Past Industrial and Applied Mathematics Seminar

The organized movement of intracellular material is part of the functioning of cells and the development of organisms. These flows can arise from the action of molecular machines on the flexible, and often transitory, scaffoldings of the cell. Understanding phenomena in this realm has necessitated the development of new simulation tools, and of new coarse-grained mathematical models to analyze and simulate. In that context, I'll discuss how a symmetry-breaking "swirling" instability of a motor-laden cytoskeleton may be an important part of the development of an oocyte, modeling active material in the spindle, and what models of active, immersed polymers tell us about chromatin dynamics in the nucleus.

The organized movement of intracellular material is part of the functioning of cells and the development of organisms. These flows can arise from the action of molecular machines on the flexible, and often transitory, scaffoldings of the cell. Understanding phenomena in this realm has necessitated the development of new simulation tools, and of new coarse-grained mathematical models to analyze and simulate. In that context, I'll discuss how a symmetry-breaking "swirling" instability of a motor-laden cytoskeleton may be an important part of the development of an oocyte, modeling active material in the spindle, and what models of active, immersed polymers tell us about chromatin dynamics in the nucleus.

Some proteins (in their folded form) are classified as being knotted.

The function of the knotting is mysterious since knotting seemingly

would make the folding process unnecessarily complicated.  To

function, proteins need to fold quickly and reproducibly, and

misfolding can have catastrophic results.  For example, Mad Cow

disease and the human analog, Creutzfeldt-Jakob disease, come from

misfolded proteins.

Traditionally, knotting is only defined for closed curves, where the

topology is trapped in the loop.  However, proteins have free ends, as

well as most of the objects that humans consider as being knotted

(like shoelaces and strings of lights).  Defining knotting in open

curves is tricky and ambiguous.  We consider some definitions of

knotting in open curves and see how one of these definitions is used

to characterize the knotting in proteins.

In this talk, I will discuss a wide range of mechanical systems,
including Hoberman’s sphere, Euler’s disk, a sliding cylinder, the
Dynabee, BB-8, and Littlewood’s hoop, and the research they inspired.
Studies of the dynamics of the cylinder ultimately led to a startup
company while studying Euler’s disk led to sponsored research with a
well-known motorcycle company.

This talk is primarily based on research performed with a number of
former students over the past three decades. including Prithvi Akella,
Antonio Bronars, Christopher Daily-Diamond, Evan Hemingway, Theresa
Honein, Patrick Kessler, Nathaniel Goldberg, Christine Gregg, Alyssa
Novelia, and Peter Varadi over the past three decades.

Synchronized activity of neurons is important for many aspects of brain function. Synchronization is affected by both network-level parameters, such as connectivity between neurons, and neuron-level parameters, such as firing rate. Many of these parameters are not static but may vary slowly in time. In this talk we focus on neuron-level parameters. Our work centres on the neurotransmitter acetylcholine, which has been shown to modulate the firing properties of several types of neurons through its affect on potassium currents such as the muscarine-sensitive M-current.  In the brain, levels of acetylcholine change with activity.  For example, acetylcholine is higher during waking and REM sleep and lower during slow wave sleep. We will show how the M-current affects the bifurcation structure of a generic conductance-based neural model and how this determines synchronization properties of the model.  We then use phase-model analysis to study the effect of a slowly varying M-current on synchronization.  This is joint work with Victoria Booth, Xueying Wang and Isam Al-Darbasah

Understanding the motion of small bodies at a fluid interface has relevance to a range of natural systems and technological applications. In this talk, we discuss two systems where capillarity and fluid inertia govern the dynamics of millimetric particles at a fluid interface.

In the first part, we present a study of superhydrophobic spheres impacting a quiescent water bath.  Under certain conditions particles may rebound completely from the interface - an outcome we characterize in detail through a synthesis of experiments, modeling, and direct numerical simulation.  In the second half, we introduce a system wherein millimetric disks trapped at a fluid interface are vertically oscillated and spontaneously self-propel.  Such "capillary surfers" interact with each other via their collective wavefield and self-assemble into a myriad of cooperative dynamic states.  Our experimental observations are well captured by a first theoretical model for their dynamics, laying the foundation for future investigations of this highly tunable active system.

Eva Antonopoulou

Surfactants in drop-on-demand inkjet printing

The rapid development of new applications for inkjet printing and increasing complexity of the inks has created a demand for in silico optimisation of the ink jetting performance. Surfactants are often added to aqueous inks to modify the surface tension. However, the time-scales for drop formation in inkjet printing are short compared to the time-scales of the surfactant diffusion resulting a non-uniform surfactant distribution along the interface leading to surface tension gradients. We present both experiments and numerical simulations of inkjet break-up and drop formation in the presence of surfactants investigating both the surfactant transport on the interface and the influence of Marangoni forces on break-up dynamics. The numerical simulations were conducted using a modified version of the Lagrangian finite element developed by our previous work by including the solution for the transport equation for the surfactants over the free surface. During the initial phase of a “pull-push-pull” drive waveform, surfactants are concentrated at the front of the main drop with the trailing ligament being almost surfactant free. The resulting Marangoni stresses act to delay and can even prevent the break-off of the main drop from the ligament. We also examine and present some initial results on the effects of surfactants on the shape oscillations  of the main drop. Although there is little change to the oscillation frequency, the presence of surfactants significantly increases the rate of decay due to the rigidification of the surface, by modifying the internal flow within the droplet and enhancing the viscous dissipation.

