Past Industrial and Applied Mathematics Seminar

Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, and optimisation. Starting with a discussion on the intrinsic structure of images and their mathematical representation, in this talk we will learn about variational models for image analysis and their connection to partial differential equations, and go all the way to the challenges of their mathematical analysis as well as the hurdles for solving these - typically non-smooth - models computationally. The talk is furnished with applications of the introduced models to image de-noising, motion estimation and segmentation, as well as their use in biomedical image reconstruction such as it appears in magnetic resonance imaging.

Boundary layers control the transport of momentum, heat, solutes and other quantities between walls and the bulk of a flow. The Prandtl-Blasius boundary layer was the first quantitative example of a flow profile near a wall and could be derived by an asymptotic expansion of the Navier-Stokes equation. For higher flow speeds we have scaling arguments and models, but no derivation from the Navier-Stokes equation. The analysis of exact coherent structures in plane Couette flow reveals ingredients of such a more rigorous description of boundary layers. I will describe how exact coherent structures can be scaled to obtain self-similar structures on ever smaller scales as the Reynolds number increases.

A quasilinear approximation allows to combine the structures self-consistently to form boundary layers. Going beyond the quasilinear approximation will then open up new approaches for controlling and manipulating boundary layers.

(Marta Lewicka)

Variational methods have been extensively used in the past decades to rigorously derive nonlinear models in the description of thin elastic films. In this context, natural growth or differential swelling-shrinking lead to models where an elastic body aims at reaching a space-dependent metric. We will describe the effect of such, generically incompatible, prestrain metrics on the singular limits' bidimensional models. We will discuss metrics that vary across the specimen in both the midplate and the thin (transversal) directions. We will also cover the case of the oscillatory prestrain, exhibit its relation to the non-oscillatory case via identifying the effective metrics, and discuss the role of the Riemann curvature tensor in the limiting models.

(Shankar Venkataramani)

Using the bidimensional models for pre-strained Elasticity, that Marta will discuss in her talk, I will discuss some contrasts between the mechanics of thin objects with non-negative curvature (plates, spherical shells, etc) and the mechanics of hyperbolic sheets, i.e. soft/thin objects with negative curvature. I will motivate the need for new "geometric" methods for discretizing the relevant equations, and present some of our preliminary work in this direction.

This is joint work with Toby Shearman and Ken Yamamoto.

J. M. Foster 1 , N. E. Courtier 2 , S. E. J. O’Kane 3 , J. M. Cave 3 , R. Niemann 4 , N. Phung 5 , A. Abate 5 , P. J. Cameron 4 , A. B. Walker 3 & G. Richardson 2 .

1 School of Mathematics & Physics, University of Portsmouth, UK. {@email}

2 School of Mathematics, University of Southampton, UK.

3 School of Physics, University of Bath, UK.

4 School of Chemistry, University of Bath, UK.

5 Helmholtz-Zentrum Berlin, Germany.

Metal halide perovskite has emerged as a highly promising photovoltaic material. Perovskite-based solar cells now exhibit power conversion efficiencies exceeding 22%; higher than that of market-leading multi-crystalline silicon, and comparable to the Shockley-Queisser limit of around 33% (the maximum obtainable efficiency for a single junction solar cell). In addition to fast electronic phenomena, occurring on timescales of nanoseconds, they also exhibit much slower dynamics on the timescales of several seconds and up to a day. One well-documented example of this is the ‘anomalous’ hysteresis observed in current-voltage scans where the applied voltage is varied whilst the output current is measured. There is now a consensus that this is caused by the motion of ions in the perovskite material affecting the internal electric field and in turn the electronic transport.

We will discuss the formulation of a drift-diffusion model for the coupled electronic and ionic transport in a perovskite solar cell as well as its systematic simplification via the method of matched asymptotic expansions. We will use the resulting reduced model to give a cogent explanation for some experimental observations including, (i) the apparent disappearance of current-voltage hysteresis for certain device architectures, and (ii) the slow fading of performance under illumination during the day and subsequent recovery in the dark overnight. Finally, we suggest ways in which materials and geometry can be chosen to reduce charge carrier recombination and improve device performance.

