Title: Approximate matrix Wiener-Hopf factorisation and applications to problems in acoustics
Abstract: In this talk I will introduce some of the techniques which are employed in the study of Helmholtz equation with various boundary conditions.
First, I will introduce the Wiener-Hopf method which extends the separation of variables technique (in Cartesian coordinate) used to investigate PDEs. It provided analytic and systematic methodology for previously unapproachable problems. One of the problems discussed will be scattering of a sound wave by an infinite periodic grating composed of rigid plates (joint work with I. D. Abrahams).
I will also talk about a matrix Wiener-Hopf problems which is motivated by studying the effect of a finite elastic trailing edge on noise production (joint work with N. Peake, L. Ayton). The approximate factorisation of this matrix with exponential phase factors is achieved using an iterative procedure which makes use of the scalar Wiener-Hopf problem arising for each junction.
Lastly, I will introduce some methods which rely on Mathieu functions, resulting from change of variable to elliptic coordinates of Helmholtz equation and the boundary conditions.
Title: Equilibrium measure for a nonlocal dislocation energy
Abstract: In this talk I will present a recent result on the characterisation of the equilibrium measure for a nonlocal and non-radial energy arising as the Gamma-limit of discrete interacting dislocations.
This is joint work with Maria Giovanna Mora and Luca Rondi.
Title: Discrete-to-continuum limits of edge dislocations in 2D
Abstract: The starting point is a 2D model for the dynamics of n dislocations, which are modelled as point particles with a positive or negative ’charge’. In the celebrated engineering paper by Groma and Balogh in 1999, the limit passage n → ∞ of these dislocation dynamics is performed in a statistical mechanics framework, which relies on a phenomenological closure assumption. In my talk, I present how to pass rigorously to the limit n → ∞ by using the theory of Wasserstein gradient flows and using advanced functional analysis on the weak form of the evolution equation. Interestingly, our conclusion for the limiting dynamics of the dislocation density differs from the conclusion in the paper by Groma and Balogh.
Title: Peeling and the growth of blisters
Abstract: The peeling of an elastic sheet away from thin layer of viscous fluid is a simply-stated and generic problem, that involves complex interactions between flow and elastic deformation on a range of length scales.
I will illustrate the possibilities by considering theoretically and experimentally the injection and spread of viscous fluid beneath a flexible elastic lid; the injected fluid forms a blister, which spreads by peeling the lid away at the perimeter of the blister. Among the many questions to be considered are the mechanisms for relieving the elastic analogue of the contact-line problem, whether peeling is "by bending" or "by pulling", the stability of the peeling front, and the effects of a capillary meniscus when peeling is by air injection. The result is a plethora of dynamical regimes and asymptotic scaling laws.
Title: Dynamic Density Functional Theory: Modelling, Analysis and Numerics
Abstract: In recent years, a number of dynamic density functional theories (DDFTs) have been developed to describe colloid particle dynamics. These DDFTs aim to overcome the high-dimensionality of systems with large numbers of particles by reducing to the dynamics of the one-body density, described by a PDE in only three spatial dimensions, independently of the number of particles. The standard derivations are via stochastic equations of motion, but there are fundamental differences in the underlying assumptions in each DDFT. I will begin by giving an overview of some DDFTs, highlighting the assumptions and range of applicability. Particular attention will be given to the inclusion of inertia and hydrodynamic interactions, both of which strongly
influence non-equilibrium properties of the system. I will then demonstrate the very good agreement with the underlying stochastic dynamics for a wide range of systems. I will also discuss an accurate and efficient numerical code, based on pseudospectral techniques, which is applicable both to the integro-PDEs of DDFT and to many other systems. Finally I will describe (i) the rigorous passage to the high-friction limit, where the one-body density satisfies a nonlinear, non-local Smoluchowski-like equation with a novel diffusion tensor and (ii) the (somewhat less rigorous) limit of being close to local equilibrium, in which we obtain a Navier-Stokes-like equation with additional non-local terms.
Joint work with Serafim Kalliadasis, Greg Pavliotis, and Andreas Nold.
Title: The effect of forest dislocations on the evolution of a phase-field model for plastic slip
Abstract: We consider a phase field model for dislocations which describes a single slip plane and consists of a Peierls potential penalising non-integer slip and a long range interaction modelling elasticity. Forest dislocations are introduced as a restriction to the allowable phase field functions: they have to vanish at the union of a number of small disks in the plane. Garroni and Müller proved large scale limits of these models in terms of Gamma-convergence, obtaining a line-tension energy for the dislocations and a bulk term penalising slip. This bulk term is a capacity stemming from the forest dislocations.
In the present work, we show that the contribution of the forest dislocations to the the viscous gradient flow evolution is small. On the other hand, of course, when adding a driving force in the direction of increasing slip, one needs to spend the energy to overcome the obstacles. Overall, this leads to an effective behaviour like a gradient flow in a wiggly potential. The forest dislocations thus act like a dissipation for increasing slip, but their effect on the propagation is absent for decreasing slip.
Ian Griffiths, Mathematical Institute, University of Oxford
Title: Wrinkly to Smoothie
Abstract: How do we make glass sheets flat enough so that they can be used for smartphone and tablet screens? How can we design a filter for a vacuum cleaner so that it never needs to be replaced? And how do we make the perfect food blender so that at the end of the day we can relax with the perfect margarita? To address all of these questions requires the development of fluid mechanics models twinned with a close partnership with industry. In this talk we will answer each of these questions, while also discussing our strategies for achieving successful outcomes.