Hadrien Oliveri

An optic ray theory for nerve durotaxis

During the development of the nervous system, neurons extend bundles of axons that grow and meet other neurons to form the neuronal network. Robust guidance mechanisms are needed for these bundles to migrate and reach their functional target. Directional information depends on external cues such as chemical or mechanical gradients. Unlike chemotaxis that has been extensively studied, the role and mechanism of durotaxis, the directed response to variations in substrate rigidity, remain unclear. We model bundle migration and guidance by rigidity gradients by using the theory of morphoelastic rods. We show that at a rigidity interface, the motion of axon bundles follows a simple behavior analogous to optic ray theory and obeys Snell’s law for refraction and reflection. We use this powerful analogy to demonstrate that axons can be guided by the equivalent of optical lenses and fibers created by regions of different stiffnesses.

Soft elastic interfaces can strongly deform under the influence of external forces, and can even exhibit elastic singularities. Here we discuss two cases where such singularities occur. First, we describe surface creases that form under compression (or swelling) of an elastic medium. Second, we consider the elastocapillary ridges that form when a soft substrate is wetted by a liquid drop. Analytical descriptions are presented and compared to experiments. We reveal that, like for liquid interfaces, the surface tension of the solid is a key factor in shaping the surface, and determines the nature of the singularity.

The Wilson–Cowan population model of neural activity has greatly influenced our understanding of the mechanisms for the generation of brain rhythms and the emergence of structured brain activity. As well as the many insights that have been obtained from its mathematical analysis, it is now widely used in the computational neuroscience community for building large scale in silico brain networks that can incorporate the increasing amount of knowledge from the Human Connectome Project. In this talk, I will introduce a new neural population model in the spirit of that originally developed by Wilson and Cowan, albeit with the added advantage that it can account for the phenomena of event related synchronisation and de-synchronisation. This derived mean field model provides a dynamic description for the evolution of synchrony, as measured by the Kuramoto order parameter, in a large population of quadratic integrate-and-fire model neurons. As in the original Wilson–Cowan framework, the population firing rate is at the heart of our new model; however, in a significant departure from the sigmoidal firing rate function approach, the population firing rate is now obtained as a real-valued function of the complex valued population synchrony measure. To highlight the usefulness of this next generation Wilson–Cowan style model I will show how it can be deployed in a number of neurobiological contexts, providing understanding of the changes in power-spectra observed in EEG/MEG neuroimaging studies of motor-cortex during movement, insights into patterns of functional-connectivity observed during rest and their disruption by transcranial magnetic stimulation, and to describe wave propagation across cortex.

Controlling film flows has long been a central target for fluid dynamicists due to its numerous applications, in fields from heat exchangers to biochemical recovery, to semiconductor manufacture. However, despite its significance in the literature, most analyses have focussed on the “forward” problem: what effect a given control has on the flow. Often these problems are already complex, incorporating the - generally multiphysical - interplay of hydrodynamic phenomena with the mechanism of control. Indeed, many systems still defy meaningful agreement between models and experiments.
 
The inverse problem - determining a suitable control scheme for producing a specified flow - is considerably harder, and much more computationally expensive (often involving thousands of calculations of the forward problem). Performing such calculations for the full Navier-Stokes problem is generally prohibitive.

We examine the use of electric fields as a control mechanism. Solving the forward problem involves deriving a low-order model that turns out to be accurate even deep into the shortwave regime. We show that the weakly-nonlinear problem is Kuramoto-Sivashinsky-like, allowing for greater analytical traction. The fully nonlinear problem can be solved numerically via the use of a rapid solver, enabling solution of both the forward and adjoint problems on sub-second timescales, allowing for both terminal and regulation optimal control studies to be implemented. Finally, we examine the feasibility of controlling direct numerical simulations using these techniques.

Noise is generated in an aerodynamic setting when flow turbulence encounters a structural edge, such as at the sharp trailing edge of an aerofoil. The generation of this noise is unavoidable, however this talk addresses various ways in which it may be mitigated through altering the design of the edge. The alterations are inspired by natural silent fliers: owls. A short review of how trailing-edge noise is modelled will be given, followed by a discussion of two independent adaptations; serrations, and porosity. The mathematical impacts of the adaptations to the basic trailing-edge model will be presented, along with the physical implications they have on noise generation and control.

The propagation of a deformable air finger or bubble into a fluid-filled channel with an imposed pressure gradient was first studied by Saffman and Taylor. Assuming large aspect ratio channels, the flow can be depth-averaged and the free-boundary problem for steady propagation solved by conformal mapping. Famously, at zero surface tension, fingers of any width may exist, but the inclusion of vanishingly small surface tension selects symmetric fingers of discrete finger widths. At finite surface tension, Vanden-Broeck later showed that other families of 'exotic' states exist, but these states are all linearly unstable.

In this talk, I will discuss the related problem of air bubble propagation into rigid channels with axially-uniform, but non-rectangular, cross-sections. By including a centred constriction in the channel, multiple modes of propagation can be stabilised, including symmetric, asymmetric and oscillatory states, with a correspondingly rich bifurcation structure. These phenomena can be predicted via depth-averaged modelling, and also observed in our experiments, with quantitative agreement between the two in appropriate parameter regimes. This agreement provides insight into the physical mechanisms underlying the observed behaviour. I will outline our efforts to understand how the system dynamics is affected by the presence of nearby unstable solution branches acting as edge states. Finally, I will discuss how feedback control and control-based continuation could be used for direct experimental observation of stable or unstable modes.

