Applied mathematics seminar series

This is only an overview page. For complete details of the venue, speakers and calendar links, please see HERE

3rd Oct
2012 Two-Phase Flow in Porous Media: Modeling, Travelling Waves and Stability
Michael Shearer (NC State University)

2:00 - Room 1.212 Alan Turing Building

Abstract (click to view)

I discuss two models of two-phase fluid flow in which undercompressive shock waves have been dis- covered recently. In the first part of talk, the focus is on two-phase flow in porous media. Plane waves are modeled by the one-dimensional Buckley-Leverett equation, a scalar conservation law. The Gray-Hassanizadeh model for rate-dependent capillary pressure adds dissipation and a BBM-type dis- persion, giving rise to undercompressive waves. Two-phase flow in porous media is notoriously subject to fingering instabilities, related to the classic Saffman-Taylor instability. However, a two dimensional linear stability analysis of sharp planar interfaces reveals a criterion predicting that weak Lax shocks may be stable or unstable to long-wave two-dimensional perturbations. This surprising result depends on the hyperbolic-elliptic nature of the system of linearized equations. Numerical simulations of the full nonlinear system of equations, including dissipation and dispersion, verify the stability predictions at the hyperbolic level. In the second part of the talk, I describe a phase field model of capillary effects in a thin tube, in which a resident fluid is displaced by injected air. PDE simulations reveal the appearance of a combination of rarefaction wave together with an undercompressive shock that terminates at the spherical cap tip of the injected air. The shock can be understood through a singular dynamical system whose trajectories yield travelling wave solutions.
9th Oct
2012 MAGIC Special Seminar: Dynamics of Social perceptions, Attitudes Norms and Applications to Digital Media and Security
Peter Grindrod (University of Reading)

4:00 - - Room 1.213 Alan Turing Building
10th Oct
2012 Multiscale modelling of biological branching structures
Tiina Roose (University of Southampton)

1:00 - - Room 1.212 Alan Turing Building
24th Oct
2012 Jet break-up in inkjet printing of complex fluids.
Oliver Harlen (University of Leeds)

2:00 - - Room 1.212 Alan Turing Building
31st Oct
2012 Modelling Ice-dammed Lake Drainage
Jonny Kingslake (University of Sheffield)

2:00 - - Room 1.212 Alan Turing Building

Abstract (click to view)

Ice-dammed lakes form next to, on the surface of, and beneath glaciers and ice sheets. Some lakes are known to drain catastrophically, creating hazards, wasting water resources and modulating the flow of the adjacent ice. My work aims to increase our understanding of such drainage. Here I will focus on lakes that form next to glaciers and drain subglacially (between ice and bedrock) through a channel. I will describe how such a system can be modelled and present results from model simulations of a lake that fills due to an input of meltwater and drains through a channel that receives a supply of meltwater along its length. Simulations yield repeating cycles of lake filling and drainage and reveal how increasing meltwater input to the system affects these cycles: enlarging or attenuating them depending on how the meltwater is apportioned between the lake and the channel. When inputs are varied with time, simulating seasonal meteorological cycles, the model simulates either regularly repeating cycles or irregular cycles that never repeat. Irregular cycles demonstrate sensitivity to initial conditions, a high density of periodic orbits and topological mixing. I will discuss how these results enhance our understanding of the mechanisms behind observed variability in these systems.
31st Oct
2012 Wrinkling of a Stretched Thin Sheet Eliot Fried (McGill University)

3:00 - - Room 1.212 Alan Turing Building

Abstract (click to view)

When a thin rectangular sheet is clamped along two opposing edges and stretched, its inability to accommodate the Poisson contraction near the clamps may lead to the formation of wrinkles with crests and troughs parallel to the axis of stretch. A variational model for this phenomenon is proposed. The underlying energy functional includes bending and membranal contributions. Motivated by work of Cerda, Ravi-Chandar and Mahadevan, the functional is minimized subject to a global constraint on the area of the mid-surface of the sheet. Analysis of a boundary-value problem for the ensuing Euler--Lagrange equation shows that wrinkled solutions exist only above a threshold of the applied stretch. A sequence of critical values of the applied stretch, each element of which corresponds to a discrete number of wrinkles, is determined. Whenever the applied stretch is sufficiently large to induce more than one wrinkle, previously proposed scaling relations for the wrinkle wavelength and root- mean-square amplitude are confirmed. Comparisons with experimental measurements and numerical results indicate that the analytical results are remarkably robust.
7th Nov
2012 Generation of the Tollmien-Schlichting Waves by Vibrations of the Wing Surface.
Anatoly Ruban (Imperial College, London)

