Forthcoming events in this series


Thu, 13 Mar 2025

12:00 - 13:00
L3

Some methods for finding vortex equilibria

Robb McDonald
(UCL)
Further Information

Robb McDonald is a Professor in the Department of Mathematics. His research falls into two areas: 

(i) geophysical fluid dynamics, including rotating stratified flows, rotating hydraulics, coastal outflows, geophysical vortices and topographic effects on geophysical flows.
(ii) complex variable methods applied to 2D free-boundary problems. This includes vortex dynamics, Loewner evolution, Hele-Shaw flows and Laplacian growth, industrial coating problems, and pattern formation in nature.

 

Abstract

Determining stationary compact configurations of vorticity described by the 2D Euler equations is a classic problem dating back to the late 19th century. The aim is to find equilibrium distributions of vorticity, in the form  of point vortices, vortex sheets, vortex patches, and hollow vortices. This endeavour has driven the development of mathematical and numerical techniques such as Hamiltonian vortex dynamics and contour dynamics.

In the case of vortex sheets, methods and results are presented for finding rotating equilibria, some in the presence of point vortices. To begin, a numerical approach based on that recently developed by Trefethen, Costa, Baddoo, and others for solving Laplace's equation in the complex plane by series and rational approximation is described. The method successfully reproduces the exact vortex sheet solutions found by O'Neil (2018) and Protas & Sakajo (2020). Some new solutions are found.

The numerical approach suggests an analytical method based on conformal mapping for finding exact closed-form vortex sheet equilibria. Examples are presented.

Finally, new numerical solutions are computed for steady, doubly-connected vortex layers of uniform vorticity surrounding a solid object such that the fluid velocity vanishes on the outer free boundary. While dynamically unrelated, these solutions have mathematical analogy and application to the industrial free boundary problem arising in the dip-coating of objects by a viscous fluid.

 

Thu, 27 Feb 2025

12:00 - 13:00
L2

Coarse-grained models for schooling swimmers in fast flows

Anand Oza
(New Jersey Institute of Technology)
Further Information

Anand Oza is Associate Professor in the Department of Mathematical Sciences as a part of the Complex Flows and Soft Matter (CFSM) Group. He is interested in fluid mechanics and nonlinear dynamics, with applications to soft matter physics and biology. His research utilizes a combination of analytical techniques and numerical simulations, collaborating with experimentalists whenever possible.

Abstract

The beautiful displays exhibited by fish schools and bird flocks have long fascinated scientists, but the role of their complex behavior remains largely unknown. In particular, the influence of hydrodynamic interactions on schooling and flocking has been the subject of debate in the scientific literature. I will present a model for flapping wings that interact hydrodynamically in an inviscid fluid, wherein each wing is represented as a plate that executes a prescribed time-periodic kinematics. The model generalizes and extends thin-airfoil theory by assuming that the flapping amplitude is small, and permits consideration of multiple wings through the use of conformal maps and multiply-connected function theory. We find that the model predictions agree well with experimental data on freely-translating, flapping wings in a water tank. The results are then used to motivate a reduced-order model for the temporally nonlocal interactions between schooling wings, which consists of a system of nonlinear delay-differential equations. We obtain a PDE as the mean-field limit of these equations, which we find supports traveling wave solutions. Generally, our results indicate how hydrodynamics may mediate schooling and flocking behavior in biological contexts.

 

Thu, 20 Feb 2025

12:00 - 13:00
L3

Advanced Effective Models in Elasticity

Claire Lestringant
(Sorbonne University)
Further Information

Dr Claire Lestringant explores new models for understanding the mechanics of thin structures under large deformations, used for example to understand morphogenesis in biological systems or for the design of multi-stable, reconfigurable space structures. She received a PhD in Mechanics from Université Pierre et Marie Curie in 2017 and worked as a post-doc in D. Kochmann’s group at ETH Zurich in Switzerland.