Caustics are places where the light intensity diverges, and where the wave front has a singularity. We use a self-similar description to derive the detailed spatial structure of a cusp singularity, from where caustic lines originate. We also study singularities of higher order, which have their own, uniquely three-dimensional structure. We use this insight to study shock formation in classical compressible Euler dynamics. The spatial structure of these shocks is that of a caustic, and is described by the same similarity equation.

Many types of patterns emerging spontaneously can be observed in systems involving thin elastic plates and subjected to external or internal stresses (compression, differential growth, shearing, tearing, etc.). These mechanical systems can sometime be seen as model systems for more complex natural systems and allow to study in detail elementary emerging patterns. One of the simplest among such systems is a bilayer composed of a thin plate resting on a thick deformable substrate. Upon slight compression, periodic undulations (wrinkles) with a well-defined wavelength emerge at the level of the thin layer. We will show that, as the compression increases, this periodic state is unstable and that a second order transition to a localized state (fold) occurs when the substrate is a dense fluid.

The desire to “freely suspend the constituents of matter” in order to study their behavior can be traced back over 200 years to the diaries of Lichtenberg. From radio-frequency ion traps to optical tweezing of colloidal particles, existing methods to trap matter in free space or solution rely on the use of external fields that often strongly perturb the integrity of a macromolecule in solution. We recently introduced the ‘electrostatic fluidic trap’, an approach that exploits equilibrium thermodynamics to realise stable, non-destructive confinement of single macromolecules in room temperature fluids, and represents a paradigm shift in a nearly century-old field. The spatio-temporal dynamics of a single electrostatically trapped object reveals fundamental information on its properties, e.g., size and electrical charge. We have demonstrated the ability to measure the electrical charge of a single macromolecule in solution with a precision much better than a single elementary charge. Since the electrical charge of a macromolecule in solution is in turn a strong function of its 3D conformation, our approach enables for the first time precise, general measurements of the relationship between 3D structure and electrical charge of a single macromolecule, in real time. I will present our most recent advances in this emerging area of molecular measurement and show how such high-precision measurement at the nanoscale may be able to unveil the presence of previously unexpected phenomena in intermolecular interactions in solution.

Locomotion strategies employed by unicellular organism are a rich source of inspiration for studying mechanisms for shape control. They are particularly interesting because they are invisible to the naked eye, and offer surprising new solutions to the question of how shape can be controlled.

In recent years, we have studied locomotion and shape control in Euglena gracilis. This unicellular protist is particularly intriguing because it can adopt different motility strategies: swimming by flagellar propulsion, or crawling thanks to large amplitude shape changes of the whole body (a behavior known as metaboly). We will survey our most recent findings within this stream of research.

In laboratories around the world, scientists use magnetic stirrers to mix solutions and dissolve powders. It is well known that at high drive rates the stir bar jumps around erratically with poor mixing, leading to its nick-name 'flea'. Investigating this behaviour, we discovered a state in which the flea levitates stably above the base of the vessel, supported by magnetic repulsion between flea and drive magnet. The vertical motion is oscillatory and the angular motion a superposition of rotation and oscillation. By solving the coupled vertical and angular equations of motion, we characterised the flea’s behaviour in terms of two dimensionless quantities: (i) the normalized drive speed and (ii) the ratio of magnetic to viscous forces. However, Earnshaw’s theorem states that levitation via any arrangement of static magnets is only possible with additional stabilising forces. In our system, we find that these forces arise from the flea’s oscillations which pump fluid radially outwards, and are only present for a narrow range of Reynold's numbers. At slower, creeping flow speeds, only viscous forces are present, whereas at higher speeds, the flow reverses direction and the flea is no longer stable. We also use both the levitating and non-levitating states to measure rheological properties of the system.

There are a huge number of nonlinear partial differential equations that do not have analytic solutions.   Often one can find similarity solutions, which reduce the number of independent variables, but still leads, generally, to a nonlinear equation.  This can, only sometimes, be solved analytically.  But always the solution is independent of the initial conditions.   What role do they play?   It is generally stated that the similarity  solution agrees with the (not determined) exact solution when (for some variable say t) obeys t >> t_1.   But what is  t_1?   How does it depend on the initial conditions?  How large must  t be for the similarity solution to be within 15, 10, 5, 1, 0.1, ….. percent of the real solution?   And how does this depend on the parameters and initial conditions of the problem?   I will explain how two such typical, but somewhat different, fundamental problems can be solved, both analytically and numerically,  and compare some of the results with small scale laboratory experiments, performed during the talk.  It will be suggested that many members of the audience could take away the ideas and apply them in their own special areas.