Biological systems provide an inspiration for creating a new paradigm
for materials synthesis. What would it take to enable inanimate material
to acquire the properties of living things? A key difference between
living and synthetic materials is that the former are programmed to
behave as they do, through interactions, energy consumption and so
forth. The nature of the program is the result of billions of years of
evolution. Understanding and emulating this program in materials that
are synthesizable in the lab is a grand challenge. At its core is an
optimization problem: how do we choose the properties of material
components that we can create in the lab to carry out complex reactions?
I will discuss our (not-yet-terribly-successful efforts)  to date to
address this problem, by designing both equiliibrium and kinetic 
properties of materials, using a combination of statistical mechanics,
kinetic modeling and ideas from machine learning.

The pandemic has had a deleterious influence on the Hollywood film
industry.  Fortunately,  however, the thin film industry continues to
flourish.  A host of effects are responsible for confined liquids
exhibiting properties that differ from their bulk counterparts. For
example, the dominant polarization and surface forces across a layered
system can control the material behavior on length scales vastly larger
than the film thickness.  This basic class of phenomena, wherein
volume-volume interactions create large pressures, are at play in,
amongst many other settings, wetting, biomaterials, ceramics, colloids,
and tribology.  When the films so created involve phase change and are
present in disequilibrium, the forces can be so large that they destroy
the setting that allowed them to form in the first place. I will
describe the connection between such films in a semi-traditional wetting
dynamics geometry and active brownian dynamics.  I then explore their
power to explain a wide range of processes from materials- to astro- to
geo-science.

We study the evolution of solid surfaces and pattern formation by
surface diffusion. Phase field models with degenerate mobilities are
frequently used to model such phenomena, and are validated by
investigating their sharp interface limits. We demonstrate by a careful
asymptotic analysis involving the matching of exponential terms that a
certain combination of degenerate mobility and a double well potential
leads to a combination of both surface and non-linear bulk diffusion to
leading order. If time permits, we will discuss implications and extensions.

Simple mathematical models have had remarkable successes in biology, framing how we understand a host of mechanisms and processes. However, with the advent of a host of new experimental technologies, the last ten years has seen an explosion in the amount and types of quantitative data now being generated. This sets a new challenge for the field – to develop, calibrate and analyse new, biologically realistic models to interpret these data. In this talk I will showcase how quantitative comparisons between models and data can help tease apart subtle details of biological mechanisms, as well as present some steps we have taken to tackle the mathematical challenges in developing models that are both identifiable and can be efficiently calibrated to quantitative data.

This talk will be three short stories on the general theme of elastic
instabilities in soft solids. First I will discuss the inflation of a
cylindrical cavity through a bulk soft solid, and show that such a
channel ultimately becomes unstable to a finite wavelength peristaltic
undulation. Secondly, I will introduce the elastic Rayleigh Plateau
instability, and explain that it is simply 1-D phase separation, much
like the inflationary instability of a cylindrical party balloon. I will
then construct a universal near-critical analytic solution for such 1-D
elastic instabilities, that is strongly reminiscent of the
Ginzberg-Landau theory of magnetism. Thirdly, and finally, I will
discuss pattern formation in layer-substrate buckling under equi-biaxial
compression, and argue, on symmetry grounds, that such buckling will
inevitably produce patterns of hexagonal dents near threshold.

In March 2020, as COVID-19 cases started to surge for the first time in the UK, a team spanning UCL engineers, University College London Hospital (UCLH) intensivists and Mercedes Formula 1 came together to design, manufacture and deploy non-invasive breathing aids for COVID-19 patients. We reverse engineered and an off-patent CPAP (continuous positive airways pressure) device, the Philips WhisperFlow, and changed its design to minimise its oxygen utilisation (given that hospital oxygen supplies are under extreme demand). The UCL-Ventura received regulatory approvals from the MHRA within 10 days, and Mercedes HPP manufactured 10,000 devices by mid-April. UCL-Ventura CPAPs are now in use in over 120 NHS hospitals.

In response to international need, the team released all blueprints open source to enable local manufacture in other countries, alongside a support package spanning technical, manufacturing, clinical and regulatory components. The designs have been downloaded 1900 times across 105 countries, and around 20 teams are now manufacturing at scale and deploying in local hospitals. We have also worked closely with NGOs, on a non-profit basis, to deliver devices directly to countries with urgent need, including Palestine, Uganda and South Africa.

Suspensions are composed of mixtures of particles and fluid and are
ubiquitous in industrial processes (e.g. waste disposal, concrete,
drilling muds, metalworking chip transport, and food processing) and in
natural phenomena (e.g. flows of slurries, debris, and lava). The
present talk focusses on the rheology of concentrated suspensions of
non-colloidal particles. It addresses the classical shear viscosity of
suspensions but also non-Newtonian behaviour such as normal-stress
differences and shear-induced migration. The rheology of dense
suspensions can be tackled via a diversity of approaches that are
introduced. In particular, the rheometry of suspensions can be
undertaken at an imposed volume fraction but also at imposed values of
particle normal stress, which is particularly well suited to yield
examination of the rheology close to the jamming transition. The
influences of particle roughness and shape are discussed.

