. We use a "stitching" approach to analytically approximate the pertinent solutions of the underlying nonlinear Schrödinger equation by matching the wave function and its derivatives at the interfaces of the nonlinearity coefficient. To accurately quantify the stability of bright and dark solitons, we adapt general tools from the theory of perturbed Hamiltonian systems. We show that stationary solitons must be centered in one of the constant regions of the piecewise-constant nonlinearity. We find both stable and unstable configurations for bright solitons and show that all dark solitons are unstable, with different instability mechanisms that depend on the soliton location. We corroborate our analytical results with numerical computations. © 2008 The American Physical Society.

- C Letellier, R Gilmore and IM Moroz (2008). Comparison of tests for embeddings. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
**78**(2): 026203. (doi:10.1103/PhysRevE.78.026203) (Abstract)It is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test. The classical tests, which depend on real numbers (fractal dimensions, Lyapunov exponents) averaged over an attractor, are compared with a topological test that depends on integers. The comparison can only be done for mappings into three dimensions. We find that the classical tests fail to predict when a mapping is an embedding and when it is not. We point out the reasons for this failure, which are not restricted to three dimensions. © 2008 The American Physical Society.

- G Kozyreff and SJ Chapman (2008). Kozyreff and Chapman reply. Physical Review Letters
**100**(4): 049402. (doi:10.1103/PhysRevLett.100.049402) - M Mcdonald, O Suleman, S Williams, S Howison and NF Johnson (2008). Impact of unexpected events, shocking news, and rumors on foreign exchange market dynamics. Physical Review E
**77**(4): 046110. (doi:10.1103/PhysRevE.77.046110) - SJ Chapman, MJ Plank, A James and B Basse (2008). A nonlinear model of age and size-structured populations with applications to cell cycles. Anziam Journal
**49**(2): 151-169. (doi:10.1017/S144618110001275X) (Abstract)The Sharpe-Lotka-McKendrick (or von Foerster) equations for an age-structured population, with a nonlinear term to represent overcrowding or competition for resources, are considered. The model is extended to include a growth term, allowing the population to be structured by size or weight rather than age, and a general solution is presented. Various examples are then considered, including the case of cell growth where cells divide at a given size. Copyright © Australian Mathematical Society 2007.

- IM Griffiths and PD Howell (2008). Mathematical modelling of non-axisymmetric capillary tube drawing. Journal of Fluid Mechanics
**605**: 181-206. (doi:10.1017/S002211200800147X)