Applied Mathematics Seminar
University of Leeds
Welcome to the Leeds Applied Mathematics Seminar
Autumn 2016
Monday 18 July 2016
Edgar Knobloch (University of California, Berkeley)
Solidification fronts in supercooled liquids: how rapid fronts can lead to disordered glassy solids
Using dynamical density functional theory we calculate the speed of solidification fronts advancing into a quenched two-dimensional model fluid of soft-core particles. We find that solidification fronts can advance via two different mechanisms, depending on the depth of the quench. For shallow quenches, the front propagation is via a nonlinear mechanism. For deep quenches, front propagation is governed by a linear mechanism and in this regime the front speed is determined via a marginal stability analysis. We find that the density modulations generated behind the advancing front have a characteristic scale that differs from the wavelength of the density modulation in thermodynamic equilibrium, i.e., the spacing between the crystal planes in an equilibrium crystal. This leads to the subsequent development of disorder in the solids that are formed. In a one-component fluid, the particles are able to rearrange to form a well-ordered crystal, with few defects. However, solidification fronts in a binary mixture exhibiting crystalline phases with square and hexagonal ordering generate solids that are unable to rearrange after the passage of the solidification front and a significant amount of disorder remains in the system.
Monday 26 September 2016
Steven Fitzgerald (University of Leeds)
Some mathematical aspects of crystal defects
We tend to think of crystals as perfectly periodic ordered arrays of atoms, yet most of the interesting properties of crystalline materials are due to the ubiquitous defects that break this symmetry. These are present naturally in almost every crystal, and can also be generated by the crystal’s interaction with its environment, e.g. applied stress and deformation, or exposure to irradiation.
The mechanics of defects has a rich mathematical structure that touches on soliton theory, non-Euclidean geometry, quantum and statistical field theory to name a few, and systems of technological interest are very often away from equilibrium. Although a wide variety of analytical and computational approaches to modelling crystal defects exists, a full mechanistic understanding of even such an everyday material as stainless steel remains elusive.
In this talk I will introduce the various types of crystal defects such as vacancies, interstitial atoms, and dislocations, and give an overview of the modelling and simulation techniques in use. I will then describe some recent progress in our understanding of defects' structure and dynamics from a theoretical perspective. Though examples will be drawn from my own background in nuclear materials modelling, I will focus on the mathematics and physics of the defects themselves rather than details of the technological applications.
Monday 10 October 2016
Laurette Tuckerman (PMMH-ESPCI, CNRS)
Can frequencies in thermosolutal convection be predicted from mean flows?
The von Karman vortex street is one of the most striking visual images in fluid dynamics. Immersed in a uniform flow of sufficient strength, a circular cylinder periodically sheds propagating vortices of alternating sign on either side of the "street". Although the von Karman vortex street can be simulated numerically with great accuracy, predicting its properties from general theoretical principles has proved elusive. It has been shown that the vortex-shedding frequency can be obtained by carrying out a linear stability analysis about the temporal mean, but there is no understanding of why the correct answer emerges from such an unorthodox procedure.
We have carried out a similar analysis of thermosolutal convection, which is driven by opposing thermal and solutal gradients. In a spatially periodic domain, branches of traveling waves and standing waves are created simultaneously by a Hopf bifurcation. We find that linearization about the mean fields of the traveling waves yields an eigenvalue whose real part is almost zero and whose imaginary part corresponds very closely to the nonlinear frequency, consistent with the cylinder wake. In marked contrast, linearization about the mean field of the standing waves yields neither zero growth nor the nonlinear frequency. It is shown that this difference can be attributed to the fact that the temporal power spectrum for the traveling waves is peaked, while that of the standing waves is broad. We give a general demonstration that the frequency of any quasi-monochromatic oscillation can be predicted from its temporal mean.
Monday 31 October 2016
Ruth Baker (University of Oxford)
Cell biology processes: model building and validation using quantitative data
Cell biology processes such as motility, proliferation and death are essential to a host of phenomena such as development, wound healing and tumour invasion, and a huge number of different modelling approaches have been applied to study them. In this talk I will explore a suite of related models for the growth and invasion of cell populations. These models take into account different levels of detail on the spatial locations of cells and, as a result, their predictions can differ depending on the relative magnitudes of the various model parameters. To this end, I will discuss how one might determine the applicability of each of these models, and the extent to which inference techniques can be used to estimate their parameters, using both cell- and population-level quantitative data.
Monday 21 November 2016
Stephen Coombes (University of Nottingham)
Next generation neural field modelling
Neural mass models have been actively used since the 1970s to model the coarse grained activity of large populations of neurons and synapses. They have proven especially fruitful for understanding brain rhythms. However, although motivated by neurobiological considerations they are phenomenological in nature, and cannot hope to recreate some of the rich repertoire of responses seen in real neuronal tissue. In this talk I will first discuss a theta-neuron network model that has recently been shown to admit to an exact mean-field description for instantaneous pulsatile interactions. I will then show that the inclusion of a more realistic synapse model leads to a mean-field model that has many of the features of a neural mass model coupled to an additional dynamical equation that describes the evolution of network synchrony. I will further show that this next generation neural mass model is ideally suited to understanding beta-rebound. This is readily observed in MEG recordings whereby hand movement causes a drop in the beta power band attributed to a loss of network synchrony. Existing neural mass models are unable to capture this phenomenon since they do not track any notion of network coherence (only firing rate). I will finish my talk by presenting some preliminary results for the spatio-temporal pattern formation properties of a neural field version of this model.
Monday 5 December 2016
Serafim Kalliadasis (Imperial College London)
From the nano- to the macroscale: bridging scales for the moving contact line problem
The moving contact line problem occurs when modelling one fluid replacing another as it moves along a solid surface, a situation widespread throughout industry and nature. Classically, the no-slip boundary condition at the solid substrate, a zero-thickness interface between the fluids, and motion at the three-phase contact line are incompatible - leading to the well-known shear-stress singularity. At the heart of the problem is its multiscale nature: a nanoscale region close to the solid boundary where the continuum hypothesis breaks down, must be resolved before effective macroscale parameters such as contact line friction and slip, often adopted to alleviate the singularity [1], can be obtained.
In this talk we will review very recent progress made by our group, considering the problem and related physics from the nano- to macroscopic length scales. In particular, to capture nanoscale properties very close to the contact line and to establish a link to the macroscale behaviour, we employ elements from the statistical mechanics of classical fluids, namely density-functional theory (DFT) [2-4], in combination with extended Navier-Stokes-like equations. Using simple models for viscosity and no slip at the wall, we compare our computations with the Molecular Kinetic Theory model for the dynamic vs. equilibrium contact angle behaviour, by extracting the contact line friction, depending on the imposed temperature of the fluid [4]. A key fluid property captured by DFT is the fluid layering at the wall-fluid interface, which has a large effect on the shearing properties of a fluid.
[1] J. Fluid Mech. 764, 445-462 (2015)
[2] J. Phys.: Condens. Matter 25, 035101 (2013)
[3] Math. Model. Nat. Phenom. 10, 111-125 (2015)
[4] Phys. Fluids 26, 072001 (2014)
[5] A. Nold, PhD Thesis, Imperial College London (2016)
[Joint work with Benjamin D. Goddard (Edinburgh), Andreas Nold (Imperial), Nikos Savva (Cardiff), David N. Sibley (Loughborough) and Peter Yatsyshin (Imperial)]