Past Junior Applied Mathematics Seminar

Unintended low energy thermal or mechanical stimuli can lead to the accidental ignition of explosive materials. During such events, described as ‘insults’ in the literature, ignition of the explosive is caused by localised regions of high temperature known as ‘hot spots’. We develop a model which helps us to understand how highly localised shear deformation, so-called shear banding, acts as a mechanism for hot spot generation. Through a boundary layer analysis, we give a deeper insight into how the additional self heating caused by chemical reactions affects the initiation and development of shear bands,  and highlight the key physical properties which control this process.

Coastal vegetation has a well-known effect of attenuating waves; however, quantifiable measures of attenuation for general wave and vegetation scenarios are not well known. On the plant scale, there are extensive studies in predicting the dynamics of a single plant in an oscillatory flow. On the coastal scale however, there are yet to be compact models which capture the dynamics of both the flow and vegetation, when the latter exists in the form of a dense canopy along the bed. In this talk, we will discuss the open questions in the field and the modelling approaches involved. In particular, we investigate how micro-scale effects can be homogenised in space and how periodic motions can be averaged in time.

Wrinkling is a universal instability occurring in a wide variety of engineering and biological materials. It has been studied extensively for many different systems but a full description is still lacking. Here, we provide a systematic analysis of the wrinkling of a thin hyperelastic film over a substrate in plane strain using stream functions. For comparison, we assume that wrinkling is generated either by the isotropic growth of the film or by the lateral compression of the entire system. We perform an exhaustive linear analysis of the wrinkling problem for all stiffness ratios and under a variety of additional boundary and material effects.

In gas-liquid two-phase pipe flows, flow regime transition is associated with changes in the micro-scale geometry of the flow. In particular, the bubbly-slug transition is associated with the coalescence and break-up of bubbles in a turbulent pipe flow. We consider a sequence of models designed to facilitate an understanding of this process. The simplest such model is a classical coalescence model in one spatial dimension. This is formulated as a stochastic process involving nucleation and subsequent growth of ‘seeds’, which coalesce as they grow. We study the evolution of the bubble size distribution both analytically and numerically. We also present some ideas concerning ways in which the model can be extended to more realistic two- and three-dimensional geometries.

As a rule of thumb, the dominant resistive force on a cyclist riding along a flat road at a speed above 10mph is aerodynamic drag; at higher speeds, this drag becomes even more influential because of its non-linear dependence on speed. Reducing drag, therefore, is of critical importance in bicycle racing, where winning margins are frequently less than a tyre's width (over a 200+km race!). I shall discuss a mathematical model of aerodynamic drag in cycling, present mathematical reasoning behind some of the decisions made by racing cyclists when attempting to minimise it, and touch upon some of the many methods of aerodynamic drag assessment.

Recovery of reaction pathways under the DCSC framework.

Structure from Motion (SfM) is a problem which asks: given photos of an object from different angles, can we reconstruct the object in 3D? This problem is important in computer vision, with applications including urban planning and autonomous navigation. A key part of SfM is bundle adjustment, where initial estimates of 3D points and camera locations are refined to match the images. This results in a high-dimensional nonlinear least-squares problem. In this talk, I will discuss how dimensionality reduction methods such as block coordinates and sketching can be used to improve solver scalability for bundle adjustment problems.

Calcination describes the heat treatment of anthracite particles in a furnace to produce a partially-graphitised material which is suitable for use in electrodes and for other met- allurgical applications. Electric current is passed through a bed of anthracite particles, here referred to as a coke bed, causing Ohmic heating and high temperatures which result in the chemical and structural transformation of the material.

Understanding the behaviour of such mechanisms on the scale of a single particle is often dealt with through the use of computational models such as DEM (Discrete Element Methods). However, because of the great discrepancy between the length scale of the particles and the length scale of the furnace, we can exploit asymptotic homogenisation theory to simplify the problem.  

In this talk, we will present some results relating to the electrical and thermal conduction through granular material which define effective quantities for the conductivities by considering a microscopic representative volume within the material. The effective quantities are then used as parameters in the homogenised macroscopic model to describe calcination of anthracite. 

It is known that in steady-state potential flows, the separation of a gravity-driven free-surface from a solid exhibits a number of peculiar characteristics. For example, it can be shown that the fluid must separate from the body so as to form one of three possible in-fluid angles: (i) 180°, (ii) 120°, or (iii) an angle such that the surface is locally perpendicular to the direction of gravity. These necessary separation conditions were notably remarked by Dagan & Tulin (1972) in the context of ship hydrodynamics [J. Fluid Mech., 51(3) pp. 520-543], but they are of crucial importance in many potential flow applications. It is not particularly well understood why there is such a drastic change in the local separation behaviours when the global flow is altered. The question that motivates this work is the following: outside a formal balance-of-terms arguments, why must (i) through (iii) occur and furthermore, what is the connections between them?

