Applied mathematics topics

Supervisors: Dr Catherine Penington and Dr Justin Tzou

Topic Description:

Cells in real biological tissue exist in a crowded environment of other cells and extra-cellular matrix. Many mathematical models and experiments in a lab only include one type of cell to investigate how quickly the cells spread out and invade other tissue. This project will use partial differential equations and probabilistic individual agent models, both existing and newly developed as part of the project, to model and better understand how more representative and complex interactions with the environment affect the spread of cells in real biological tissue. There will (hopefully) be opportunities to work with experimental data on melanoma cells. A background in applied maths is important, but the biology can be learnt as part of the project and is not a prerequisite.

Supervisor: Dr Chris Lustri

Topic Description:

This project involves the study of singularly-perturbed water wave problems. It has been shown that the behaviour of certain classes of gravity waves and capillary waves require exponential asymptotic techniques in order to be calculated; however, many avenues of study remain open. Possible directions of research include gravity-capillary wave behaviour in three dimensions, waves on two-layer flows with internal interfaces, and higher-order nonlinear wave equations.

Supervisor: Dr Chris Lustri

Topic Description:

Many physical systems can be modelled as chains on particles with behaviour governed by particle interactions. These systems are valuable due to their highly tuneable nonlinear response. This project involves studying the dynamics of waves and pulses propagating through particle chains. Some particle systems support solitary waves, while others produce less robust generalized solitary waves, with oscillations that draw energy from the main pulse. This project involves the asymptotic and numerical study of solitary waves in these systems.

Supervisor: Dr Justin Tzou

Topic Description:

The study of pattern formation in reaction-diffusion systems, which model phenomena such as animal coat markings and patterned vegetation, uses various analytic and numerical methods for analyzing partial differential equations. Many such patterns can exhibit sharp gradients or large deviations from the steady state. The technical difficulties caused by these characteristics have constrained their study to domains with simple geometries, such as disks and spheres. As a result, the effect of curvature on the dynamics and stability of patterns is not yet fully understood. This project will use finite difference, asymptotic, and local analytic methods to formally characterise how localised patterns behave on surfaces of non-constant curvature. The work may involve external collaboration(s).

Supervisor: Dr Christian Thomas

Topic Description:

As fluid flows over a surface, it transitions from a smooth laminar state to a turbulent one. This process is often excited by small two- and three-dimensional disturbances that occur naturally on the wings of aircraft and in applications involving a rotating body. Recent theoretical and experimental results have shown that the right sort of surface roughness can suppress the onset of these disturbances and delay transition, potentially leading to significant reductions in fuel usage and greenhouse gas emissions. However, current theoretical models for surface roughness are based on an approximation of the surface conditions. This project will develop new innovative numerical methods for optimising laminar flow control via surface roughness.

Supervisor: Dr Christian Thomas

Topic Description:

In many physical and industrial processes, the fluid’s viscosity is non-constant and is said to be non-Newtonian. Honey, paint and blood are some examples of non-Newtonian fluids, with applications in the food processing, pharmaceutical and oil/gas industries. Generalised Newtonian fluids are one particular class of non-Newtonian fluids, in which the fluid viscosity is only dependent on the shear rate of the flow. Recent studies on laminar-turbulent transition in rotating bodies, with applications to turbomachinery, have focussed on the development of crossflow. However, these investigations have neglected several flow features that can bring about significant changes in flow mechanisms. This project involves developing our understanding of flow processes in non-Newtonian fluids past rotating bodies.