Applied Mathematics Seminars Autumn 2022

Seminars (unless otherwise stated) usually 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).

Tuesday 18 October 2022

Speaker: André Nachbin (Instituto Nacional Pura e Aplicada, Rio de Janeiro, Brazil)

Title: The effect of isolation on two-particle correlations in pilot-wave hydrodynamics

Couder and Fort (Nature 2005, PRL 2006) discovered that a fluid droplet bouncing on the surface of a vertically vibrating silicon oil bath, forms a wave-particle association referred to as a hydrodynamic pilot-wave system. Such an object was only imagined in the quantum realm. Much research has been done since this discovery. The main theme of this talk regards nonlinear oscillators which can be correlated at a distance. The droplet-dynamics is mediated by a background vibrating field, namely the Faraday waves. We present several regimes where two oscillating droplets are confined to separate wells. Exploring through numerical simulations we detect “coherence” when the bouncing droplets spontaneously synchronize, as in the celebrated Kuramoto model for phase oscillators. We also discover a new regime where “coherence” emerges in a statistical fashion (Chaos, 2018). More recently we found a regime of cooperative tunneling leading to a hydrodynamic analogue of superradiance (Nature, Comm. Phys. 2022). Also, we studied the effect of isolation on two droplets which previously interacted and were correlated. The dynamics and statistics are quite different had the two droplets been isolated at the onset (Phys. Rev. Fluids, September 2022). Information of the pre-existing correlation is somehow stored in the wave-field. Additional very recent results will be presented if time permits.

Tuesday 25 October 2022

Speaker: Dr Toby Kirk (Chapman Fellow, Department of Mathematics, Imperial College London)

Title: Asymptotic solutions for forced convection from ridged superhydrophobic surfaces: protruding liquid-gas interfaces and Marangoni effects

The use of superhydrophobic surfaces has attracted significant research in recent years, due to their ability to reduce flow resistance, giving a range of applications in microfluidics. They are partially non-wetting surfaces consisting of microscale roughness that resists a liquid fully wetting the solid substrate due to surface tension. A natural example is the lotus leaf, but artificial surfaces include arrays of microscale grooves, ridges or pillars. The focus of this talk is on the application of such surfaces to thermal transport, which has received relatively little attention but has consequences for, e.g., liquid cooling of microelectronics. In particular, we consider forced convection from a heated substrate textured with parallel ridges aligned with the flow direction. The reduced contact area between the solid and liquid enhances the flow rate (measured by an effective slip length) but reduces the convective heat transfer (quantified by a Nusselt number). These competing effects both contribute to the total heat transfer, which can be enhanced or diminished depending on the ridge geometry and liquid properties. One of the most important geometric effects is the curvature of the liquid-gas interfaces (menisci) that span the ridge tips.

In the first part of the talk, we analyse the periodic flow and thermal problems accounting for curved menisci using boundary perturbation methods, and provide semi-analytical series solutions for the slip length and Nusselt number. In the second part, we consider the additional impact of thermal Marangoni stresses due to temperature variations along the menisci and derive asymptotic expressions for the slip length in a suite of limits using matched asymptotic expansions and conformal mapping techniques. These formulae are compared against full numerical simulations, and found to have a wide range of validity, spanning much of the relevant parameter space.   

Tuesday 01 November 2022

Speaker: Saleh Tanveer* (The Ohio State University)

Title: Analysis of a non-local model for the interfacial problem in a two fluid Couette flow

Following Kalogirou et al (2016) work on deriving a thin-film approximation of two-layer Couette flow, we consider the mathematical properties of this nonlocal model, including local and global existence and existence of finite amplitude waves. We also consider the variation of this problem that introduces the effect of a small slip at the interface as appropriate for modeling hydrophobic surfaces. We find surprising singular effect of arbitrarily small slip on the stability of otherwise stable thin layer arrangement. The global existence and finite amplitude wave analysis extends to this problem as well.

*Work with D. Papageorgiou.

Tuesday 08 November 2022



Tuesday 15 November 2022

Speaker: Jon Chapman (Oxford University)

Title: Blowup in the nonlinear Schrodinger equation

We systematically derive a normal form for the emergence of radially symmetric blowup solutions from stationary ones in the nonlinear Schrodinger equation. The derivation uses the methodology of asymptotics beyond all orders, and the resulting normal form applies when either the power law or dimension is used as the bifurcation parameter. It yields excellent agreement with numerics in both leading and higher-order effects, is applicable to both infinite and finite domains, and is valid in both critical and supercritical regimes.

Tuesday 22 November 2022

Speaker: Suraj Shankar (Harvard University, USA)

Title: Controlling active matter - from dRops to defects

We are active matter. From the subcellular processes occurring within a living cell to the large-scale collective dynamics of human crowds or animal flocks, systems driven far from equilibrium by a sustained flux of energy through its constituents routinely exhibit stunning emergent phenomena that pose fundamental challenges to our understanding of the natural world. While we know much about the complex patterns and dynamics exhibited by active systems, little is known about the inverse problem of controlling active matter in space-time for functional purposes. After briefly reviewing some of the tools employed to understand paradigmatic active phenomena, I will discuss current work on searching for design principles to control autonomous dynamics of localized excitations, drops etc., in active materials. I will conclude by highlighting future directions for embodying function, programmable response and computation in biological and synthetic active systems.

Tuesday 29 November 2022

Speaker: Wooyoung Choi (New Jersey Institute of Technology, USA)

Title: High-order long wave approximation and solitary wave solution in shallow water

Nonlinear long waves in shallow water have been often described by weakly nonlinear models. Considering that the number of extreme events increases, strongly nonlinear long wave models have recently attracted much attention. In this talk, a high-order strongly nonlinear long wave model will be introduced and its well-posedness and solitary wave solution will be discussed. When the high-order long wave model is solved numerically, it is found that a finite difference scheme could be unstable due to discretization errors in approximating high-order derivatives and, therefore, a more accurate numerical scheme is required. Here a pseudo-spectral method is adopted for spatial discretization and  some numerical solutions  of the high-order model will be presented in comparison with laboratory observations. In addition, a similar extension to a two-layer system is introduced. 

Tuesday 06 December 2022

Speaker: John Billingham, University of Nottingham

Title: Nonlocal reaction-diffusion equations: Travelling waves and steady states in the strongly nonlocal limit

Nonlocal reaction-diffusion equations are used in population modelling, modelling of cell motion, neuroscience and genetics. In this talk I will discuss solutions of perhaps the simplest nonlocal reaction-diffusion equation, the nonlocal Fisher-KPP equation, u_t = Du_xx + u(1-phi*u), where phi(y) is a kernel and phi*u is a spatial convolution. I will focus on the case D<<1, the strongly nonlocal limit, and discuss (i) which properties of phi affect the structure of travelling wave solutions that connect the two uniform steady states, u=0 and u=1, (ii) periodic steady states when phi is a top hat function, (iii) some other problems I am currently working on, including nonlocal Stefan problems, nonlocal Lotka-Volterra equations and the cubic nonlocal FKPP equation, u_t = Du_xx + u^2(1-phi*u).

Tuesday 13 December 2022

Speaker: Jerry Bona, University of Illinois at Chicago

Title: Systems of nonlinear, dispersive wave equations

The discussion will be centered around coupled systems of nonlinear wave equations.  Such systems arise in a number of different applications.  We will examine a class of such systems with an eye toward local and global well posedness, and existence of solitary waves and their stability.  One particular system is singled out for more detailed commentary.  The lecture will conclude with some illuminating numerical simulations.