Seminars (unless otherwise stated) will take place online on **Tuesdays at 3.00pm on Zoom via the link **https://ucl.zoom.us/j/99614222402**. **If you require any more information on the Applied seminars please contact Prof Jean-Marc Vanden-Broeck (e-mail: j.vanden-broeck AT ucl.ac.uk or tel: 020-7679-2835) or Prof Ilia Kamotski (e-mail: i.kamotski AT ucl.ac.uk or tel: 020-7679-3937).

## 12 October 2021

### Speaker: Robert Timms, Oxford University

##### Title: Multiscale modelling of lithium-ion batteries

**Abstract: **

Lithium-ion batteries (LIBs) are one of the most popular forms of energy storage for many modern devices, with applications ranging from portable electronics to electric vehicles. Improving both the performance and lifetime of LIBs by design changes that increase capacity, reduce losses and delay degradation effects is a key engineering challenge. Mathematical modelling is an invaluable tool for tackling this challenge: accurate and efficient models play a key role in the design, management, and safe operation of batteries. In order to be useful for real-time diagnostics and end-of-life prediction, models must be computationally efficient and include the most important physical effects.

Models of LIBs can be broadly categorised into two groups: equivalent circuit models, which aim to describe battery behaviour by making an analogy with traditional circuit components such as resistors and capacitors; and electrochemical models, which aim to describe the physical processes of mass, charge and heat transport within the cell. By exploiting various small parameters we can “bridge the gap”, and derive models whose computational complexity is comparable to that of equivalent circuit models, but whose fidelity approaches that of a detailed physics-based model.

## 19 October 2021

### Speaker: Zihua (Grace) Liu, Woods Hole Oceanographic Institution (USA)

##### Title: Vertical structure of barotropic-to-baroclinic energy conversion on a continental slope

**Abstract: **

Horizontal distribution of the barotropic-to-baroclinic energy conversion has been widely studied to examine the generation of internal tides. The vertical structure that provides insights into the dynamics of conversion, however, is masked by this depth-integrated energy conversion. Here, we reveal the vertical profile of energy conversion by employing an idealized ocean model in a slope-shelf context forced by $M_2$ barotropic tidal flow. The model shows two vertically separated hotspots of energy conversion, near the sloping bottom and at the thermocline. The two hotspots result from stronger vertical velocity associated with the barotropic flow on the sloping bottom and enhancement of the tidally-driven density anomaly at the thermocline, respectively. For a supercritical slope, conversion is concentrated in a narrow region along the slope, while for a subcritical slope, it spreads out in the cross-shelf direction with a negative and positive distribution. Isolation of the energy conversion hotspots demonstrates that baroclinic energy generated in the lower layer radiates toward onshore and offshore primarily in the form of internal wave beams, whereas that generated at the thermocline propagates away in the form of internal wave modes. Although energy converted in the upper layer contributes only a small amount to the total energy conversion, it plays an important role in onshore baroclinic energy radiation and can be significantly affected by the internal wave activity at the seafloor. With a fixed bottom topography, two energy conversion rates, one integrated over the entire water column and the other integrated over the thermocline layer only, are linearly related to a body force exerted by the barotropic tidal flow over the topography. This link provides a convenient way to estimate the overall barotropic-to-baroclinic energy conversion over a continental slope and shelf in the real ocean by measuring the energy conversion in the thermocline only.

## 26 October 2021

### Speaker: Alan Champneys, University of Bristol

##### Title: Cell polarity and localised pattern formation, Turing instability revisited

**Abstract: **

Recent biological theories of cellular level polarity formation and morphogenesis have focused on the dynamics of small G-proteins which exist in either inactive or active membrane-bound forms. The mechanism is preserved across all eukaryotic cells, but we are focus on model problems in plants, namely root hair formation and jigsaw instability of leaf pavement cells.

Models of their spatio-temporal action naturally fit into the framework of activator-inhibitor reaction-diffusion systems that are susceptible to pattern-forming or Turing instabilities. Except in all the parameter regions of interest, the Turing bifurcation is found to be sub-critical. In this case the stable patterned states tend to be localised rather than spatially periodic.

This leads us to study activator-inhibitor systems on large domains using a a variety of analytical and numerical methods. A canonical form is considered that contains known models such as the Schnackenberg and Brusselator systems with cubic autocatalytic nonlinearity. Such models all display similar behaviour which connects together several different asymptotic theories; so-called homoclinic snaking, semi-strong interaction theory, and an unusual kind of wave-pinning. Temporal dynamics of these localised patterned states is found to strongly depend on the diffusion ratio, with a delicate interplay between global and "breathing" Hopf instabilities.

The theory is extended to include models, such as Gray-Scott, and Gierer-Meinhardt with bistability of homogeneous equilibria and more realistic models of G-protein interaction where the wave-pinning limit leads to more complex structures that can be studied using singularity theory. We also iscuss extensions to more general systems and more realistic geometries.

