Past Differential Equations and Applications Seminar

18 February 2010
16:30
Cameron Hall (OCCAM)
Abstract
If an ideal elastic spring is greatly stretched, it will develop large stresses. However, solid biological tissues are able to grow without developing such large stresses. This is because the cells within such tissues are able to lay down new fibres and remove old ones, fundamentally changing the mechanical structure of the tissue. In many ways, this is analogous to classical plasticity, where materials stretched beyond their yield point begin to flow and the unloaded state of the material changes. Unfortunately, biological tissues are not closed systems and so we are not able to use standard plasticity techniques where we require the flow to be mass conserving and energetically passive. In this talk, a general framework will be presented for modelling the changing zero stress state of a biological tissue (or any other material). Working from the multiplicative decomposition of the deformation gradient, we show that the rate of 'desired' growth can represented using a tensor that describes both the total rate of growth and any directional biases. This can be used to give an evolution equation for the effective strain (a measure of the difference between the current state and the zero stress state). We conclude by looking at a perhaps surprising application for this theory as a method for deriving the constitutive laws of a viscoelastic fluid.
  • Differential Equations and Applications Seminar
11 February 2010
16:30
Peter Howell (OCIAM)
Abstract
We study the axisymmetric stretching of a thin sheet of viscous fluid driven by a centrifugal body force. Time-dependent simulations show that the sheet radius tends to infinity in finite time. As the critical time is approached, the sheet becomes partitioned into a very thin central region and a relatively thick rim. A net momentum and mass balance in the rim leads to a prediction for the sheet radius near the singularity that agrees with the numerical simulations. By asymptotically matching the dynamics of the sheet with the rim, we find that the thickness in the central region is described by a similarity solution of the second kind. For non-zero surface tension, we find that the similarity exponent depends on the rotational Bond number B, and increases to infinity at a critical value B=1/4. For B>1/4, surface tension defeats the centrifugal force, causing the sheet to retract rather than stretch, with the limiting behaviour described by a similarity solution of the first kind.
  • Differential Equations and Applications Seminar
4 February 2010
16:30
Jonathan Wattis
Abstract
The FPU lattice is a coupled system of ordinary differential equations in which each atom in a chain is coupled to its nearest neighbour by way of a nonlinear spring. After summarising the properties of travelling waves (kinks) we use asymptotic analysis to describe more complicate envelope solutions (breathers). The interaction of breathers and kinks will then be analysed. If time permits, the method will be extended to two-dimensional lattices.
  • Differential Equations and Applications Seminar
28 January 2010
16:30
Abstract
Some years ago Hall and Smith in a number of papers developed a theory governing the interaction of vortices and waves in shear flows. In recent years immense interest has been focused on so-called self-sustained processes in turbulent shear flows where the importance of waves interacting with streamwise vortex flows has been elucidated in a number of; see for example the work of Waleffe and colleagues, Kerswell, Gibson, etc. These processes have a striking resemblance to coherent structures observed in turbulent shear flow and for that reason they are often referred to as exact coherent structures. It is shown that the structures associated with the so-called 'lower branch' state, which has been shown to play a crucial role in these self-sustained process, is nothing but a Rayleigh wave vortex interaction with a wave system generating streamwise vortices inside a critical layer. The theory enables the reduction of the 3D Navier Stokes equations to a coupled system for a steady streamwise vortex and an inviscid wave system. The reduced system for the streamwise vortices must be solved with jump conditions in the shear across the critical layer and the position of that layer constitutes a nonlinear pde eigenvalue problem. Remarkable agreement between the asymptotic theory and numerical simulations is found thereby demonstrating the importance of vortex-wave interaction theory in the mathematical description of coherent structures in turbulent shear flows. The theory offers the possibility of drag reduction in turbulent shear flows by controlling the flow to the neighborhood of the lower branch state. The relevance of the work to more general shear flows is also discussed.
