PhD Projects in Geometry Mechanics and Fluids
Research in this area studies a geometric approach to Hamiltonian systems and their manifestation in areas such as mechanics and fluid dynamics. Topics range from fundamental aspects of symplectic and multi-symplectic geometry to their applications to (quasi-)periodically forced systems, bifurcation theory, structure preserving discretizations and data assimilation, and to studies of specific systems such as N-body problems, electronic circuits, Josephson junctions, weather prediction and oceanographic waves. A particular emphasis is placed on studying the consequences of symmetries possessed by these systems. Cross-disciplinary research topics includes the dynamics of spacecraft (with the Surrey Space Centre) and environmental fluid flows (with the Fluid Dynamics Group and the Centre for Environmental Strategy). More details, including staff members working in this area, can be found on the Geometry, Mechanics and Fluids research group pages. Below are some examples of PhD projects in geometry, mechanics and fluids. Alternative projects might be formulated following discussions with individual staff members, just contact the staff member or PhD admissions tutor Dr Gianne Derks.
Numerical weather prediction: multi-scale variational data assimilation and high-resolution models (Supervisor: Prof. Ian Roulstone)
Assessing risk from extremes of weather and climate, evaluating sustainable energy resources, and generally improving our ability to adapt to a changing climate, all require detailed knowledge of how weather patterns will affect our lives in the future. Reliable weather forecasts are currently issued for 3 to 4 days ahead at continental and global scales, though more detailed local/regional information from higher-resolution weather prediction models requires a radical new approach (because relatively rapid, unstable nonlinear processes become more important).
This project will contribute to the development of high-resolution numerical weather prediction systems (which are recognized as an important part of evaluating the impact of climate change at local or regional scales) and their use to generate meaningful information for end users e.g. where forecasts are used as part of a flood warning system. Success will lead to immediate benefits for all users of forecasts (e.g. Environment Agencies, Aviation Authorities (e.g. CAA), etc.).
Theory and numerics of reversing symmetry breaking bifurcations of Hamiltonian relative periodic orbits (Supervisor: Dr Claudia Wulff)
Relative periodic orbits (RPOs) of symmetric dynamical systems are periodic orbits for the symmetry reduced system. Both the theory and numerics of periodic orbits of general dynamical systems is well-developed. But additional structure such as time-reversing and preserving symmetries and symplecticity changes the generic behaviour of dynamical systems dramatically. Recently there has been a lot of progress in the development of a bifurcation theory for symmetric and Hamiltonian systems, but the theory is far from being complete. In particular the bifurcation theory of reversible symmetric periodic and relative periodic orbits, is still in its beginning, as is the numerical analysis of symmmetry breaking bifurcations of reversible periodic orbits.
The topic of this project is to analyze reversing symmetry breaking bifurcations of RPOs theoretically and to derive and implement numerical methods for their detection and computation within the package SYMPERCON of Wulff, Schebesch, Schilder. The results will be applied to various symmetric Hamiltonian systems, in particular to N-body systems.
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.
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.
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.
Soliton Switching in Fibres (Supervisor: Dr Gianne Derks)
For the optical transmission of data across a cable, one can use two (or more) coupled fibre cables. Experiments have shown that if a certain type of signal is put at one end of the cable, it will go to the other end of this cable and hardly anything happens in the other cable. However, if one puts other types of signals on the cable, the signal will switch to the other cable. This gives a convenient way of sending data consisting of zeros and ones. In this project we will aim for a better understanding of this experimentally observed process by investigating the family of soliton-like solutions, especially issues like existence, stability, bifurcations and invariant manifolds will be investigated.
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.
Multi-symplectic structures and nonlinear PDEs (Supervisor: Prof Thomas J. Bridges)
An understanding of nonlinear PDEs is one of the great challenges for the twenty-first century. In this project the geometry of nonlinear PDEs is studied. Multi-symplectic structure, which generalises classical symplectic geometry, is used as a backbone for the analysis. The project has three parts:
- Develop the geometry of the total exterior algebra bundle for smooth two-dimensional manifolds and construct Dirac operators on this bundle as a basis for canonical hyperbolic and elliptic PDEs.
- Use the calculus of variations to find solutions of multi-symplectic Hamiltonian PDEs, using the Mountain Pass Theorem and related functional analytical tools.
- Apply numerical methods to either part (I) or part (II).