Abstract

I will discuss two classes of effective, macroscopic models in elasticity: (i) 1D models applicable to thin structures, and (ii) homogenized 2D or 3D continua applicable to materials with a periodic microstructure. In both systems, the separation of scales calls for the definition of macroscopic models that slave fine-scale fluctuations to an effective, macroscopic deformation field. I will show how such models can be established in a systematic and rigorous way based on a two-scale expansion that accounts for nonlinear and higher-order (i.e. deformation gradient) effects. I will further demonstrate that the resulting models accurately predict nonlinear effects, finite size effects and localization for a set of examples. Finally, I will discuss two challenges that arise when solving these effective models: (1) missed boundary layer effects and (2) negative stiffness associated with higher-order terms.

Thu, 13 Feb 2025

12:00 - 13:00
L3

Various

Various Speakers from OCIAM Year 2 Graduates
(Mathematical Institute)
Thu, 13 Feb 2025
12:00
L3

OCIAM TBC

OCIAM TBC
Thu, 06 Feb 2025

12:00 - 13:00
L3

Modelling flying formations and vortex ring motions

Christiana Mavroyiakoumou
( Courant Institute of Mathematical Sciences)
Further Information

Christiana is an Assistant Professor at the Courant Institute of Mathematical Sciences (New York University) working in the Applied Math Lab, primarily with Leif Ristroph and Jun Zhang. Her interests are in using modeling, numerical simulations, and experiments to study fluid dynamical problems, with an emphasis on fluid-structure interactions.

Currently Christiana is working on understanding the role of flow interactions in flying bird formations and the hydrodynamics of swimming fish.

Abstract

We consider two problems in fluid dynamics: the collective locomotion of flying animals and the interaction of vortex rings with fluid interfaces. First, we present a model of formation flight, viewing the group as a material whose properties arise from the flow-mediated interactions among its members. This aerodynamic model explains how flapping flyers produce vortex wakes and how they are influenced by the wakes of others. Long in-line arrays show that the group behaves as a soft, excitable "crystal" with regularly ordered member "atoms" whose positioning is susceptible to deformations and dynamical instabilities. Second, we delve into the phenomenon of vortex ring reflections at water-air interfaces. Experimental observations reveal reflections analogous to total internal reflection of a light beam. We present a vortex-pair--vortex-sheet model to simulate this phenomenon, offering insights into the fundamental interactions of vortex rings with free surfaces.

Thu, 30 Jan 2025

12:00 - 13:00
L3

Spontaneous shape transformations of active surfaces

Alexander Mietke
(Department of Physics)
Further Information

Alexander Mietke is a theoretical physicist working on active and living matter. He frequently collaborates with experimentalists who study processes at the cell, tissue and organism scale to identify minimal physical principles that guide these processes. This often inspires new theoretical work on topics in non-equilibrium soft matter physics, more broadly in the self-organization of mechanical and chemical patterns in active matter, the emergent shape dynamics of membranes and active surfaces, liquid crystals in complex geometries, chirality in active systems, as well as in developing coarse-graining and inference approaches that are directly applicable to experimental data. 

Abstract

Biological matter has the fascinating ability to autonomously generate material deformations via intrinsic active forces, where the latter are often present within effectively two-dimensional structures. The dynamics of such “active surfaces” inevitably entails a complex, self-organized interplay between geometry of a surface and its mechanical interactions with the surrounding. The impact of these factors on the self-organization capacity of surfaces made of an active material, and how related effects are exploited in biological systems, is largely unknown.

In this talk, I will first discuss general numerical challenges in analysing self-organising active surfaces and the bifurcation structure of emergent shape spaces. I will then focus on active surfaces with broken up-down symmetry, of which the eukaryotic cell cortex and epithelial tissues are highly abundant biological examples. In such surfaces, a natural interplay arises between active stresses and surface curvature. We demonstrate that this interplay leads to a comprehensive library of spontaneous shape transformations that resemble stereotypical morphogenetic processes. These include cell-division-like invaginations and the autonomous formation of tubular surfaces of arbitrary length, both of which robustly overcome well-known shape instabilities that would arise in analogue passive systems.