Contagion models on networks can be used to describe the spread of information, rumours, opinions, and (more topically) diseases through a population. In the simplest contagion models, each node represents an individual that can be in one of a number of states (e.g. Susceptible, Infected, or Recovered), and the states of the nodes evolve according to specified rules. Even with simple Markovian models of transmission and recovery, it can be difficult to compute the dynamics of contagion on large networks: running simulations can be slow, and the system of master equations is typically too large to be tractable.

 One approach to approximating contagion dynamics is to assume that each node state is independent of the neighbouring node states; this leads to a system of ODEs for the node state probabilities (the “first-order approximation”) that always overestimates the speed of infection spread. This approach can be made more sophisticated by introducing pair approximations or higher-order moment closures, but this dramatically increases the size of the system and slows computations. In this talk, I will present some alternative node-based approximations for contagion dynamics. The first of these is exact on trees but will always underestimate the speed of infection spread on a network with loops. I will show how this can be combined with the classic first-order node-based approximation to obtain a node-based approximation that has similar accuracy to the pair approximation, but which is considerably faster to solve.

Kirigami, the relatively unheralded cousin of origami, is the art of cutting paper to articulate and deploy it as a whole. By varying the number, size, orientation and coordination of the cuts, artists have used their imagination and intuition to create remarkable sculptures in 2 and 3 dimensions. I will describe some of our attempts to quantify the inverse problem that artists routinely solve, combining elementary mathematical ideas, with computations and physical models. 

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Daniel M. Anderson

Department of Mathematical Sciences, George Mason University

Applied and Computational Mathematics Division, NIST

Binary and multicomponent alloy solidification occurs in many industrial materials science applications as well as in geophysical systems such as sea ice. These processes involve heat and mass transfer coupled with phase transformation dynamics and can involve the formation of mixed phase regions known as mushy layers.  The understanding of transport mechanisms within mushy layers has important consequences for how these regions interact with the surrounding liquid and solid regions.  Through linear stability analyses and numerical calculations of mathematical models, convective instabilities that occur in solidifying ternary alloys will be explored.  Novel fluid dynamical phenomena that are predicted for these systems will be discussed.

Amy Kent

Multiscale Mathematical Models for Tendon Tissue Engineering

Tendon tissue engineering aims to grow functional tendon in vitro. In bioreactor chambers, cells growing on a solid scaffold are fed with nutrient-rich media and stimulated by mechanical loads. The Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences is developing a Humanoid Robotic Bioreactor, where cells grow on a flexible fibrous scaffold actuated by a robotic shoulder. Tendon cells modulate their behaviour in response to shear stresses - experimentally, it is desirable to design robotic loading regimes that mimic physiological loads. The shear stresses are generated by flowing cell media; this flow induces deformation of the scaffold which in turn modulates the flow. Here, we capture this fluid-structure interaction using a homogenised model of fluid flow and scaffold deformation in a simplified bioreactor geometry. The homogenised model admits analytical solutions for a broad class of forces representing robotic loading. Given the solution to the microscale problem, we can determine microscale shear stresses at any point in the domain. In this presentation, we will outline the model derivation and discuss the experimental implications of model predictions.

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Michael Negus

High-Speed Droplet Impact Onto Deformable Substrates: Analysis And Simulations

The impact of a high-speed droplet onto a substrate is a highly non-linear, multiscale phenomenon and poses a formidable challenge to model. In addition, when the substrate is deformable, such as a spring-suspended plate or an elastic sheet, the fluid-structure interaction introduces an additional layer of complexity. We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behaviour during the early stages of the impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct direct numerical simulations designed to both validate the analytical framework and provide insight into the later times of impact. Through both methods, we are able to observe how the properties of the substrate, such as elasticity, affect the behaviour of the flow. We conclude by showing how these methods are complementary, as a combination of both can lead to a thorough understanding of the droplet impact across timescales.

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Edwina Yeo

Modelling of Magnetically Targeted Stem Cell Delivery

Targeting delivery of stem cells to the site of an injury is a key challenge in regenerative medicine. One possible approach is to inject cells implanted withmagnetic nanoparticles into the blood stream. Cells can then be targeted to the delivery site by an external magnetic field. At the injury site, it is of criticalimportance that the cells do not form an aggregate which could significantly occlude the vessel.We develop a model for the transport of magnetically tagged cells in blood under the action of an external magnetic field. We consider a system of blood and stem cells in a single vessel.  We exploit the small aspect ratio of the vessel to examine the system asymptotically. We consider the system for a range of magnetic field strengths and varying strengths of the diffusion coefficient of the stem cells. We explore the different regimes of the model and determine the optimal conditions for the effective delivery of stem cells while minimising vessel occlusion.

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Helen Zha

Mathematical model of a valve-controlled, gravity driven bioreactor for platelet production

Hospitals sometimes experience shortages of donor blood platelet supplies, motivating research into~\textit{in vitro}~production of platelets. We model a novel platelet bioreactor described in Shepherd et al [1]. The bioreactor consists of an upper channel, a lower channel, and a cell-seeded porous collagen scaffold situated between the two. Flow is driven by gravity, and controlled by valves on the four inlets and outlets. The bioreactor is long relative to its width, a feature which we exploit to derive a lubrication reduction of unsteady Stokes flow coupled to Darcy. As the shear stress experienced by cells influences platelet production, we use our model to quantify the effect of varying pressure head and valve dynamics on shear stress.

