Second AGM Meeting on Geometric Fluid Dynamics 2015-2016
22 January 2016, University of Surrey
Lecture Theatre A (LTA), Lecture Theatre Block
[IMPORTANT! -- All non-speaker attendees are requested to register by writing to Paul Skerritt]
This one-day meeting will cover various disciplines related to geometric fluid dynamics. Specific topics will include:
-- complex fluids,
-- geometry of fluid flows,
-- geometric integration,
-- multiscale fluid models,
-- stochastic methods.
Timetable:
10.00 -- 10.45 Peter Hydon (Kent)
10.45 -- 11.15 coffee break
11.15 -- 12.00 François Gay-Balmaz (CNRS/ENS Paris)
12.00 -- 12.45 Emanuele Tassi (CNRS/CPT Marseille)
12.45 -- 14.15 lunch
14.15 -- 14.45 Matthew Turner (Surrey)
14.45 -- 15.15 Andrea Natale (Imperial)
15.15 -- 15.45 coffee break
15.45 -- 16.15 Paul Skerritt (Surrey)
16.15 -- 16.45 Tomasz Tyranowski (Imperial)
Titles and Abstracts
Peter Hydon
Conservation laws: what’s the difference?
Conservation laws are among the most fundamental attributes of a given system of partial differential equations, yet the theory of finite difference conservation laws has been developed only recently. This talk compares differential and difference conservation laws, which have analogous structures in very different contexts. One useful outcome is a result that bridges Noether’s First and Second Theorems, yielding both conservation laws and Bianchi-type identities. Another is a new approach to obtaining finite difference approximations that preserve multiple conservation laws of a given differential equation.
François Gay-Balmaz
Free and moving boundary fluids: variational and geometric properties
The equations of motion of free boundary continuum mechanics can be
equivalently formulated in the material, spatial, or convective
representations. In this talk I will present the relation between these
three formulations by using the material and spatial symmetries of the
continuum system, in a general setting that includes both fluid dynamics
and nonlinear elasticity.
I will then mention how the spatial and convective representations are
both crucial for the study of fluid-structure interaction problems.
Emanuele Tassi
Hamiltonian reduced fluid models for plasmas
Progress in the understanding of astrophysical and laboratory plasma dynamics often relies on the use of fluid models. Such models are typically derived starting from kinetic equations, taking moments of distribution functions and truncating the resulting infinite hierarchy of fluid equations by imposing a closure relation. When dissipative effects are neglected, the original kinetic model is supposed to possess a Hamiltonian structure and the closure relation should be such to preserve a Hamiltonian structure in the resulting fluid model. In this talk I will present closure relations that permit to obtain Hamiltonian reduced fluid models starting from a Hamiltonian drift-kinetic system, which applies to plasmas in the presence of an intense magnetic field.
Matthew Turner
Two-layer shallow water sloshing with a rigid lid
In this talk we investigate two layer shallow water sloshing in a rectangular container with a rigid lid. This problem has the constraint that h_1(x,t)+h_2(x,t)=d where h_1 is the fluid height in layer 1, h_2 is the fluid height in layer 2 and d is the fixed height of the container. This constraint has to hold for all spatial positions x and time t. This is an Eulerian constraint, but the Eulerian form of the Lagrangian for this problem is not immediately conducive to energy preserving numerical schemes. In this talk we show how to convert the Eulerian Lagrangian into a Lagrangian Particle Path formulation, while still satisfying the Eulerian constraint above. The Lagrangian in this Particle Path formulation is conducive to energy preserving numerical schemes. Some numerical results will be presented.
Andrea Natale
Lie derivative discretisations for perfect fluids
The discretisation of the Lie derivative is a crucial part of devising numerical methods for fluid models. As a matter of fact, much of the mathematical structure of such models is hidden in this operator and in its analytical and algebraic properties. In this talk I will explain the main challenges arising when devising discretisation methods for the Lie derivative in the case of perfect fluids. Moreover, I will develop on the work of Holger Heumann on advection of differential forms to present a promising discretisation approach.
Paul Skerritt
The Frame Bundle Approach to Liquid Crystals
In this talk we discuss a formulation of liquid crystal dynamics in terms of frame bundles. In particular, we explain how Eringen's micropolar theory of liquid crystals has a natural interpretation as a gauge theory in the orthonormal frame bundle of the liquid crystal domain, with Eringen's wryness tensor playing the role of a connection. The frame bundle picture is a reformulation of prior work of Holm, Tronci, Gay-Balmaz, and Ratiu on the gauge theory of liquid crystals.
Tomasz Tyranowski
Variational Principles for Stochastic Soliton Dynamics
We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension we numerically simulate singular solutions (peakons) of the stochastically perturbed Camassa-Holm (CH) equation derived using this method. These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to finite dimensional solutions of stochastic partial differential equations (SPDE). In particular, some choices of stochastic perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian stochastic deformations, or CH-SD) allow peaks to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler-Poincaré structure of the CH equation (parametric stochastic deformations, or
P-SD), and it also does not occur for peakon solutions of the unperturbed deterministic CH equation. The discussion raises issues about the science of stochastic deformations of finite-dimensional approximations of evolutionary PDE and the sensitivity of the resulting solutions to the choices made in stochastic modelling.
Directions to Campus
At Guildford rail station, take the University exit (not the town centre exit) and turn right as soon as you get out. Then, take the second right (the first is a cul-de-sac). Walk by the parking lot until you enter campus. Keep going along the same road, which continues to the left, next to a big pond. At the end of the road, go between the buildings, past the laundrette and up two flights of stairs. The side entrance to the Lecture Block will be ahead on the right, and Lecture Theatre A is just inside.
The Lecture Theatre Block is labelled LT on this campus map (in square H5).
You can view the entire route from the station on Google Maps here.
Edit 22/01/2016: If you'd prefer not to walk in the miserable weather this morning, there is a bus stop across the road from the station University exit. Buses 27, 36, and 37 all go to campus. The nearest bus stop to the Lecture Theatre Block is Senate Square.
Contacts:
Paul Skerritt (main organizer)