 

 

Thu, 23 Jan 2025

12:00 - 13:00
L3

Optimal design of odd active solids

Anton Souslov
(University of Cambridge)
Further Information

Anton Souslov is an Associate Professor of Theoretical Statistical Physics working on the theory of soft materials, including mechanical metamaterials, active matter, topological states, and polymer physics.

Abstract

Active solids consume energy to allow for actuation and shape change not possible in equilibrium. I will first introduce active solids in comparison with their active fluid counterparts. I will then focus on active solids composed of non-reciprocal springs and show how so-called odd elastic moduli arise in these materials. Odd active solids have counter-intuitive elastic properties and require new design principles for optimal response. For example, in floppy lattices, zero modes couple to microscopic non-reciprocity, which destroys odd moduli entirely in a phenomenon reminiscent of rigidity percolation. Instead, an optimal odd lattice will be sufficiently soft to activate elastic deformations, but not too soft. These results provide a theoretical underpinning for recent experiments and point to the design of novel soft machines.

 

 

Thu, 05 Dec 2024

12:00 - 13:00
L3

Chaotic flows in polymer solutions: what’s new?

Prof. Rich Kerswell
(University of Cambridge)
Further Information

Rich Kerswell is a professor in the Department of Applied Mathematics and Theoretical Physics (DAMTP) at the University of Cambridge. His research focuses on fluid dynamics, particularly in the transition to turbulence, geophysical fluid flows, and nonlinear dynamics. Kerswell is known for studying how simple fluid systems can exhibit complex, chaotic behavior and has contributed to understanding turbulence's onset and sustainment in various contexts, including pipes and planetary atmospheres. His work integrates mathematical modeling, theoretical analysis, and computational simulations to explore instabilities and the fundamental mechanisms governing fluid behavior in nature and industry.

Abstract

It is well known that adding even small amounts of  long chain polymers (e.g. few parts per million) to Newtonian solvents can drastically change the flow behaviour by introducing elasticity. In particular,  two decades ago, experiments in curved geometries  demonstrated  that polymer flows can be  chaotic even at vanishingly small Reynolds numbers. The situation in `straight’ flows  such as pressure-driven flow down a channel is less clear  and hence an area of current focus. I will discuss recent progress.

Thu, 21 Nov 2024

12:00 - 13:00
L3

Tension-induced giant actuation in elastic sheets (Marc Sune) Deciphering Alzheimer's Disease: A Modelling Framework for In Silico Drug Trials (Georgia Brennan)

Dr Marc Suñé & Dr Georgia Brennan
(Mathematical Institute)
Abstract

Tension-induced giant actuation in elastic sheets

Dr. Marc Suñé

Buckling is normally associated with a compressive load applied to a slender structure; from railway tracks in extreme heat to microtubules in cytoplasm, axial compression is relieved by out-of-plane buckling. However, recent studies have demonstrated that tension applied to structured thin sheets leads to transverse motion that may be harnessed for novel applications, such as kirigami grippers, multi-stable `groovy-sheets', and elastic ribbed sheets that close into tubes. Qualitatively similar behaviour has also been observed in simulations of thermalized graphene sheets, where clamping along one edge leads to tilting in the transverse direction. I will discuss how this counter-intuitive behaviour is, in fact, generic for thin sheets that have a relatively low stretching modulus compared to the bending modulus, which allows `giant actuation' with moderate strain.