[1] Shepherd, J.H., Howard, D., Waller, A.K., Foster, H.R., Mueller, A., Moreau, T., Evans, A.L., Arumugam, M., Chalon, G.B., Vriend, E. and Davidenko, N., 2018. Structurally graduated collagen scaffolds applied to the ex vivo generation of platelets from human pluripotent stem cell-derived megakaryocytes: enhancing production and purity. Biomaterials.

Materials made from a mixture of liquid and solid are, instinctively, very obviously complex. From dilatancy (the reason wet sand becomes dry when you step on it) to extreme shear-thinning (quicksand) or shear-thickening (cornflour oobleck) there is a wide range of behaviours to explain and predict. I'll discuss the seemingly simple case of solid spheres suspended in a Newtonian fluid matrix, which still has plenty of surprises up its sleeve.

Artificial microswimmers are an emerging field of research, attracting
interest as testing beds for physical theories of complex biological
entities, as inspiration for the design of smart materials, and for the
sheer elegance, and often quite counterintuitive phenomena of
experimental nonlinear dynamics.

Self-propelling droplets are among the most simplified swimmer models
imaginable, requiring just three components (oil, water, surfactant). In
this talk, I will show how these inherently stupid objects can make
surprisingly smart decisions based on interactions with microfluidic
structures and self-generated and external chemical fields.

We revisit the tears of wine problem for thin films in
water-ethanol mixtures and present a new model for the climbing
dynamics. The new formulation includes a Marangoni stress balanced by
both the normal and tangential components of gravity as well as surface
tension which lead to distinctly different behavior. The combined
physics can be modeled mathematically by a scalar conservation law with
a nonconvex flux and a fourth order regularization due to the bulk
surface tension. Without the fourth order term, shock solutions must
satisfy an entropy condition - in which characteristics impinge on the
shock from both sides. However, in the case of a nonconvex flux, the
fourth order term is a singular perturbation that allows for the
possibility of undercompressive shocks in which characteristics travel
through the shock. We present computational and experimental evidence
that such shocks can happen in the tears of wine problem, with a
protocol for how to observe this in a real life setting.

There are many examples of thin-film flows in fluid dynamics, and in many cases similarity solutions are possible. In the typical, well-known case the thin-film shape is described by a nonlinear partial differential equation in two independent variables (say x and t), which upon recognition of a similarity variable, reduces the problem to a nonlinear ODE. In this talk I describe work we have done on 1) Marangoni-driven spreading on pre-wetted films, where the thickness of the pre-wetted film affects the dynamics, and 2) the drainage of a film on a vertical substrate of finite width. In the latter case we find experimentally a structure to the film shape near the edge, which is a function of time and two space variables. Analysis of the corresponding thin-film equation shows that there is a similarity solution, collapsing three independent variables to one similarity variable, so that the PDE becomes an ODE. The solution is in excellent agreement with the experimental measurements.

Tissue folding during animal development involves an intricate interplay
of cell shape changes, cell division, cell migration, cell
intercalation, and cell differentiation that obfuscates the underlying
mechanical principles. However, a simpler instance of tissue folding
arises in the green alga Volvox: its spherical embryos turn themselves
inside out at the close of their development. This inversion arises from
cell shape changes only.

In this talk, I will present a model of tissue folding in which these
cell shape changes appear as variations of the intrinsic stretches and
curvatures of an elastic shell. I will show how this model reproduces
Volvox inversion quantitatively, explains mechanically the arrest of
inversion observed in mutants, and reveals the spatio-temporal
regulation of different biological driving processes. I will close with
two examples illustrating the challenges of nonlinearity in tissue
folding: (i) constitutive nonlinearity leading to nonlocal elasticity in
the continuum limit of discrete cell sheet models; (ii) geometric
nonlinearity in large bending deformations of morphoelastic shells.
 

This week Professor Dominic Vella will talk about wrinkling  

In this talk I will provide an overview of recent work on the wrinkling of thin elastic objects. In particular, the focus of the talk will be on answering questions that arise in recent applications that seek not to avoid, but rather, exploit wrinkling. Such applications usually take place far beyond the threshold of instability and so key themes will be the limitations of “standard” instability analysis, as well as what analysis should be performed instead. I will discuss the essential ingredients of this ‘Far-from-Threshold’ analysis, as well as outlining some open questions.  

Abstract

This talk will provide an overview of mathematical modelling applied to the behaviour of ice sheets and their role in the climate system.  I’ll provide some motivation and background, describe simple approaches to modelling the evolution of the ice sheets as a fluid-flow problem, and discuss some particular aspects of the problem that are active areas of current research.  The talk will involve a variety of interesting continuum-mechanical models and approximations that have analogues in other areas of applied mathematics.

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The many facets of community detection on networks 

Community detection, the decomposition of a graph into essential building blocks, has been a core research topic in network science over the past years. Since a precise notion of what consti- tutes a community has remained evasive, community detection algorithms have often been com- pared on benchmark graphs with a particular form of assortative community structure and classified based on the mathematical techniques they employ. However, this comparison can be misleading because apparent similarities in their mathematical machinery can disguise different goals and rea- sons for why we want to employ community detection in the first place. Here we provide a focused review of these different motivations that underpin community detection. This problem-driven classification is useful in applied network science, where it is important to select an appropriate algorithm for the given purpose. Moreover, highlighting the different facets of community detection also delineates the many lines of research and points out open directions and avenues for future research.