              In this work, we seek to explain the transitions between the three cases in terms of the singularity structure of the associated solutions once they are extended into the complex plane. A numerical scheme is presented for the analytic continuation of a vertical jet (or alternatively a rising bubble). It will be shown that the transition between the three cases can be predicted by observing the coalescence of singularities as the speed of the jet is modified. A scaling law is derived for the coalescence rate of singularities.

Over the past decade, the randomized singular value decomposition (RSVD)
algorithm has proven to be an efficient, reliable alternative to classical
algorithms for computing low-rank approximations in a number of applications.
However, in cases where no information is available on the singular value
decay of the data matrix or the data matrix is known to be close to full-rank,
the RSVD is ineffective. In recent years, there has been great interest in
randomized algorithms for computing full factorizations that excel in this
regime.  In this talk, we will give a brief overview of some key ideas in
randomized numerical linear algebra and introduce a new randomized algorithm for
computing a full, rank-revealing URV factorization.

Collective neural crest (NC) cell migration determines the formation of peripheral tissues during vertebrate development. If NC cells fail to reach a target or populate an incorrect location, improper cell differentiation or uncontrolled cell proliferation can occur. Therefore, knowledge of embryonic cell migration is important for understanding birth defects and tumour formation. However, the response of NC cells to different stimuli, and their ability to migrate to distant targets, are still poorly understood. Recently, experimental and computational studies have provided evidence that there are at least two subpopulations of NC cells, namely “leading” and “trailing” cells, with potential further differentiation between the cells in these subpopulations [1,2]. The main difference between these two cell types is the mechanism driving motility and invasion: the leaders follow the gradient of a chemoattractant, while the trailing cells follow “gradients” of the leaders. The precise mechanisms underlying these leader-follower interactions are still unclear.

We develop and apply innovative multi-scale modelling frameworks to analyse signalling effects on NC cell dynamics. We consider different potential scenarios and investigate them using an individual-based model for the cell motility and reaction-diffusion model to describe chemoattractant dynamics. More specifically, we use a discrete self-propelled particle model [3] to capture the interactions between the cells and incorporate volume exclusion. Streaming migration is represented using an off-lattice model to generate realistic cell arrangements and incorporate nonlinear behaviour of the system, for example the coattraction between cells at various distances. The simulations are performed using Aboria, which is a C++ library for the implementation of particle-based numerical methods [4]. The source of chemoattractant, the characteristics of domain growth, and types of boundary conditions are some other important factors that affect migration. We present results on how robust/sensitive cells invasion is to these key biological processes and suggest further avenues of experimental research.

[1] R. McLennan, L. Dyson, K. W. Prather, J. A. Morrison, R.E. Baker, P. K. Maini and P. M. Kulesa. (2012). Multiscale mechanisms of cell migration during development: theory and experiment, Development, 139, 2935-2944.

[2] R. McLennan, L. J. Schumacher, J. A. Morrison, J. M. Teddy, D. A. Ridenour, A. C. Box, C. L. Semerad, H. Li, W. McDowell, D. Kay, P. K. Maini, R. E. Baker and P. M. Kulesa. (2015). Neural crest migration is driven by a few trailblazer cells with a unique molecular signature narrowly confined to the invasive front, Development, 142, 2014-2025.

[3] G. Grégoire, H. Chaté and Y Tu. (2003). Moving and staying together without a leader, Physica D: Nonlinear Phenomena, 181, 157-170.

[4] M. Robinson and M. Bruna. (2017). Particle-based and meshless methods with Aboria, SoftwareX, 6, 172-178. Online documentation https://github.com/martinjrobins/Aboria.

Protein interaction networks (PINs) allow the representation and analysis of biological processes in cells. Because cells are dynamic and adaptive, these processes change over time. Thus far, research has focused either on the static PIN analysis or the temporal nature of gene expression. By analysing temporal PINs using multilayer networks, we want to link these efforts. The analysis of temporal PINs gives insights into how proteins, individually and in their entirety, change their biological functions. We present a general procedure that integrates temporal gene expression information with a monolayer PIN to a temporal PIN and allows the detection of modular structure using multilayer modularity maximisation.