Thu, 07 Nov 2024

12:00 - 13:00
L3

Translational Applications of Mathematical and Computational Modeling in Respiratory and Critical Care Medicine

Prof. Samir Ghadiali
((Imperial College)
Further Information

Samir Ghadiali is Professor and Chair/Head of the Department of Biomedical Engineering at the Ohio State University (OSU) and a Professor of Pulmonary and Critical Care Medicine at the OSU Wexner Medical Center. Dr. Ghadiali is a Fellow of the American Institute of Medical and Biological Engineering, the Biomedical Engineering Society and is a Parker B. Francis Fellow in Pulmonary Research. He is a member of the Davis Heart & Lung Research Institute and the Biophysics Graduate Program at OSU, and his internationally recognized research program uses biomedical engineering tools to develop novel diagnostic platforms and drug/gene therapies for cardiovascular and respiratory disorders. His research has been funded by the National Science Foundation, National Institutes of Health, the American Heart Association, and the United States Department of Defense and he has mentored over 35 pre-doctoral and post-doctoral trainees who have gone on to successful academic, industrial and research careers. 

Abstract

The global COVID19 pandemic has highlighted the lethality and morbidity associated with infectious respiratory diseases. These diseases can lead to devastating syndrome known as the acute respiratory distress syndrome (ARDS) where bacterial/viral infections cause excessive lung inflammation, pulmonary edema, and severe hypoxemia (shortness of breath). Although ARDS patients require artificial mechanical ventilation, the complex biofluid and biomechanical forces generated by the ventilator exacerbates lung injury leading to high mortality. My group has used mathematical and computational modeling to both characterize the complex mechanics of lung injury during ventilation and to identify novel ways to prevent injury at the cellular level. We have used in-vitro and in-vivo studies to validate our mathematical predictions and have used engineering tools to understand the biological consequences of the mechanical forces generated during ventilation. In this talk I will specifically describe how our mathematical/computational approach has led to novel cytoskeletal based therapies and how coupling mathematics and molecular biology has led to the discovery of a gene regulatory mechanisms that can minimize ventilation induced lung injury. I will also describe how we are currently using nanotechnology and gene/drug delivery systems to enhance the lung’s native regulatory responses and thereby prevent lung injury during ARDS.

Thu, 31 Oct 2024

12:00 - 13:00
L3

Volcanic fissure localisation and lava delta formation: Modelling of volcanic flows undergoing rheological evolution

Jesse Taylor-West
(University of Bristol)
Abstract
In this talk, I will present two volcanologically motivated modelling problems.  In the first, I will detail how thermoviscous localisation of volcanic eruptions is influenced by the irregular geometry of natural volcanic fissures. Fissure eruptions typically start with the opening of a linear fissure that erupts along its entire length, following which activity localises to one or more isolated vents within a few hours or days. Previous work has proposed that localisation can arise through a thermoviscous fingering instability driven by the strongly temperature dependent viscosity of the rising magma. I will show that, even for relatively modest variations of the fissure width, a non-planar geometry supports strongly localised steady states, in which the wider parts of the fissure host faster, hotter flow, and the narrower parts of the fissure host slower, cooler flow. This geometrically-driven localisation is different from, and typically more potent than, the thermoviscous fingering localisation observed in planar geometries.  
 
The second problem concerns lava delta formation. A lava delta arises when a volcanic lava flow enters a body of water, extending the pre-eruption shoreline via the creation of new, flat land. A combination of cooling induced rheological changes and the reduction in gravitational driving forces controls the morphology and evolution of the delta. I will present shallow-layer continuum models for this process, highlighting how different modes of delta formation manifest in different late-time behaviours.
Thu, 24 Oct 2024

12:00 - 13:00
L3

Effective elasticity and dynamics of helical filaments under distributed loads

Michael Gomez
(Kings College London)
Abstract

Slender elastic filaments with intrinsic helical geometry are encountered in a wide range of physical and biological settings, ranging from coil springs in engineering to bacteria flagellar filaments. The equilibrium configurations of helical filaments under a variety of loading types have been well studied in the framework of the Kirchhoff rod equations. These equations are geometrically nonlinear and so can account for large, global displacements of the rod. This geometric nonlinearity also makes a mathematical analysis of the rod equations extremely difficult, so that much is still unknown about the dynamic behaviour of helical rods under external loading.