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no known cure. Yet a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies. In this talk, I will show that by linking new mathematical theories to recent progress in imaging, we can unravel some of the universal features associated with dementia and, more generally, brain functions. In particular, I will outline interesting mathematical problems and ideas that naturally appear in the process.

Heterogeneity in Space and Time: Novel Dispersion Relations in Morphogenesis

Dr. Andrew Krause

Motivated by recent work with biologists, I will showcase some results on Turing instabilities in complex domains. This is scientifically related to understanding developmental tuning in the whiskers of mice, and in synthetic quorum-sensing patterning of bacteria. Such phenomena are typically modelled using reaction-diffusion systems of morphogens, and one is often interested in emergent spatial and spatiotemporal patterns resulting from instabilities of a homogeneous equilibrium. In comparison to the well-known effects of how advection or manifold structure impacts the modes which may become unstable in such systems, I will present results on instabilities in heterogeneous systems, reaction-diffusion systems on evolving manifolds, as well as layered reaction-diffusion systems. These contexts require novel formulations of classical dispersion relations, and may have applications beyond developmental biology, such as in understanding niche formation for populations of animals in heterogeneous environments. These approaches also help close the vast gap between the simplistic theory of instability-driven pattern formation, and the messy reality of biological development, though there is still much work to be done in concretely demonstrating such a theory's applicability in real biological systems.
 

Cavity flow characteristics and applications to kidney stone removal

Dr. Jessica Williams

Ureteroscopy is a minimally invasive surgical procedure for the removal of kidney stones. A ureteroscope, containing a hollow, cylindrical working channel, is inserted into the patient's kidney. The renal space proximal to the scope tip is irrigated, to clear stone particles and debris, with a saline solution that flows in through the working channel. We consider the fluid dynamics of irrigation fluid within the renal pelvis, resulting from the emerging jet through the working channel and return flow through an access sheath . Representing the renal pelvis as a two-dimensional rectangular cavity, we investigate the effects of flow rate and cavity size on flow structure and subsequent clearance time of debris. Fluid flow is modelled with the steady incompressible Navier-Stokes equations, with an imposed Poiseuille profile at the inlet boundary to model the jet of saline, and zero-stress conditions on the outlets. The resulting flow patterns in the cavity contain multiple vortical structures. We demonstrate the existence of multiple solutions dependent on the Reynolds number of the flow and the aspect ratio of the cavity using complementary numerical simulations and PIV experiments. The clearance of an initial debris cloud is simulated via solutions to an advection-diffusion equation and we characterise the effects of the initial position of the debris cloud within the vortical flow and the Péclet number on clearance time. With only weak diffusion, debris that initiates within closed streamlines can become trapped. We discuss a flow manipulation strategy to extract debris from vortices and decrease washout time.

 

Your brain has its own waterscape: whether you are reading or sleeping, fluid flows around or through the brain tissue and clears waste in the process. These physiological processes are crucial for the well-being of the brain. In spite of their importance we understand them but little. Mathematics and numerics could play a crucial role in gaining new insight. Indeed, medical doctors express an urgent need for modeling of water transport through the brain, to overcome limitations in traditional techniques. Surprisingly little attention has been paid to the numerics of the brain’s waterscape however, and fundamental knowledge is missing. In this talk, I will discuss mathematical models and numerical methods for the brain's waterscape across scales - from viewing the brain as a poroelastic medium at the macroscale and zooming in to studying electrical, chemical and mechanical interactions between brain cells at the microscale.
 

Nonlinear generalizations of the Schrödinger equation are of wide applicability to a range of areas including atomic and optical systems, 
plasma physics and water waves.  In this  talk we revisit some principal excitations in atomic and optical systems (such as Bose-Einstein condensates and photo-refractive crystals), namely dark solitonic fronts in single-component systems, and dark-bright waves in multi-component systems. Upon introducing them and explaining their existence and stability properties in one spatial dimension, we will extend them both in the form of stripes and in that rings in two-dimensions, presenting an alternative (adiabatic-invariant based) formulation of their stability and excitations. We will explore their filamentary dynamics, as well as the states that emerge from their transverse (snaking) instability. Then, we will consider these structures even in three dimensions, in the form of planar, as well as spherical shell wave patterns and generalize our adiabatic invariant formulation there. Finally, time permitting, we will give some glimpses of how some of these dynamical features in 1d and 2d generalize in a multi-orbital, time-dependent quantum setting.

The flow of a thin film down an inclined plane is an important physical phenomenon appearing in many industrial applications, such as coating (where it is desirable to maintain the fluid interface flat) or heat transfer (where a larger interfacial area is beneficial). These applications lead to the need of reliably manipulating the flow in order to obtain a desired interfacial shape. The interface of such thin films can be described by a number of models, each of them exhibiting instabilities for certain parameter regimes. In this talk, I will propose a feedback control methodology based on same-fluid blowing and suction. I use the Kuramoto–Sivashinsky (KS) equation to model interface perturbations and to derive the controls. I will show that one can use a finite number of point-actuated controls based on observations of the interface to stabilise both the flat solution and any chosen nontrivial solution of the KS equation. Furthermore, I will investigate the robustness of the designed controls to uncertain observations and parameter values, and study the effect of the controls across a hierarchy of models for the interface, which include the KS equation, (nonlinear) long-wave models and the full Navier–Stokes equations.