We are all familiar with the need for continuum mechanics-based models in physical applications. In this case, we are interested in large-scale water-wave problems, such as coastal flows and dam breaks.
When modelling these problems, we inevitably wish to solve them on a finite domain, and require boundary conditions to do so. Ideally, we would recreate the semi-infinite nature of a coastline by allowing any generated waves to flow out of the domain, as opposed to them reflecting off the far-field boundary and disrupting the remainder of our simulation. However, applying an appropriate boundary condition is not as straightforward as we might think.
In this talk, we aim to evaluate alternatives to so-called 'active boundary condition' absorption. We will derive a toy model of a shallow-water wavetank, and consider the implementation and efficacy of two 'passive' absorption techniques.
 

Snap-through buckling is a type of instability in which an elastic object rapidly jumps from one state to another, just as an umbrella flips upwards in a gust of wind. While snap-through under dry, mechanical loads has already been harnessed in engineering to generate fast motions between two states, the mechanisms underlying snapping in bulk fluid flows remain relatively unexplored. In this talk we demonstrate how elastic snap-through may be used to passively control fluid flows at low Reynolds number, in contrast to some pre-existing valves that rely on active control. We study viscous flow through a channel in which one of the bounding walls is an elastic arch. By performing experiments at the macroscopic scale, we show that snap-through of the arch rapidly changes the channel from a constricted to an unconstricted state, increasing the hydraulic conductivity by up to an order of magnitude. We also observe nonlinear pressure-flux characteristics away from snapping due to the coupling between the driving flow and elasticity. This behaviour is confirmed by a mathematical model that also shows the device may readily be scaled down for microfluidic applications. Finally, we demonstrate that such a device may be used to create a fluidic analogue of a fuse: the fluid flux through a channel may not rise above a given value. 

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 significantly 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.

Many metallurgical processes involve the heat treatment of granular material due to large alternating currents. To understand how the current propagates through the material, one must understand the bulk resistivity, that is, the resistivity of the granular material as a whole. The literature suggests that the resistance due to contacts between particles contributes significantly to the bulk resistivity, therefore one must pay particular attention to these contacts. 

My work is focused on understanding the precise impact of small contacts on the current propagation. The scale of the contacts is several order of magnitude smaller than that of the furnace itself, therefore we apply matched asymptotics methods to study how the current varies with the size of the contact.

One of the greatest challenges in developing renewable energy sources is finding an efficient energy storage solution to smooth out the inherently fluctuating supply. One cheap solution is lead-acid batteries, which are used to provide off-grid solar energy in developing countries. However, modelling of this technology has fallen behind other types of battery; the state-of-the-art models are either overly simplistic, fitting black-box functions to current and voltage data, or overly complicated, requiring complex and time-consuming numerical simulations. Neither of these methods offers great insight into the chemical behaviour at the micro-scale.

In our research, we use asymptotic methods to explore the Newman porous-electrode model for a constant-current discharge at low current densities, a good estimate for real-life applications. In this limit, we obtain a simple yet accurate formula for the cell voltage as a function of current density and time. We also gain quantitative insight into the effect of various parameters on this voltage. Further, our model allows us to quantitatively investigate the effect of ohmic resistance and mass transport limitations, as a correction to the leading order cell voltage. Finally, we explore the effect on cell voltage of other secondary phenomena, such as growth of a discharge-product layer in the pores and reaction-induced volume changes in the electrolyte.

In this talk we consider the limiting behaviour of the strong solution of the Navier--Stokes equation as the viscosity goes to zero, on a three--dimensional region with curved boundary. Under the Navier and kinematic boundary conditions, we show that the solution converges to that of the Euler equation (in suitable topologies). The proof is based on energy estimates and differential--geometric considerations. This is a joint work with Profs. Gui-Qiang Chen and Zhongmin Qian, both at Oxford. 

Multi-agent systems in nature oftentimes exhibit emergent behaviour, i.e. the formation of patterns in the absence of a leader or external stimuli such as light or food sources. We present a non-local two species crossinteraction model with cross-diffusion and explore its long-time behaviour. We observe a rich zoology of behaviours exhibiting phenomena such as mixing and/or segregation of both species and the formation of travelling pulses.

In this presentation, we give an overview of the numerical methods used in commercial oil and gas reservoir simulation. The models are described by flow through porous media and are solved using a series of nested numerical methods. Most of the computational effort resides in solving large linear systems resulting from Newton iterations. Therefore, we will go in greater detail about the iterative linear solvers and preconditioning techniques.

Note: This talk will cover similar topics to the InFoMM group meeting talks on Friday 28th April, but I will discuss more mathematical details for this JAMS talk.

In this talk, I will talk about my current approach to model customer movements and in particular congestion inside supermarkets using queuing networks. As the research question for my project is ‘How should one design supermarkets to minimize congestion?’, I will then talk about my current progress in understanding how the network structure can affect this dynamics.