Statements in media about record wave heights being measured are more and more common, the latest being about a record wave of almost 24m in the Southern Ocean on 9 May 2018. We will review some of these wave measurements and the various techniques to measure waves. Then we will explain the various mechanisms that can produce extreme waves both in wave tanks and in the ocean. We will conclude by providing the mechanism that, we believe, explains some of the famous extreme waves. Note that extreme waves are not necessarily rogue waves and that rogue waves are not necessarily extreme waves.

The peeling of an elastic sheet away from thin layer of viscous fluid is a simply-stated and generic problem, that involves complex interactions between flow and elastic deformation on a range of length scales. 

I will illustrate the possibilities by considering theoretically and experimentally the injection and spread of viscous fluid beneath a flexible elastic lid; the injected fluid forms a blister, which spreads by peeling the lid away at the  perimeter of the blister. Among the many questions to be considered are the mechanisms for relieving the elastic analogue of the contact-line problem, whether peeling is "by bending" or "by pulling", the stability of the peeling front, and the effects of a capillary meniscus when peeling is by air injection. The result is a plethora of dynamical regimes and asymptotic scaling laws.

Superhydrophobic surfaces, formed by air entrapment within the cavities of a hydrophobic solid substrate, offer a promising potential for drag reduction in small-scale flows. It turns out that low-drag configurations are associated with singular limits, which to date have typically been addressed using numerical schemes. I will discuss the application of singular perturbations to several of the canonical problems in the field. 

Elastic gridshells arise from the buckling of an initially planar grid of rods. Architectural elastic gridshells first appeared in the 1970’s. However, to date, only a limited number of examples have been constructed around the world, primarily due to the challenges involved in their structural design. Yet, elastic gridshells are highly appealing: they can cover wide spans with low self-weight, they allow for aesthetically pleasing shapes and their construction is typically simple and rapid. A more mundane example is the classic pasta strainer, which, with its remarkably simple design, is a must-have in every kitchen.

This talk will focus on the geometry-driven nature of elastic gridshells. We use a geometric model based on the theory of discrete Chebyshev nets (originally developed for woven fabric) to rationalize their actuated shapes. Validation is provided by precision experiments and rod-based simulations. We also investigate the linear mechanical response (rigidity) and the non-local behavior of these discrete shells under point-load indentation. Combining experiments, simulations, and scaling analysis leads to a master curve that relates the structural rigidity to the underlying geometric and material properties. Our results indicate that the mechanical response of elastic gridshells, and their underlying characteristic forces, are dictated by Euler's elastica rather than by shell-related quantities. The prominence of geometry that we identify in elastic gridshells should allow for our results to transfer across length scales: from architectural structures to micro/nano–1-df mechanical actuators and self-assembly systems.

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Tubing issues: 

- Moving a sphere in a narrow pipe

What is the force required to move an object inside a narrow elastic pipe? The constriction by the tube induces a normal force on the sphere. In the case of solid friction, the pulling force may  be simply deduced from Coulomb’s law. How does is such force modified by the addition of a lubricant? This coupled problem between elasticity and viscous flow results in a non-linear dependence of the force with the traction speed.

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- Baromorphs

When a bicycle tyre is inflated the cross section of the pipe increases much more than its circumference. Can we use this effect to induce non-isotropic growth in a plate?  We developed, through standard casting techniques, flat plates imbedded with a network of channels of controlled geometry. How are such plates deformed as pressure is applied to this network? Using a simplified mechanical model, 3D complex shapes can be programmed and dynamically actuated. 