2:00 - - Room 1.212 Alan Turing Building
14th Nov
2012 A minimal two-phase flow model for cell motility.
Laura Kimpton (University of Oxford)

2:00 - - Room 1.212 Alan Turing Building

Abstract (click to view)

The interplay between mechanical forces and biochemical control mechanisms make understanding cell motility a rich and exciting challenge for mathematical modelling. In the talk I shall present a stripped-down version of the two-phase, poroviscous, reactive flow framework used in the literature to describe crawling cells. Working with the simplest of such models enables us to gain insight into the behaviour of the model through both linear stability analysis and numerical simulation. The model displays various types of behaviour relevant to cell crawling. It highlights the importance of polarization of a stationary cell prior to the onset of motion and yields qualitatively distinct families of travelling wave solutions. The numerical solutions also capture the experimentally observed behaviour that cells crawl fastest when the surface they crawl over is neither too sticky nor too slippy.
21st Nov
2012 Fluid Rings and Floating Plates
Stephen Wilson (University of Stratclyde)

2:00 - - Room 1.212 Alan Turing Building

Abstract (click to view)

In this talk I shall describe two rather different, but not entirely unrelated, problems involving thin-film flow of a viscous fluid which I have found of interest and which may have some application to a number of practical situations, including condensation in heat exchangers and microfluidics. The first problem, which is joint work with Adam Leslie and Brian Duffy at the University of Strathclyde, concerns the steady three-dimensional flow of a thin, slowly varying ring of fluid on either the outside or the inside of a uniformly rotating large horizontal cylinder. Specifically, we study "full-ring" solutions, corresponding to a ring of continuous, finite and non-zero thickness that extends all the way around the cylinder. These full-ring solutions may be thought of as a three-dimensional generalisation of the "full-film" solutions described by Moffatt (1977) for the corresponding two-dimensional problem. We describe the behaviour of both the critical and non-critical full-ring solutions. In particular, we show that, while for most values of the rotation speed and the load the azimuthal velocity is in the same direction as the rotation of the cylinder, there is a region of parameter space close to the critical solution for sufficiently small rotation speed in which backflow occurs in a small region on the upward-moving side of the cylinder. The second problem, which is joint work with Phil Trinh and Howard Stone at Princeton University, concerns a rigid plate moving steadily on the free surface of a thin film of fluid. Specifically, we study two problems involving a rigid flat (but not, in general, horizontal) plate: the pinned problem, in which the upstream end of plate is pinned at a fixed position, the fluid pressure at the upstream end of the plate takes a prescribed value and there is a free surface downstream of the plate, and the free problem, in which the plate is freely floating and there are free surfaces both upstream and downstream of the plate. For both problems, the motion of the fluid and the position of the plate (and, in particular, its angle of tilt to the horizontal) depend in a non-trivial manner on the competing effects of the relative motion of the plate and the substrate, the surface tension of the free surface, and of the viscosity of the fluid, together with the value of the prescribed pressure in the pinned case. Specifically, for the pinned problem we show that, depending on the value of an appropriately defined capillary number and on the value of the prescribed fluid pressure, there can be either none, one, two or three equilibrium solutions with non-zero tilt angle. Furthermore, for the free problem we show that the solutions with a horizontal plate (i.e. zero tilt angle) conjectured by Moriarty and Terrill (1996) do not, in general, exist, and in fact there is a unique equilibrium solution with, in general, a non-zero tilt angle for all values of the capillary number. Finally, if time permits some preliminary results for an elastic plate will be presented.
12th Dec
2012 On adaptive methods for solving systems of evolution equations
Andreas Dedner (University of Warwick)

2:00 - - Room 1.212 Alan Turing Building

Abstract (click to view)

For a better understanding of a wide range of problems arising in both physical and industrial applications, complex systems of evolution equations have to be solved, including non-linear advection, diffusion, and reaction terms. Such models arise for example in weather prediction (the incompressible Navier-Stokes equations), in the modeling of flooding events (the viscous shallow water equations) and many more. At least two regimes have to be considered, the one where advection is dominant, the other where the diffusion processes dominate. These regimes require very different tools for analysis and methods for solving the equations. The smoothness of the solution can vary greatly in the domain, ranging from very smooth to discontinuous. This makes the use of high order scheme challenging since they are highly efficient in smooth regions and mostly unstable near discontinuities. Thus hp adaptive scheme, which lead to low order schemes on refine grids near discontinuities but use high order on coarse grids elsewhere, can lead to very effective methods. In this talk I will present some aspects of the design and analysis of hp adaptive methods for this type of advection-diffusion problems, focusing on the advection dominated case. The scheme is based on the Discontinuous Galerkin method. I will demonstrate both the effectiveness and the efficiency of this method, comparing it to a production code developed at the German Metoffice.

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