  • Differential Equations and Applications Seminar
21 January 2010
16:30
Abstract
Patterns of sources and sinks in the complex Ginzburg-Landau equation Jonathan Sherratt, Heriot-Watt University The complex Ginzburg-Landau equation is a prototype model for self-oscillatory systems such as binary fluid convection, chemical oscillators, and cyclic predator-prey systems. In one space dimension, many boundary conditions that arise naturally in applications generate wavetrain solutions. In some contexts, the wavetrain is unstable as a solution of the original equation, and it proves necessary to distinguish between two different types of instability, which I will explain: convective and absolute. When the wavetrain is absolutely unstable, the selected wavetrain breaks up into spatiotemporal chaos. But when it is only convectively stable, there is a different behaviour, with bands of wavetrains separated by sharp interfaces known as "sources" and "sinks". These have been studied in great detail as isolated objects, but there has been very little work on patterns of alternating sources and sinks, which is what one typically sees in simulations. I will discuss new results on source-sink patterns, which show that the separation distances between sources and sinks are constrained to a discrete set of possible values, because of a phase-locking condition. I will present results from numerical simulations that confirm the results, and I will briefly discuss applications and the future challenges. The work that I will describe has been done in collaboration with Matthew Smith (Microsoft Research) and Jens Rademacher (CWI, Amsterdam). ------------------------------
  • Differential Equations and Applications Seminar
3 December 2009
16:30
Charlie Elliott
Abstract
Evolutionary PDEs on stationary and moving surfaces appear in many applications such as the diffusion of surfactants on fluid interfaces, surface pattern formation on growing domains, segmentation on curved surfaces and phase separation on biomembranes and dissolving alloy surfaces. In this talk I discuss three numerical approaches based on:- (I) Surface Finite Elements and Triangulated Surfaces, (II)Level Set Method and Implicit Surface PDEs and (III) Phase Field Approaches and Diffuse Surfaces.
  • Differential Equations and Applications Seminar
26 November 2009
16:30
Tim Myers
Abstract
Modelling phase change in the presence of a flowing thin liquid film There are numerous physical phenomena that involve a melting solid surrounded by a thin layer of liquid, or alternatively a solid forming from a thin liquid layer. This talk will involve two such problems, namely contact melting and the Leidenfrost phenomenon. Contact melting occurs, for example, when a solid is placed on a surface that is maintained at a temperature above the solid melting temperature. Consequently the solid melts, while the melt layer is squeezed out from under the solid due to its weight. This process has applications in metallurgy, geology and nuclear technology, and also describes a piece of ice melting on a table. Leidenfrost is similar, but involves a liquid droplet evaporating after being placed on a hot substrate. This has applications in cooling systems and combustion of fuel or a drop of water on a hot frying pan. The talk will begin with a brief introduction into one-dimensional Stefan problems before moving on to the problem of melting coupled to flow. Mathematical models will be developed, analysed and compared with experimental results. Along the way the Heat Balance Integral Method (HBIM) will be introduced. This is a well-known method primarily used by engineers to approximate the solution of thermal problems. However, it has not proved so popular with mathematicians, due to the arbitrary choice of approximating function and a lack of accuracy. The method will be demonstrated on a simple example, then it will be shown how it may be modified to significantly improve the accuracy. In fact, in the large Stefan number limit the modified method can be shown to be more accurate than the asymptotic solution to second order.
  • Differential Equations and Applications Seminar
19 November 2009
16:30
Stephen Creagh
Abstract
Whispering gallery modes in optical resonators have received a lot of attention as a mechanism for constructing small, directional lasers. They are also potentially important as passive optical components in schemes for coupling and filtering signals in optical fibres, in sensing devices and in other applications. In this talk it is argued that the evanescent field outside resonators that are very slightly deformed from circular or spherical is surprising in a couple of respects. First, even very small deformations seem to be capable of leading to highly directional emission patterns. Second, even though the undelying ray families are very regular and hardly differ from the integrable circular or spherical limit inside the resonator, a calculation of the evanescent field outside it is not straightforward. This is because even very slight nonintegrability has a profound effect on the complexified ray families which guide the external wave to asymptopia. An approach to describing the emitted wave is described which is based on canonical perturbation theory applied to the ray families and extended to comeplx phase space.
  • Differential Equations and Applications Seminar

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