An important class of simplified models consists of 'equivalent-column' theories. These model the helical filament as a naturally-straight beam (aligned with the helix axis) for which the extensional and torsional deformations are coupled. Such theories have long been used in engineering to describe the free vibrations of helical coil springs, though their validity remains unclear, particularly when distributed forces and moments are present. In this talk, we show how such an effective theory can be derived systematically from the Kirchhoff rod equations using the method of multiple scales. Importantly, our analysis is asymptotically exact in the small-wavelength limit and can account for large, unsteady displacements. We then illustrate our theory with two loading scenarios: (i) a heavy helical rod deforming under its own weight; and (ii) axial rotation (twirling) in viscous fluid, which may be considered as a simple model for a bacteria flagellar filament. More broadly, our analysis provides a framework to develop reduced models of helical rods in a wide variety of physical and biological settings, as well as yielding analytical insight into their tensile instabilities.

Thu, 17 Oct 2024

12:00 - 13:00
L3

Microswimmer motility and natural robustness in pattern formation: the emergence and explanation of non-standard multiscale phenomena

Mohit Dalwadi
(Mathematical Institute)
Abstract
In this talk I use applied mathematics to understand emergent multiscale phenomena arising in two fundamental problems in fluids and biology.
 
In the first part, I discuss an overarching question in developmental biology: how is it that cells are able to decode spatio-temporally varying signals into functionally robust patterns in the presence of confounding effects caused by unpredictable or heterogeneous environments? This is linked to the general idea first explored by Alan Turing in the 1950s. I present a general theory of pattern formation in the presence of spatio-temporal input variations, and use multiscale mathematics to show how biological systems can generate non-standard dynamic robustness for 'free' over physiologically relevant timescales. This work also has applications in pattern formation more generally.
 
In the second part, I investigate how the rapid motion of 3D microswimmers affects their emergent trajectories in shear flow. This is an active version of the classic fluid mechanics result of Jeffery's orbits for inert spheroids, first explored by George Jeffery in the 1920s. I show that the rapid short-scale motion exhibited by many microswimmers can have a significant effect on longer-scale trajectories, despite the common neglect of this motion in some mathematical models, and how to systematically incorporate this effect into modified versions of Jeffery's original equations.
Thu, 13 Jun 2024

12:00 - 13:00
L3

The mechanics of physical knots: from shoelaces to surgical sutures

Pedro M. Reis
(EPFL)
Further Information

 

Pedro M. Reis

Flexible Structures Laboratory, 

Institute of Mechanical Engineering,

Ecole Polytechnique Fédérale de Lausanne (EPFL), 

Pedro Miguel Reis is a Professor of Mechanical Engineering at the École Polytechnique Fédérale de Lausanne (EPFL) in Switzerland. Prof. Reis received a B.Sc. in Physics from the University of Manchester, UK (1999), a Certificate of Advanced Studies in Mathematics (Part III Maths) from St. John’s College and DAMTP, University of Cambridge (2000), and a Ph.D. in physics from the University of Manchester (2004). He was a postdoc at the City College of New York (2004-2005) and at the CNRS/ESPCI in Paris (2005-2007). He joined MIT in 2007 as an Instructor in Applied Mathematics. In 2010, he moved to MIT’s School of Engineering, with dual appointments in Mechanical Engineering and Civil & Environmental Engineering, first as the Esther and Harold E. Edgerton Assistant Professor and, after 2014, as Gilbert W. Winslow Associate Professor. In October 2013, the Popular Science magazine named Prof. Reis to its 2013 “Brilliant 10” list of young stars in Science and Technology. In 2021, he was the President of the Society of Engineering Science (SES). Prof. Reis has also received the 2014 CAREER Award (NSF), the 2016 Thomas J.R. Hughes Young Investigator Award (Applied Mechanics Division of the ASME), the 2016 GSOFT Early Career Award for Soft Matter Research (APS), and he is a Fellow of the American Physical Society (APS).

