It is thought that the hairy legs of water walking arthropods are able to remain clean and dry because the flexibility of the hairs spontaneously moves drops off the hairs. We present a mathematical model of this bending-induced motion, or bendotaxis, and study how it performs for wetting and non-wetting drops. Crucially, we show that both wetting and non-wetting droplets move in the same direction (using physical arguments and numerical solutions). This suggests that a surface covered in elastic filaments (such as the hairy leg of insects) may be able to universally self-clean. To quantify the efficiency of this effect, we explore the conditions under which drops leave the structure by ‘spreading’ rather than translating and also how long it takes to do so.

# Past Industrial and Applied Mathematics Seminar

Many species of insects adhere to vertical and inverted surfaces using footpads that secrete thin films of a mediating fluid. The fluid bridges the gap between the foot and the target surface. The precise role of this liquid is still subject to debate, but it is thought that the contribution of surface tension to the adhesive force may be significant. It is also known that the footpad is soft, suggesting that capillary forces might deform its surface. Inspired by these physical ingredients, we study a model problem in which a thin, deformable membrane under tension is adhered to a flat, rigid surface by a liquid droplet. We find that there can be multiple possible equilibrium states, with the number depending on the applied tension and aspect ratio of the system. The presence of elastic deformation ignificantly enhances the adhesion force compared to a rigid footpad. A mathematical model shows that the equilibria of the system can be controlled via two key parameters depending on the imposed separation of the foot and target surface, and the tension applied to the membrane. We confirm this finding experimentally and show that the system may transition rapidly between two states as the two parameters are varied. This suggests that different strategies may be used to adhere strongly and then detach quickly.

We present a framework for the design, analysis and application of computational multiscale methods for slow-fast high-dimensional stochastic processes. We call these processes "microscopic'', and assume existence of an approximate "macroscopic'' model that captures the slow behaviour of a selected set of macroscopic state variables. The methodology combines short bursts of microscopic simulation with extrapolation at the macroscopic level. The methodology requires the careful study of a few key algorithmic ingredients. First, we need to properly initialise the microscopic system, based on a given macroscopic state and (possibly) a prior microscopic state that contains additional information about the system. Second, we need to control the variance of the noise that originates from the microscopic Monte Carlo simulation. Third, we need to analyse stability of the extrapolation step. We will discuss these aspects on two types of model problems -- scale-separated SDEs and kinetic equations -- and show the efficacity of the resulting methods in diverse applications, ranging from tumor growth to fusion energy.

The statistical physics governing phase-ordering dynamics following a symmetry breaking rst-order phase transition is an area of active research. The Coarsening/Ageing of the ensemble of phase domains, wherein irreversible annihilation or joining of domains yields a growing characteristic domain length, is an omniprescent feature whose universal characteristics one would wish to understand. Driven kinetic Ising models and growing nano-faceted crystals are theoretically important examples of such Coarsening (Ageing) Dynamical Systems (CDS), since they additionally break thermodynamic uctuation-dissipation relations. Power-laws for the growth in time of the characteristic size of domains, and a concomitant scale-invariance of associated length distributions, have so frequently been empirically observed that their presence has acquired the status of a principle; the so-called Dynamic-Scaling Hypothesis. But the dynamical symmetries of a given CDS- its Coarsening Group G - may include more than the global spatio-temporal scalings underlying the Dynamic Scaling Hypothesis. In this talk, I will present a recently developed theoretical framework (Ref.[1]) that shows how the symmetry group G of a Coarsening (ageing) Dynamical System necessarily yields G-equivariance (covariance) of its universal statistical observables. We exhibit this theory for a variety of model systems, of both thermodynamic and driven type, with symmetries that may also be Emergent (Ref. [2,3]) and/or Hidden. We will close with a magical theoretical coarsening law that combines Lorentzian and Parabolic symmetries!

References

[1] Lorentzian symmetry predicts universality beyond scaling laws, SJ Watson, EPL 118 (5), 56001, (Aug.2, 2017) Editor's Choice

[2] Emergent parabolic scaling of nano-faceting crystal growth Stephen J. Watson, Proc. R. Soc. A 471: 20140560 (2015)

[3] Scaling Theory and Morphometrics for a Coarsening Multiscale Surface, via a Principle of Maximal Dissipation", Stephen

Phytoplankton moving in the ocean, spermatozoa making their way through the female reproductive tract and harmful bacteria that form biofilms on implanted medical devices interact with a surrounding fluid. Their length scales are small enough so that viscous effects dominate inertial effects allowing the resulting fluid dynamics to be described by the linear Stokes equations. However, nonlinear behavior can occur because these structures are flexible and their form evolves with the flow. In addition, the fluid environment may also be complex because of embedded microstructures that further complicate the dynamics. We will discuss recent successes and challenges in describing these elastohydrodynamic systems.