## 02 November 2021

### Speaker: Darren Crowdy, Imperial College London

##### Title: Marangoni flows, surfactant dynamics and the complex Burgers equation

**Abstract: **

This talk will survey a number of new mathematical results on insoluble surfactant dynamics, and the resulting Marangoni effects, on the free surface of a viscous fluid. Starting with a discussion of some basic transport problems involving ``viscous Marangoni propulsion'' (or ``Marangoni surfers'') we will build up the theoretical approach to describe the unsteady formation of ``stagnant caps'' when surface diffusion is negligible. The effects of reaction kinetics are also incorporated into the model. All this leads up to the main aim of the talk: to show the (surprising) relevance of the complex Burgers equation to the dynamics of insoluble surfactants at arbitrary surface Peclet numbers -- when surface diffusion and advection are both present -- leading to a remarkable linearization of this nonlinear problem having a number of mathematical, and physical, ramifications only so far partially explored.

## 09 November 2021

**NO SEMINAR - READING WEEK**

## 16 November 2021

### Speaker: Michael Siegel, New Jersey Institute of Technology (USA)

##### Title: Finite-time singularity formation in the generalized Constantin-Lax-Majda equation

**Abstract: **

The question of finite-time singularity formation for solutions to the generalized Constantin-Lax-Majda (gCLM) equation is considered. This equation was first introduced by Constantin, Lax and Majda as a simplified model for singularity formation in the 3D incompressible Euler equations. It was later generalized by Okamoto, Sakajo and Wensch to include an advection term with parameter a, which allows different relative weights for advection and vortex stretching. There has been intense interest in the gCLM equation, but little is known about singularity formation over the full range of a. In this talk we provide such information via a combination of analysis and numerical computations. For solutions on the real line we find a new critical value ac = 0.689066 . . . below which there is finite time singularity formation that has a form of self-similar collapse, with the spatial extent of blow-up shrinking to zero. For ac < a ≤ 1, we observe a different singularity type in which the spatial extent of the blow-up region expands infinitely fast at the singularity time. For larger a we find that the solution exists globally in time. We also discuss the case of periodic boundary conditions.

## 23 November 2021

### Speaker: Matthew Turner, University of Surrey

##### Title: Vortex Leapfrogging external to a circular cylinder

**Abstract: **

In this talk we investigate the interaction of two line vortices of differing strengths in the presence of a circular cylinder. We will show that depending upon the initial positions of the vortices they will either undergo a periodic leapfrogging motion as they rotate around the cylinder, or they will move around the cylinder with no leapfrogging. We will derive an explicit criteria which separates these different behaviours of the system. Numerical results for initial vortex positions which do, and do not satisfy this criteria are presented to demonstrate the different motions available, as well as the robustness of the criteria.

## 30 November 2021

### Speaker: Philip Pearce, UCL

##### Title: Biological pattern formation in spatio-temporally fluctuating environments

**Abstract: **

The study of biological pattern-forming systems has revealed generic features that promote robustness with respect to variations between cells or populations in morphogen and protein production rates. By contrast, less is understood about how such systems respond to spatio-temporal fluctuations in morphogen concentrations within individual cells or populations over timescales faster than or similar to growth. Such fluctuations can be caused by growth, motility or changes in the external environment. Here, we formulate a general theory of biological pattern formation in spatio-temporally fluctuating environments. Using our framework, we assess the robustness of several pattern-forming systems.

## 07 December 2021

### Speaker: Ryan Palmer, University of Bristol

##### Title: Aerial electroreception: uncovering a “sixth-sense”

**Abstract: **

Recent investigations have highlighted the sensory possibility of electroreception within arthropods (e.g. insects, spiders) through charged mechanosensory hairs. With such new discoveries, there are many questions that arise that require both theoretical and empirical examination.

My aim is to introduce the fundamentals of this sensory modality, and begin to uncover some of the mechanical and sensory complexities.

Firstly, I will present a study into the mechanics of mechanoreceptor hairs in response to electro- and acousto-stimuli and show the physical, biological and parametric feasibility of electroreception. Secondly, I will introduce the concept of a sensitivity contour (regions of the solution space where hairs deflect to a given sensory threshold), and examine how a contours shape and size, and the overall hair behaviour changes due to electrical interaction between hairs. Finally, I will discuss some of the sensory possibilities of electroreception (e.g. object identification and location detection) and the biological implications of this (e.g. foraging decisions, predator-prey behaviour).

## 14 December 2021

### Speaker: Miles Wheeler, University of Bath

##### Title: Overhanging water waves with constant vorticity

**Abstract: **

We consider steady waves propagating along the surface of an incompressible fluid, with constant vorticity and under the influence of gravity. In the irrotational case, the surface is necessarily the graph of a single-valued function. With nonzero vorticity, on the other hand, numerics have long predicted so-called overhanging waves with multi-valued height. In the first part of the talk, I will show how to construct periodic overhanging waves as perturbations of a new family of explicit solutions with zero gravity. These explicit solutions have the same surfaces as Crapper's celebrated irrotational capillary waves, but the flow beneath the surface, as well as the governing equations, are completely different. In the second part of the talk, I will present new existence results for solitary waves, which are based on a reformulation of the problem as an elliptic system for two real-valued functions, one describing the conformal mapping of the fluid domain and another describing the motion inside the fluid.

This is joint work with Vera Hur and with Susanna Haziot.