Abstract

Even though most of us tie our shoelaces "wrongly," knots in ropes and filaments have been used as functional structures for millennia, from sailing and climbing to dewing and surgery. However, knowledge of the mechanics of physical knots is largely empirical, and there is much need for physics-based predictive models. Tight knots exhibit highly nonlinear and coupled behavior due to their intricate 3D geometry, large deformations, self-contact, friction, and even elasto-plasticity. Additionally, tight knots do not show separation of the relevant length scales, preventing the use of centerline-based rod models. In this talk, I will present an overview of recent work from our research group, combining precision experiments, Finite Element simulations, and theoretical analyses. First, we study the mechanics of two elastic fibers in frictional contact. Second, we explore several different knotted structures, including the overhand, figure-8, clove-hitch, and bowline knots. These knots serve various functions in practical settings, from shoelaces to climbing and sailing. Lastly, we focus on surgical knots, with a particularly high risk of failure in clinical settingsincluding complications such as massive bleeding or the unraveling of high-tension closures. Our research reveals a striking and robust power law, with a general exponent, between the mechanical strength of surgical knots, the applied pre-tension, and the number of throws, providing new insights into their operational and safety limits. These findings could have potential applications in the training of surgeons and enhanced control of robotic-assisted surgical devices.

 

Thu, 06 Jun 2024

12:00 - 13:00
L3

Isolating internal waves using on-the-fly Lagrangian filtering in numerical simulations

Lois Baker
(University of Edinburgh, School of Mathematics)
Further Information

Dr Lois Baker is the Flora Philip Fellow and EPSRC National Fellow in Fluid Dynamicsa in the School of Mathematics at the University of Edinburgh. Her research involves using mathematical and numerical models to understand oceanic fluid dynamics. Baker is particularly interested in the interactions of internal waves and submesoscale vortices that are generated in the deep and upper ocean.

Abstract

 

In geophysical and astrophysical flows, we are often interested in understanding the impact of internal waves on the non-wavelike flow. For example, oceanic internal waves generated at the surface and the seafloor transfer energy from the large scale flow to dissipative scales, thereby influencing the global ocean state. A primary challenge in the study of wave-flow interactions is how to separate these processes – since waves and non-wavelike flows can vary on similar spatial and temporal scales in the Eulerian frame. However, in a Lagrangian flow-following frame, temporal filtering offers a convenient way to isolate waves. Here, I will discuss a recently developed method for evolving Lagrangian mean fields alongside the governing equations in a numerical simulation, and extend this theory to allow effective filtering of waves from non-wavelike processes.

 

Thu, 30 May 2024

12:00 - 13:00
L3

Patterned illumination for complex spatio-temporal morphing of LCE sheets

John Biggins
(University of Cambridge)
Further Information

Biography

John Biggins read natural sciences at Cambridge University. He specialized in experimental and theoretical physics, and was the top ranked student in his cohort. He then did a PhD in the theory of condensed matter group under the supervision of Prof Mark Warner FRS, working on the exotic elasticity of a new phase of soft matter known as a liquid crystal elastomer (LCE). During his PhD he made an extended visit to Caltech to work with Prof Kaushik Bhattacharya on analogies between LCEs and shape memory alloys.

After his PhD, John won an 1851 Royal Commission Fellowship and traveled to Harvard to work with Prof L. Mahadevan on instabilities in soft solids and biological tissues, including creasing, fingering and brain folding. He then returned to Cambridge, first as Walter Scott Research Fellow at Trinity Hall and then as an early career lecturer in the tcm group at the Cavendish Laboratory. During this time, he explained the viral youtube phenomena of the chain fountain, and explored how surface tension can sculpt soft solids, leading to a solid analogue of the Plateau–Rayleigh instability. He also taught first year oscillations, and a third year course "theoretical physics 1."

In 2017, John was appointed to an Assistant Professorship of applied mechanics in Cambridge Engineering Department, where he teaches mechanics and variational methods. In 2019 he won a UKRI Future Leaders Fellowship on "Liquid Crystal Elastomers, from new materials via new mechanics to new machines." This grant added an exciting experimental component to the group, and underpins our current focus on using LCEs as artificial muscles in soft mechanical devices.

