Industrial and Applied Mathematics PhD projects

This page provides a (partial) list of specific (and not so specific) PhD projects currently offered around the Industrial and Applied Mathematics topic.

Identifying an interesting, worthwhile and do-able PhD project is not a trivial task since it depends crucially on the interests and abilities of the student (and the supervisor!) The list below therefore contains a mixture of very specific projects and fairly general descriptions of research interests across the School. In either case you should feel free to contact the potential supervisors to find out more if you're interested.

You should also explore the School's research groups' pages, and have a look around the homepages of individual members of staff to find out more about their research interests. You may also contact us for general enquiries.

Title

Thermo-visco-acoustic metamaterials for underwater applications

Group Industrial and Applied Mathematics
Supervisor
Description

The ability to control underwater noise has been of practical interest for decades. Such noise, radiating from e.g. offshore wind farms, turbines, and merchant vessels, frequently needs to be attenuated artificially given the close proximity of its generation to sensitive marine environments for example.

Over the last century a number of materials have been designed to assist with underwater noise attenuation. However, recently there has been an explosion of interest in the topic of acoustic metamaterials and metasurfaces. Such media have special microstructures, designed to provide overall (dynamic) material properties that natural materials can never hope to attain and lead to the potential of negative refraction, wave redirection and the holy grail of cloaking. Many of the mechanisms to create these artificial materials rely on the notion of resonance, which in turn gives rise to the possibility of low frequency sound attenuation. This is extremely difficult to achieve with normal materials.

The mechanisms of sound attenuation, i.e. thermal and viscous, have not yet been properly understood for the many metamaterials under study, particularly in an underwater context. The aim of this project is to study this aspect via mathematical analysis and then to optimize designs in order to design and employ metamaterials for use in underwater noise reduction applications. Although there has been some initial interest over the last few years in the “in-air” context, the parameter regime underwater gives rise to new effects that need to be explored and understood thoroughly.

Initially canonical geometries such as simple apertures and infinite and semi-infinite ducts shall be considered before moving on to more complex, realistic scenarios and geometries where resonance plays a key role.

Mathematical modelling using the method of matched asymptotics shall be employed. This is ideally suited to the scenarios considered given the low frequency regime. Comparisons shall be drawn with direct numerical simulations using finite element methods in e.g. COMSOL.

Title

Convective mass transfer for cleaning and decontamination

Group Industrial and Applied Mathematics
Supervisors
Description

Cleaning and decontamination processes can rely on different mechanisms to remove a patch of alien substance attached to a substrate. A shear flow covering the substrate can remove the substance through mechanical forces, potentially combined with chemical surfactant agent decreasing the adhesion of the substance onto the surface. However, this project is concerned with a second type of mechanism which is based on the dissolution of the substance into the cleaning fluid flow covering the substance.

This second type of cleaning process establishes a convective mass transfer between the alien phase and the cleaning phase. Several applications rely on this process, particularly when the dispersion of the substance is unwanted, such as in the decontamination process of toxic chemical spills. In our daily life, the cleaning mechanism more and more favoured in dishwashers relies also on a convective mass transfer as it has been shown empirically to reduce energy and water consumption.

This project will focus on the case of a film flow covering a single droplet containing several substances. Many fundamental questions are still unresolved in this multiphase convective mass transfer problem. In particular, we will study how advection processes inside the drop can influence the convective mass transfer. Effect of solubility and surface tension on the overall mass transfer can also be analysed. The project will explore these questions using a combination of experimentation, numerical simulations and theoretical analysis.

The project is suitable for an enthusiastic and creative candidate who has good knowledge in fluid mechanics and some experience in experimentation and numerical simulations.

Reference: J. R. Landel, A. L. Thomas, H. McEvoy and S. B. Dalziel. Convective mass transfer from a submerged drop in a thin falling film, Journal of Fluid Mechanics, 2016.

Title

Turbulent particle-laden jets

Group Industrial and Applied Mathematics
Supervisors
Description

Turbulent particle-laden jets are relevant to many geophysical and industrial applications: from volcanic eruptions, to sediment resuspension, fluidisation processes and chemical reactors. Much work has been done on the dilute regime of these two-phase flows, where the particles have a small impact on the fluid and can often be considered as passive tracers. In this experimental project, we focus on the poorly understood dense regime, where the coupling between the solid particles and the fluid is more complex.

