PhD projects in Geometrical Fluid Dynamics
This research area studies atmospheric, oceanographic and physiological fluid flow, using geometric mathematical methods and the calculus of variations. Problems of interest include numerical weather prediction, analysis of models for atmospheric dynamics, mathematical meteorology, singularities in the calculus of variations, optimal transport, the study of waves on the ocean surface, geophysical fluid dynamics, data assimilation, symmetry of fluid dynamics, and geometric numerical methods. Geometric mathematics used includes symplectic geometry, calculus of variations, Hamiltonian mechanics, symmetry methods, stochastic analysis, multi-symplectic geometry, differential geometry of manifolds, and the theory of Lie groups. More details, including staff members working in this area, can be found on the geometrical fluid dynamics pages.
Below are some examples of PhD projects in geometrical fluid dynamics. Alternative projects might be formulated following discussions with individual staff members, just contact the staff member or PhD admissions tutor Dr Gianne Derks.
Granular matter (Supervisor: Dr Anne Skeldon)
The equations that govern the motion of fluids are well-established. Less well understood is the motion of particulate materials, such as sand, soil or powders: sometimes these flow like fluids--you can pour gravel out of a bucket; and sometimes they behave like a solid--you can make a castle out of sand but not out of water. This project is joint with Prof Ugur Tuzun in Chemical and Process Engineering at Surrey to develop data analysis techniques/understand the physics of granular media.
Faraday waves (Supervisor: Dr Anne Skeldon)
Patterns can be made to form on the surface of a container of fluid by shaking the container up and down. The shaking has to be at the right frequency and right amplitude for any patterns to form--if you move a container of fluid up and down very slowly then the surface of the fluid will remain flat. There are a wealth of experimental results showing a variety of different possible patterns. On the theoretical front, arguments based on symmetries and the way different patterns interact has led to some knowledge of the mechanisms that cause particular patterns to become dominant. Although these have been tested on mathematical models describing the fluid, there are still significant gaps in our understanding. The aim of this project is to investigate further the theory underlying pattern selection in the Faraday and related problems. This will require using a variety of different dynamical systems techniques both theoretical and computational.
Attractors for hydrodynamical problems in unbounded domains (Supervisor: Dr Sergey Zelik)
The concept of a global attractor plays a very important role in the modern theory of dissipative systems generated by PDEs. For the case of equations in bounded domains (like a square, disk or ball), this attractor is usually finite-dimensional. Thus, despite the infinite-dimensionality of the initial phase space, the limit dynamics reduced to the attractor occurs finite-dimensional and can be studied by the methods of the classical dynamical systems theory. The situation is more difficult for the case where the underlying domain is unbounded (like R2 or R3) since the dimension of the attractor is typically infinite and the finite-dimensional reduction is no more possible. Nevertheless, a reasonable theory has been recently developed for large class of such systems. The aim of this project is to extend/apply this theory to hydrodynamical problems in unbounded (cylindrical) domains. Such extension was problematic during a long time since even the well-posedness of the Navier-Stokes equation in 2D unbounded domains in the proper classes of spatially non-decaying solutions was not known. The situation is changed now due to recent results on the solvability and dissipativity of 2D and 3D Navier-Stokes equations in cylindrical domains.
Chaotic Advection in Large Airways (Supervisor: Prof Peter Hydon)
Babies who are born very prematurely usually need mechanical help to breathe. One particularly effective type of mechanical ventilator uses High Frequency Ventilation (HFV), in which a low volume of air is delivered to the lung at a frequency of 10-15 Hz. However the volume of air used is so small that the conventional theory of breathing does not explain how adequate gas transport can be achieved. It has recently been shown that a type of stirring called`chaotic advection' greatly enhances transport in small airways. The purpose of this project is to find out the extent to which chaotic advection operates in the large airways, and thus to develop strategies for obtaining the best possible transport at various frequencies. This project combines mathematical biology, dynamical systems theory and computational fluid dynamics. It is of direct relevance to medicine, and the student will have the opportunity to work with some of the UK's leading medical specialists in HFV.
Methods for Obtaining Discrete Symmetries (Supervisor: Prof Peter Hydon)
Differential equations are widely used as models of real systems. To understand the behaviour of a system, it is usually necessary to find the symmetries of the model differential equation. Recent work at Surrey has led to the first systematic technique for obtaining discrete symmetries. The purpose of this project is to simplify and extend the technique, and to create a Maple package that will systematically obtain the discrete symmetries of a given differential equation. This project combines differential equation techniques, linear algebra, group theory and Maple programming. It is expected that the resulting package will be distributed with future releases of Maple, and will be used by those working in many areas of mathematics and theoretical physics.