 

from http://www.eng.cam.ac.uk/profiles/jsb56 

Abstract

Liquid crystal elastomers are rubbery solids containing molecular LC rods that align along a common director. On heating, the alignment is disrupted, leading to a substantial (~50%) contraction along the director.  In recent years, there has been a great deal of interest in fabrication LCE sheets with a bespoke alignment pattern. On heating, these patterns generate  corresponding patterns of contraction that can morph a sheet into a bespoke curved surface such as a cone or face. Moreover, LCEs can also be activated by light, either photothermally or photochemically, leading to similarly large contractions. Stimulation by light also introduces an important new possibility: using spatio-temporal patterns of illumination to morph a single LCE sample into a range of different surfaces. Such stimulation can enable non-reciprocal actuation for viscous swimming or pumping, and control over the whole path taken by the sheet through shape-space rather than just the final destination. In this talk, I will start by with an experimental example of a spatio-temporal pattern of illumination being used to actuate an LCE peristaltic pump. I will then introduce a second set of experiments, in which a monodomain sheet morphs first into a cone, an anti-cone and then an array of cones upon exposure to different patterns of illumination. Finally, I will then discuss the general problem of how to choose a pattern of illumination to morph a director-patterned sheet into an arbitrary surface, first analytically for axisymmetric cases, then numerically for low symmetry cases. This last study exceeds our current experimental capacity, but highlights how, with full spatio-temporal control over the stimulation magnitude, one can choreograph an LCE sheet to undergo almost any pattern of morphing.

Thu, 23 May 2024

12:00 - 13:00
L3

Mathematical models for biological cooperation: lessons from bacteria

Maria Tatulea-Codrean
(University of Cambridge)
Further Information

Maria is a member of the Biological Fluid Mechanics group. Her current research interests revolve around the themes of flows (flows around and in between filaments, flows in membranes), motors (in particular, bacterial flagellar motors) and oscillators (synchronization of coupled non-linear oscillators, and biological rhythms more broadly).

Abstract
 
Cooperation occurs at all scales in the natural world, from the cooperative binding of ligands on
the molecular scale, to the coordinated migration of animals across continents. To understand
the key principles and mechanisms underlying cooperative behaviours, researchers tend to
focus on understanding a small selection of model organisms. In this talk, we will look through a
mathematician’s lens at one of the most well-studied model organisms in biology—the multiflagellated bacterium Escherichia coli.
 
First, we will introduce the basic features of swimming at the microscopic scale, both biological
(the flagellum) and mathematical (the Stokes equations). Then, we will describe two recent
theoretical developments on the cooperative dynamics of bacterial flagella: an
elastohydrodynamic mechanism that enables independent bacterial flagella to coordinate their
rotation, and a load-sharing mechanism through which multiple flagellar motors split the
burden of torque generation in a swimming bacterium. These results are built on a foundation of
classical asymptotic approaches (e.g., multiple-scale analysis) and prominent mathematical
models (e.g., Adler’s equation) that will be familiar to mathematicians working in many areas of

applied mathematics.

Thu, 16 May 2024

12:00 - 13:00
L3

Modelling liquid infiltration in a porous medium: perils of oversimplification

​Doireann O'Kiely
(University of Limerick)
Abstract

Mathematical modelling can support decontamination processes in a variety of ways.  In this talk, we focus on the contamination step: understanding how much of a chemical spill has seeped into the Earth or a building material, and how far it has travelled, are essential for making good decisions about how to clean it up.  

We consider an infiltration problem in which a chemical is poured on an initially unsaturated porous medium, and seeps into it via capillary action. Capillarity-driven flow through partially-saturated porous media is often modelled using Richards’ equation, which is a simplification of the Buckingham-Darcy equation in the limit where the infiltrating phase is much more viscous than the receding phase.  In this talk, I will explore the limitations of Richards equation, and discuss some scenarios in which predictions for small-but-finite viscosity ratios are very different to the Richards simplification.