The Navier-Stokes paradigm does not capture thermal fluctuations that drive familiar effects such as Brownian motion and are seen to be key to understanding counter-intuitive phenomena in nanoscale interfacial flows.  On the other hand, molecular simulations naturally account for these fluctuations but are limited to exceptionally short time scales. A framework that incorporates thermal noise is provided by fluctuating hydrodynamics, based on the so-called Landau-Lifshitz-Navier-Stokes equations, and in this talk we shall exploit these equations to gain insight into nanoscale free surface flows.  Particular attention will be given to flows with topological changes, such as the coalescence of drops, breakup of jets and rupture of thin liquid films for which both analytic linear stability results and numerical simulations will be presented and compared to the results of molecular dynamics.

We present analysis and computations of a non-local thin film model developed by Kalogirou et al (2016) for a perturbed two-layer Couette flow when the thickness of the more viscous fluid layer next to the stationary wall is small compared to the thickness of the less viscous fluid. Travelling wave solutions and their stability are determined numerically, and secondary bifurcation points identified in the process. We also determine regions in parameter space where bistability is observed with two branches being linearly stable at the same time. The travelling wave solutions are mathematically justified through a quasi-solution analysis in a neighbourhood of an empirically constructed approximate solution. This relies in part on precise asymptotics of integrals of Airy functions for large wave numbers. The primary bifurcation about the trivial state is shown rigorously to be supercritical, and the dependence of bifurcation points, as a function of Reynolds number R and the primary wavelength 2πν−1/2 of the disturbance, is determined analytically. We also present recent results on time periodic solutions arising from Hoof-Bifurcation of the primary solution branch.

(This work is in collaboration with D. Papageorgiou & E. Oliveira ) 
 

The problem of a bubble moving steadily in a Hele-Shaw cell goes back to Taylor and Saffman in 1959.  It is analogous to the well-known selection problem for Saffman-Taylor fingers in a Hele-Shaw channel.   We apply techniques in exponential asymptotics to study the bubble problem in the limit of vanishing surface tension, confirming previous numerical results, including a previously predicted surface tension scaling law.  Our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane. 

In biological cells, genomic DNA is complexed with proteins, forming so-called chromatin structure, and packed into the nucleus. Not only the nucleotide (A, T, G, C) sequence of DNA but also the 3D structure affects the genomic function. For example, certain regions of DNA are tightly packed with proteins (heterochromatin), which inhibits expression of genes coded there. The structure sometimes changes drastically depending on the state (e.g. cell cycle or developmental stage) of the cell. Hence, the structural dynamics of chromatin is now attracting attention in cell biology and medicine. However, it is difficult to experimentally observe the motion of the entire structure in detail. To combine and interpret data from different modes of observation (such as live imaging and electron micrograph) and predict the behavior, structural models of chromatin are needed. Although we can use molecular dynamics simulation at a microscopic level (~ kilo base-pairs) and for a short time (~ microseconds), we cannot reproduce long-term behavior of the entire nucleus. Mesoscopic models are wanted for that purpose, however hard to develop (there are fundamental difficulties).

In this seminar, I will introduce our recent theoretical/computational studies of chromatin structure, either microscopic (molecular dynamics of DNA or single nucleosomes) or abstract (polymer models and reaction-diffusion processes), toward development of such a mesoscopic model including local "states" of DNA and binding proteins.

References:

T. Kameda, A. Awazu, Y. Togashi, "Histone Tail Dynamics in Partially Disassembled Nucleosomes During Chromatin Remodeling", Front. Mol. Biosci., in press (2019).

Y. Togashi, "Modeling of Nanomachine/Micromachine Crowds: Interplay between the Internal State and Surroundings", J. Phys. Chem. B 123, 1481-1490 (2019).

E. Rolls, Y. Togashi, R. Erban, "Varying the Resolution of the Rouse Model on Temporal and Spatial Scales: Application to Multiscale Modelling of DNA Dynamics", Multiscale Model. Simul. 15, 1672-1693 (2017).

S. Shinkai, T. Nozaki, K. Maeshima, Y. Togashi, "Dynamic Nucleosome Movement Provides Structural Information of Topological Chromatin Domains in Living Human Cells", PLoS Comput. Biol. 12, e1005136 (2016).

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The dynamics of many physical systems often evolve to asymptotic states that exhibit periodic spatial and temporal variations in their properties such as density, temperature, etc. Such regular patterns look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. In contrast, an aperiodic crystal displays long range spatial order but no translational symmetry. 

Recently, quasicrystals which are related to aperiodic crystals have been observed to form in diverse physical systems such as metallic alloys (atomic scale) and dendritic-, star-, and block co-polymers (molecular scale). Such quasicrystals lack the lattice symmetries of regular crystals, yet have discrete Fourier spectra. We look to understand the minimal mechanism which promotes the formation of such quasicrystalline structures using a phase field crystal model. Direct numerical simulations combined with weakly nonlinear analysis highlight the parameter values where the quasicrystals are the global minimum energy state and help determine the phase diagram. 

By locating parameter values where multiple patterned states possess the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a quasicrystal) is present in the background of another (for example, the homogeneous liquid state) in the form of spatially localized dodecagonal (in 2D) and icosahedral (in 3D) quasicrystals. In two dimensions, we compute several families of spatially localized quasicrystals with dodecagonal structure and investigate their properties as a function of the system parameters. The presence of such meta-stable localized quasicrystals is significant as they may affect the dynamics of the crystallisation in soft matter.

