# Applied Mathematics Topics

## Applied Mathematics Topics

### Topic Title: Self-prioritising queues

Supervisor: Dr David Bulger

Topic Description:

This project seeks to build and optimise models for dynamic differential pricing of self-prioritising queue systems. Many business models today offer express service at a premium rate. Such differential pricing allows a flexibility of service which is responsive to a broader range of customer priorities. In rapidly changing queue systems such as supermarkets and road traffic networks, modern information systems have the potential to adjust and announce prices (or tolls) and projected wait (or transit) times in real time, and thereby increase predictability and satisfaction for customers.

### Topic Title: The effect of super-hydrophobic coatings on controlling boundary-layer separation and transition to turbulence

Supervisor: Dr Sophie Calabretto

Topic Description:

During the development of modern theories of fluid mechanics, there was considerable controversy over the correct physical behaviour of fluids near surfaces. The controversy was a simple one; did a fluid “stick” to the surface (yielding a “no-slip” boundary condition) or “slip” past the surface. The weight of opinion, backed up the experiments and theory available at that time, fell in the court of the no-slip boundary condition. Almost 150 years on, we now know that both are indeed possible. Indeed, through developments in surface chemistry we are now in a position where slip surfaces (most notably the so-called “super-hydrophobic surfaces”) can be readily manufactured. This project will explore how slip surfaces can be used to control boundary-layer separation and transition to turbulence.

### Topic Title: The effect of surface heating on fluid in boundary-layer flows

Supervisor: Dr Sophie Calabretto

Topic Description:

When fluid flows over a heated surface, the heating-induced changes in fluid density can serve to drive a buoyancy-induced streamwise pressure gradient. This streamwise pressure gradient can then accelerate the fluid near to the surface, leading to a region of velocity-overshoot. The physical phenomena that give rise to this velocity maximum also occur if the fluid within the boundary layer is compressible. One could conjecture then that the stability characteristics of a compressible boundary layer with velocity overshoot would closely follow that of its buoyant cousin. This does not appear to be the case. This project will explore, using advanced asymptotic methods, the reasons behind this important difference in flow behaviour.

### Topic Title: Large Eddy Simulation of transition to turbulence in rotating flows

Supervisor: Dr Sophie Calabretto

Topic Description:

Fluid flows impacted by system rotation are abundant in many engineering and technological applications. An important question for such flows is how, where, and when they undergo a transition to turbulence. One of the major challenges with understanding this process of transition in such flows is that they are usually developing both (a) spatially and (b) temporally. Classical methods to explore transition to turbulence generally require a separation between the spatial (or temporal) scale over which the instabilities develop, and that of the underlying spatially (or temporally) developing base flow. This project will exploit Large Eddy Simulations, coupled with a spectral element approach, to explore transition to turbulence in rotating flows.

### Topic Title: Simulation of the flow in particle-laden jets

Supervisor: Dr Sophie Calabretto

Topic Description:

Jet flows are ubiquitous in many engineering and technological applications. They are used in a wide variety of applications such as high-speed water jets, used for precision cutting and, of course, in all applications of jet propulsion. In many of these applications, the fluid in the jet (whether it be water, air or some other more exotic fluid) can often contain particulate matter. In many applications it is important to be able to control the transition from a laminar to a turbulent state in the jet. The addition of particulate matter to the flow can significantly impact upon how such flows transition to turbulence. This problem has received considerable attention in the past, with the main focus of research being on contained flows. However, in the case of jet flows, the underlying flow is one which develops both temporally (that is, with time) and spatially (that is, in space). Even without the additional complication of the presence of the particulate matter, classical methods to explore transition to turbulence generally require a separation between the spatial (or temporal) scale over which the instabilities develop, and that of the underlying spatially (or temporally) developing base flow.

### Topic Title: The spontaneous flow of active matter in 3D confinement

Supervisor: Dr Lyndon Koens

Topic Description:

Suspensions of bacteria or micro-machines in a viscous fluid forms an exotic state of matter called active matter. These systems inject energy into the fluid on a microscopic scale, thereby making them non-equilibrium by nature and allowing the spontaneous formation of large scale flows. Recently the Dogic lab, showed that an active system confined to a cylinder could develop large scale circulations (Wu et al. (2017) Science, 355:6331). However, this circulation was lost when the height of the cylinder became smaller than the cylinder's radius. In this project we will develop simplified viscous flow models to explore and understand the physics of these transitions in three dimensional confinement.

### Topic Title: Solitary and near-solitary waves in particle systems

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.

### Topic Title: Exponential Asymptotics in Water Wave Problems

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.

### Topic Title: Cell spreading through a crowded environment

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.

### Topic Title: The stability and dynamics of patterns on curved surfaces

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).

### Topic Title: New mathematical theory for fluid motion on surfaces with holes

Supervisor: Dr Chris Green

Topic Description:

The purpose of this project is to explore and develop new methods and theory for ideal fluid motion on surfaces with holes (compact Riemann surfaces of genus greater than zero). There are many possible directions the project could take depending on the particular interests of the student. The project will touch on a mix of mathematical topics, including complex analysis, special function theory, potential flows, and differential geometry

### Topic Title: Disorder to order in programmable matter

Supervisor: Dr Lyndon Koens

Topic Description:

Collections of microscopic moving objects display many complex behaviours.  The study of these systems are often key to understanding the infection of bacteria, the dynamics of cells and how to manipulate and the control of the microscopic machines. In these systems, small changes in the individual members can lead to large changes in the collective motion, sometimes reminiscent of the phase transitions between solids, liquids and gasses. This project will create models to explore the transition between disordered (liquid-like) states and ordered (solid-like) states in an experimental collection of microscopic rotating disks. This will improve our understanding of these complex dynamical systems and guide the creation of new programmable matter.

### Topic Title: Flows through arrays of complex shapes

Supervisor: Dr Lyndon Koens

Topic Description:

Periodic arrays of objects occur in many soils and microscopic technologies. These technologies often aim to produce portable medical devices, that could sort bodily fluids, like blood, on-site. The sorting of these devices depends critically on how the object moves through the given array. However, in many cases, even the flow through the array is not well understood. This project will develop approximate representations of the flow-through arrays of cylinders with different cross-sections. This will guide the development of future sorting microscopic technologies.

### Topic Title: Equivalent representations of viscous flows

Supervisor: Dr Lyndon Koens

Topic Description:

There are many different ways to determine the behaviour of viscous flows. Though ultimately equivalent, the method to convert one representation into another is not always obvious. As such some of these methods are deemed 'less physical' than others. This project will bridge the gap between two such representations.  With the aid of a bipolar coordinate system, it will connect two representations of the flow around a point force above a wall. This will change the way we view the many representations of viscous flows and open a multitude of new avenues for evaluating viscous problems.