Thu, 09 May 2024

12:00 - 13:00
L1

Models of viscous anisotropy

Daniel Richards
(University of Tasmania)
Abstract

What do fiber polymers and ice sheets have in common? They both flow with a directionally dependent - anisotropic - viscosity. This behaviour occurs in other geophysical flows, such as the Earth's mantle, where a material's microstructure affects its large-scale flow. In ice, the alignment of crystal orientations can cause the viscosity to vary by an order of magnitude, consequently having a strong impact on the flow of ice sheets and glaciers. However, the effect of anisotropy on large-scale flow is not well understood, due to a lack of understanding of a) the best physical approximations to model crystal orientations, and b) how crystal orientations affect rheology. In this work, we aim to address both these questions by linking rheology to crystal orientation predictions, and testing a range of models against observations from the Greenland ice sheet. The results show assuming all grains experience approximately the same stress provides realistic predictions, and we suggest a set of equations and parameters which can be used in large-scale models of ice sheets. 

Thu, 02 May 2024

12:00 - 13:00
L3

Path integral formulation of stochastic processes

Steve Fitzgerald
(University of Leeds)
Abstract

Traditionally, stochastic processes are modelled one of two ways: a continuum Fokker-Planck approach, where a PDE is solved to determine the time evolution of the probability density, or a Langevin approach, where the SDE describing the system is sampled, and multiple simulations are used to collect statistics. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also have much to offer classical stochastic processes (and statistical physics).  

In this talk I will introduce the formalism at a physicist’s level of rigour, and focus on determining the dominant contribution to the path integral when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian dynamics in an effective potential [1,2,3]. I will then discuss some applications, including reaction pathways conditioned on finite time [2]. We demonstrate that the most probable pathway at a finite time may be very different from the usual minimum energy path used to calculate the average reaction rate. If time permits, I will also discuss the extremely nonlinear crystal dislocation response to applied stress [4].  

[1] Ge, Hao, and Hong Qian. Int. J. Mod. Phys. B 26.24 1230012 (2012)     

[2] Fitzgerald, Steve, et al. J. Chem. Phys. 158.12 (2023).

[3] Honour, Tom and Fitzgerald, Steve. in press J. Phys. A (2024)

[4] Fitzgerald, Steve. Sci. Rep. 6 (1) 39708 (2016)

 

Thu, 25 Apr 2024

12:00 - 13:00
L3

Static friction models, buckling and lift-off for a rod deforming on a cylinder

Rehan Shah
(Queen Mary, University of London)
Further Information

Dr. Rehan Shah, Lecturer (Assistant Professor) in Mathematics and Engineering Education, Queen Mary University of London

Abstract

We develop a comprehensive geometrically-exact theory for an end-loaded elastic rod constrained to deform on a cylindrical surface. By viewing the rod-cylinder system as a special case of an elastic braid, we are able to obtain all forces and moments imparted by the deforming rod to the cylinder as well as all contact reactions. This framework allows us to give a complete treatment of static friction consistent with force and moment balance. In addition to the commonly considered model of hard frictionless contact, we analyse two friction models in which the rod, possibly with intrinsic curvature, experiences either lateral or tangential friction. As applications of the theory we study buckling of the constrained rod under compressive and torsional loads, finding critical loads to depend on Coulomb-like friction parameters, as well as the tendency of the rod to lift off the cylinder under further loading. The cylinder can also have arbitrary orientation relative to the direction of gravity. The cases of a horizontal and vertical cylinder, with gravity having only a lateral or axial component, are amenable to exact analysis, while numerical results map out the transition in buckling mechanism between the two extremes. Weight has a stabilising effect for near-horizontal cylinders, while for near-vertical cylinders it introduces the possibility of buckling purely due to self-weight. Our results are relevant for many engineering and medical applications in which a slender structure winds inside or outside a cylindrical boundary.