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The talk originates from two studies on the dynamic properties of one-dimensional elastic quasicrystalline solids. The first one refers to a detailed investigation of scaling and self-similarity of the spectrum of an axial waveguide composed of repeated elementary cells designed by adopting the family of generalised Fibonacci substitution rules corresponding to the so-called precious means. For those, an invariant function of the circular frequency, the Kohmoto's invariant, governs self-similarity and scaling of the stop/pass band layout within defined ranges of frequencies at increasing generation index. The Kohmoto's invariant also explains the existence of particular frequencies, named canonical frequencies, associated with closed orbits on the geometrical three-dimensional representation of the invariant. The second part shows the negative refraction properties of a Fibonacci-generated quasicrystalline laminate and how the tuning of this phenomenon can be controlled by selecting the generation index of the sequence.

Most motile bacteria are equipped with multiple helical flagella, slender appendages whose rotation in viscous fluids allow the cells to self-propel. We highlight in this talk two consequences of hydrodynamics for bacteria. We first show how the swimming of cells with multiple flagella is enabled by an elastohydrodynamic instability. We next demonstrate how interactions between flagellar filaments mediated by the fluid govern the ability of the cells to reorient. 

What if one desires to have a World perfectly slippery to water? What are the strategies that can be adopted? And how can smart slippery surfaces be created? In this seminar, I will outline approaches to creating slippery surfaces, which all involve reducing or removing droplet contact with the solid, whilst still supporting the droplet. The first concept is to decorate the droplet surface with particles, thus creating liquid marbles and converting the droplet-solid contact into a solid-solid contact. The second concept is to use the Leidenfrost effect to instantly vaporize a layer of water, thus creating a film of vapor and converting the droplet-solid contact into vapor-solid contact. The third concept is to infuse oil into the surface, thus creating a layer of oil and converting the droplet-solid contact into a lubricant-solid contact. I will also explain how we design such to have smart functionality whilst retaining and using the mobility of contact lines and droplets. I will show how Leidenfrost levitation can lead to new types of heat engines [1], how a microsystems approach to the Leidenfrost effect can reduce energy input and lead to a new type of droplet microfluidics [2] (Fig. 1a) and how liquid diodes can be created [3]. I will explain how lubricant impregnated surfaces lead to apparent contact angles [4] and how the large retained footprint of the droplet allows droplet transport and microfluidics using energy coupled via a surface acoustic wave (SAW) [5]. I will argue that droplets confined between reconfigurable slippery boundaries can be continuously translated in an energy invariant manner [6] (Fig. 1b). I will show that a droplet Cheerios effect induced by the menisci arising from structuring the underlying lubricated surface or by droplets in close proximity to each other can be used to guide and position droplets [7] (Fig. 1c). Finally, I will show that active control of droplet spreading by electric field induced control of droplet spreading, via electrowetting or dielectrowetting, can be achieved with little hysteresis [8] and can be a new method to investigate the dewetting of surfaces [9].

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Figure 1 Transportation and positioning of droplets using slippery surfaces: (a) Localized Leidenfrost effect, (b) Reconfigurable boundaries, and (c) Droplet Cheerio’s effect.

Acknowledgements The financial support of the UK Engineering & Physical Sciences Research Council (EPSRC) and Reece Innovation ltd is gratefully acknowledged. Many collaborators at Durham, Edinburgh, Nottingham Trent and Northumbria Universities were instrumental in the work described.

[1] G.G. Wells, R. Ledesma-Aguilar, G. McHale and K.A. Sefiane, Nature Communications, 2015, 6, 6390.

[2] L.E. Dodd, D. Wood, N.R. Geraldi, G.G. Wells, et al., ACS Applied & Materials Interfaces, 2016, 8, 22658.

[3] J. Li, X. Zhou , J. Li, L. Che, J. Yao, G. McHale, et al., Science Advances, 2017, 3, eaao3530.

[4] C. Semprebon, G. McHale, and H. Kusumaatmaja, Soft Matter, 2017, 13, 101.

[5] J.T. Luo, N.R. Geraldi, J.H. Guan, G. McHale, et al., Physical Review Applied, 2017, 7, 014017.

[6] É. Ruiz-Gutiérrez, J.H. Guan, B.B. Xu, G. McHale, et al., Physical Review Letters, 2017, 118, 218003.

[7] J.H. Guan, É. Ruiz-Gutiérrez, B.B. Xu, D. Wood, G. McHale, et al., Soft Matter, 2017, 13, 3404.

[8] Z. Brabcová, G. McHale, G.G. Wells, et al., Applied Physics Letters, 2017, 110, 121603.