Brain convolutions are specificity of mammals. Varying in intensity according to the animal species, it is measured by an index called the gyrification index, ratio between the effective surface of the cortex compared to its apparent surface. Its value is closed to 1 for rodents (smooth brain), 2.6 for new-borns and 5 for dolphins. For humans, any significant deviation is a signature of a pathology occurring in fetal life, which can be detected now by magnetic resonance imaging (MRI). We propose a simple model of growth for a bilayer made of the grey and white matter, the grey matter being in cortical position. We analytically solved the Neo-Hookean approximation in the short and large wavelength limits. When the upper layer is softer than the bottom layer (possibly, the case of the human brain), the selection mechanism is dominated by the physical properties of the upper layer. When the anisotropy favours the growth tangentially as for the human brain, it decreases the threshold value for gyri formation. The gyrification index is predicted by a post-buckling analysis and compared with experimental data. We also discuss some pathologies in the model framework.

Discontinuous solutions, such as cracks or cavities, can suddenly appear in elastic solids when a limiting condition is reached. Similarly, self-contacting folds can nucleate at a free surface of a soft material subjected to a critical compression. Unlike other elastic instabilities, such as buckling and wrinkling, creasing is still poorly understood. Being invisible to linearization techniques, crease nucleation is a problem of high mathematical complexity.

In this talk, I will discuss some recent theoretical insights solving the quest for both the nucleation threshold and the emerging crease morphology. The analytic predictions are in agreement with experimental and numerical data. They prove a fundamental insight either for understanding the creasing onset in living matter, e.g. brain convolutions, or for guiding engineering applications, e.g. morphable meta-materials.

In this talk we consider a system of interacting Brownian particles. When diffusing particles interact with each other their motions are correlated, and the configuration space is of very high dimension. Often an equation for the one-particle density function (the concentration) is sought by integrating out the positions of all the others. This leads to the classic problem of closure, since the equation for the concentration so derived depends on the two-particle correlation function. We discuss two common closures, the mean-field (MFA) and the Kirkwood-superposition approximations, as well as an alternative approach, which is entirely systematic, using matched asymptotic expansions (MAE). We compare the resulting (nonlinear) diffusion models with Monte Carlo simulations of the stochastic particle system, and discuss for which types of interactions (short- or long-range) each model works best.

Understanding the evolution of a solidification front is important in the study of solidification processes. Mathematically, self-similar solutions exist to the Stefan problem when the liquid domain is assumed semi-infinite, and such solutions have been extensively studied in the literature. However, in the case where the liquid region is finite and sufficiently small, such of solutions no longer hold, as in this case two solidification fronts will move toward each other and interact. We present an asymptotic analysis for the two-front Stefan problem with a small amount of constitutional supercooling and compare the asymptotic results with numerical simulations. We finally discuss ongoing work on the same problem near the time when the two fronts are close to colliding.

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Silicon is produced from quartz rock in electrode-heated furnaces by using carbon as a reduction agent. We present a model of the heat and mass transfer in an experimental pilot furnace and perform an asymptotic analysis of this model. First, by prescribing a steady state temperature profile in the furnace we explore the leading order reactions in different spatial regions. We next utilise the dominant behaviour when temperature is prescribed to reduce the full model to two coupled partial differential equations for the time-variable temperature profile within the furnace and the concentration of solid quartz. These equations account for diffusion, an endothermic reaction, and the external heating input to the system. A moving boundary is found and the behaviour on either side of this boundary explored in the asymptotic limit of small diffusion. We note how the simplifications derived may be useful for industrial furnace operation.

Understanding the spatial distribution of organisms throughout an environment is an important topic in population ecology. We briefly review ecological questions underpinning certain mathematical work that has been done in this area, before presenting a few examples of spatially structured population models. As a first example, we consider a model of two species aggregation and clustering in two-dimensional domains in the presence of heterogeneity, and demonstrate novel aggregation mechanisms in this setting. We next consider a second example consisting of a predator-prey-subsidy model in a spatially continuous domain where the spatial distribution of the subsidy influences the stability and spatial structure of steady states of the system. Finally, we discuss ongoing work on extending such results to network-structured domains, and discuss how and when the presence of a subsidy can stabilize predator-prey dynamics."

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Compaction is a primary process in the evolution of a sedimentary basin. Various 1D models exist to model a basin compacting due to overburden load. We explore a multi-dimensional model for a basin undergoing mechanical and chemical compaction. We discuss some properties of our model. Some test cases in the presence of geological features are considered, with appropriate numerical techniques presented.