The talk will begin with an introduction to the science of what determines the behaviour of a liquid on a on a surface and giving an overview of some of the different theories that can be used to describe the shape and structure of the liquid in the drop. These include microscopic density functional theory (DFT), which describes the liquid structure on the scale of the individual liquid molecules, and mesoscopic thin film equation (PDE) and kinetic Monte-Carlo models. A DFT based method for calculating the binding potential ?(h) for a film of liquid on a solid surface, where h is the thickness of the liquid film, will be presented. The form of ?(h) determines whether or not the liquid wets the surface. Calculating drop profiles using both DFT and also from inputting ?(h) into the mesoscopic theory and comparing quantities such as the contact angle and the shape of the drops, we find good agreement between the two methods, validating the coarse-graining. The talk will conclude with a discussion of some recent work on modelling evaporating drops with applications to inkjet printing.

One of the key unsolved challenges at the interface of physical and life sciences is to formulate comprehensive computational modeling of cells of higher organisms that is based on microscopic molecular principles of chemistry and physics. Towards addressing this problem, we have developed a unique reactive mechanochemical force-field and software, called MEDYAN (Mechanochemical Dynamics of Active Networks: http://medyan.org).  MEDYAN integrates dynamics of multiple mutually interacting phases: 1) a spatially resolved solution phase is treated using a reaction-diffusion master equation; 2) a polymeric gel phase is both chemically reactive and also undergoes complex mechanical deformations; 3) flexible membrane boundaries interact mechanically and chemically with both solution and gel phases.  In this talk, I will first outline our recent progress in simulating multi-micron scale cytosolic/cytoskeletal dynamics at 1000 seconds timescale, and also highlight the outstanding challenges in bringing about the capability for routine molecular modeling of eukaryotic cells. I will also report on MEDYAN’s applications, in particular, on developing a theory of contractility of actomyosin networks and also characterizing dissipation in cytoskeletal dynamics. With regard to the latter, we devised a new algorithm for quantifying dissipation in cytoskeletal dynamics, finding that simulation trajectories of entropy production provide deep insights into structural evolution and self-organization of actin networks, uncovering earthquake-like processes of gradual stress accumulation followed by sudden rupture and subsequent network remodeling.
 

Thin film flows of nematic liquid crystal will be considered, using the Leslie-Ericksen formulation for nematics. Our model can account for variations in substrate anchoring, which may exert a strong influence on patterns that arise in the flow. A number of simulations will be presented using an "in house" code, developed to run on a GPU. Current modeling directions involving flow over interlaced electrodes, so-called "dielectrowetting", will be discussed.

During this seminar, we will present a new mathematical model describing the transport of nitric oxide (NO) in a realistic geometrical representation of the lungs. Nitric oxide (NO) is naturally produced in the bronchial region of the lungs. It is a physiological molecule that has antimicrobial properties and allows the relaxation of muscles. It is well known that the measurement of the molar fraction of NO in the exhaled air, the so-called FeNO, allows a monitoring of asthmatic patients, since the production of this molecule in the lungs is increased in case of inflammation. However, recent clinical studies have shown that the amount of NO in the exhaled air can also be affected by « non-inflammatory » processes, such as the action of a bronchodilator or a respiratory physiotherapy session for a patient with cystic fibrosis. Using our new model, we will highlight the complex interplay between different transport phenomena in the lungs. More specifically, we will show why changes taking place in the deepest part of the lungs are expected to impact the FeNO. This gives a new light on the clinical studies mentioned below, allowing to confer a new role to the NO for the management of various pulmonary pathologies.

Tensors are higher dimensional analogues of matrices; they are used to record data with multiple changing variables. Interpreting tensor data requires finding low rank structure, and the structure depends on the application or context. Often tensors of interest define semi-algebraic sets, given by polynomial equations and inequalities. I'll give a characterization of the set of tensors of real rank two, and answer questions about statistical models using probability tensors and semi-algebraic statistics. I will also describe work on learning a path from its three-dimensional signature tensor. This talk is based on joint work with Guido Montúfar, Max Pfeffer, and Bernd Sturmfels.

When a liquid drop is deposited over a solid surface whose temperature is sufficiently above the boiling point of the liquid, the drop does not experience nucleate boiling but rather levitates over a thin layer of its own vapor. This is known as the Leidenfrost effect. Whilst highly undesirable in certain cooling applications, because of a drastic decrease of the energy transferred between the solid and the evaporating liquid due to poor heat conductivity of the vapor, this effect can be of great interest in many other processes profiting from this absence of contact with the surface that considerably reduces the friction and confers an extreme mobility on the drop. During this presentation, I hope to provide a good vision of some of the knowledge on this subject through some recent studies that we have done. First, I will present a simple fitting-parameter-free theory of the Leidenfrost effect, successfully validated with experiments, covering the full range of stable shapes, i.e., from small quasi-spherical droplets to larger puddles floating on a pocketlike vapor film. Then, I will discuss the end of life of these drops that appear either to explode or to take-off. Finally, I will show that the Leidenfrost effect can also be observed over hot baths of non-volatile liquids. The understanding of the latter situation, compare to the classical Leidenfrost effect on solid substrate, provides new insights on the phenomenon, whether it concerns levitation or its threshold.