[9] A.M.J. Edwards, R. Ledesma-Aguilar, et al., Science Advances, 2016, 2, e1600183

A polymer, or microscopic elastic filament, is often modelled as a linear chain of rigid bodies interacting both with themselves and a heat bath. Then the classic notions of persistence length are related to how certain correlations decay with separation along the chain. I will introduce these standard notions in mathematical terms suitable for non specialists, and describe the standard results that apply in the simplest cases of wormlike chain models that have a straight, minimum energy (or ground or intrinsic) shape. Then I will introduce an appropriate  splitting of a matrix recursion in the group SE(3) which deconvolves the distinct effects of stiffness and intrinsic shape in the more complicated behaviours of correlations that arise when the polymer is not intrinsically straight. The new theory will be illustrated by fully implementing it within a simple sequence-dependent rigid base pair model of DNA. In that particular context, the persistence matrix factorisation generalises and justifies the prior scalar notions of static and dynamic persistence lengths.

When soft ferromagnetic particles are suspended at air-water interfaces in the presence of a vertical magnetic field, dipole-dipole repulsion competes with capillary attraction such that 2d structures self-assemble. The complex arrangements of such floating bodies are emphasized. The equilibrium distance between particles exhibits hysteresis when the applied magnetic field is modified. Irreversible processes are evidenced. By adding a horizontal and oscillating magnetic field, periodic deformations of the assembly are induced. We show herein that collective particle motions induce locomotion at low Reynolds number. The physical mechanisms and geometrical ingredients behind this cooperative locomotion are identified. These physical mechanisms can be exploited to much smaller scales, offering the possibility to create artificial and versatile microscopic swimmers.

Moreover, we show that it is possible to generate complex structures that are able to capture particles, perform cargo transport, fluid mixing, etc.

In this talk, I will present some recent results exploring the connections between dynamical systems and network science. I will particularly focus on large-scale structures and their dynamical interpretation. Those may correspond to communities/clusters or classes of dynamically equivalent nodes. If time allows, I will also present results where the underlying network structure is unknown and where communities are directly inferred from time series observed on the nodes.

How do organisms cope with cellular variability to achieve well-defined morphologies and architectures? We are addressing this question by combining experiments with live plants and analyses of (stochastic) models that integrate cell-cell communication and tissue mechanics. During the talk, I will survey our results concerning plant architecture (phyllotaxis) and organ morphogenesis.

Network models may be applied to describe many complex systems, and in the era of online social networks the study of dynamics on networks is an important branch of computational social science.  Cascade dynamics can occur when the state of a node is affected by the states of its neighbours in the network, for example when a Twitter user is inspired to retweet a message that she received from a user she follows, with one event (the retweet) potentially causing further events (retweets by followers of followers) in a chain reaction. In this talk I will review some simple models that can help us understand how social contagion (the spread of cultural fads and the viral diffusion of information) depends upon the structure of the social network and on the dynamics of human behaviour. Although the models are simple enough to allow for mathematical analysis, I will show examples where they can also provide good matches to empirical observations of cascades on social networks.

Using synchronized high-speed video camera, hydrophone and microphone we investigated flow patterns, the impact and secondary sound pulses emitted by oscillating bubbles. On the submerging  drop found short capillary waves produced by small secondary impact droplets. Picturesque filament and grid structures produced by colour drop of mixing fluid registered on the surface of the cavity and crown. Physical model includes discussion of the potential surface energy effects.

Inelastic surface growth associated with continuous creation of incompatibility on the boundary of an evolving body is behind a variety of both natural processes (embryonic development,  tree growth) and technological processes (dam construction, 3D printing). Despite the ubiquity of such processes, the mechanical aspects of surface growth are still not fully understood. In this talk we present  a new approach to surface growth that allows one to address inelastic effects,  path dependence of the growth process and the resulting geometric frustration. In particular, we show that incompatibility developed during deposition can be fine-tuned to ensure a particular behaviour of the system in physiological (or working) conditions. As an illustration, we compute an explicit deposition protocol aimed at "printing" arteries, that guarantees the attainment of desired stress distributions in physiological conditions. Another illustration is the growth starategy for explosive plants, allowing a complete release of residual elastic energy with a single cut.