Differential Equations and Applications Seminar
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Thu, 21/01/2010 16:30 |
Jonathan Sherratt (Herriot-Watt University, Edinburgh) |
Differential Equations and Applications Seminar  |
DH 1st floor SR |
| 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). —————————— | |||
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Thu, 28/01/2010 16:30 |
Phil Hall (Imperial College London) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| 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. | |||
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Thu, 04/02/2010 16:30 |
Jonathan Wattis (Nottingham) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| 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. | |||
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Thu, 11/02/2010 16:30 |
Peter Howell (OCIAM) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| 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. | |||
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Thu, 18/02/2010 16:30 |
Cameron Hall (OCCAM) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| 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. | |||
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Thu, 25/02/2010 16:30 |
Sam Howison (Oxford) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
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Thu, 04/03/2010 16:30 |
Andrea Bertozzi, UCLA, USA |
Differential Equations and Applications Seminar |
OCCAM Common Room (RI2.28) |
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Thu, 11/03/2010 16:30 |
Richard Braun (University of Delaware) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| We study dynamics from models for the human tear film in one and two dimensional domains. The tear film is roughly a few microns thick over a domain on a centimeter scale; this separation of scales makes lubrication models desirable. Results on one-dimensional blinking domains are presented for multiple blink cycles. Results on two-dimensional stationary domains are presented for different boundary conditions. In all cases, the results are sensitive to the boundary conditions; this is intuitively satisfying since the tear film seems to be controlled primarily from the boundary and its motion. Quantitative comparison with in vivo measurement will be given in some cases. Some discussion of tear film properties will also be given, and results for non-Newtonian models will be given as available, as well as some future directions. | |||