Many fundamental questions, of high relevance to the applications mentioned above, are still unresolved. This project will explore the impact of the particle density on turbulent entrainment processes. Entrainment processes during an explosive volcanic eruption have a considerable impact on the extent of the damages. They determine whether the eruption will collapse and form a pyroclastic flow, with local implications, or whether the eruption column will rise and form an ash cloud spreading over extended regions, such as in the case of the 2010 eruption of the Icelandic volcano Eyjafjallajökul. This project will also explore the effect on mixing processes, which are very important for instance in chemical reactors where the efficiency of the reaction depends strongly on the efficiency of the mixing.

These dense particle-laden jets are still poorly understood due to the considerable challenges faced analytically and numerically. Technical difficulties have also prevented progress on the experimental side for a long time. New experimental techniques, based on novel experimental design and imaging techniques, recently developed in the laboratory have allowed to probe much further into the complex dynamics of these dense particle laden jet. The main goal of this project is to pursue the development of these techniques in order to address the questions on entrainment and mixing described above.

The project is suitable for an enthusiastic and creative candidate who has some experience in experimentation and good knowledge in fluid mechanics. Some knowledge in imaging analysis technique is desired but not necessary. The motivation and readiness of the candidate to learn new techniques and develop them to explore fundamental scientific questions will be key to the success of this project.

Title

Segmentation and mathematical modelling of cerebrospinal fluid in the vertebral column

Group Industrial and Applied Mathematics
Supervisor
Description

In the human body, the spinal cord, transmitting bi-directional information between brain and body, is found within the bony vertebral column. It is contained inside a membranous sac, the dura, and bathed in cerebrospinal fluid (CSF), which circulates throughout the central nervous system in the subarachnoid space between dura and cord. Blood vessels and nerves enter and exit through the walls of the dural sac along its length. During each cardiac cycle a cyclic flow and pressure change has been observed within the CSF having the same period as the cardiac cycle. However, little is known about the precise nature or mechanism of this “CSF circulation”, including how the geometry of the subarachnoid space may affect the pressure of the CSF and its flow pattern. This knowledge would further our understanding of several pathological conditions of the central nervous system.

With this in mind, the objective of this project is to understand the dynamics of the CSF, contained within the subarachnoid space of the spinal column, throughout a cardiac cycle. To achieve this goal the student will use 4D phase-contrast MR images, which are capable of detecting the velocity field of the CSF. In the beginning stages the focus will be on developing segmentation methods capable of creating a three-dimensional model of the subarachnoid space allowing visualization of the changes throughout a cycle. This segmentation is difficult due to the complicated geometry of the subarachnoid space including vasculature and exiting nerves. Once this first stage is complete the student will use the resulting segmentation to produce a computational model for the flow of CSF throughout the cardiac cycle. The goal of this modeling will be to measure local relative pressure changes of the CSF, as well as to understand the interaction between the CSF pressure and the geometry of the subarachnoid space.

Though based in the school of mathematics, this project will also involve close collaboration with researchers at the Royal Preston Hospital’s Neurosurgery and Neuroradiology departments who will be providing the MR images, produced as part of an ongoing study into CSF circulation in health and disease. 

Title

Efficient solvers for multi-physics problems in drilling

Group Industrial and Applied Mathematics
Supervisor
Description

This EPSRC Industrial Case Studentship is suitable for students with a strong applied mathematics background, particular in fluid and/or solid mechanics, and good computer programming skills.

Drilling mud motors work in extreme environments with strong coupling between complex fluid (mud) and mechanical components.  Efficient numerical simulation is a key engineering design tool, but fast solvers for these multi-physics systems are lacking. The aim of this project is to develop such solvers for modern multi-core desktops and supercomputers.

The solvers will be developed within the framework of our in-house open-source finite element library (oomph-lib).

Title

Modelling the formation of uranium hydride blisters

Group Industrial and Applied Mathematics
Supervisor
Description

An industrially-sponsored applied mathematics project that aims to develop quantitative models describing the corrosion of uranium by hydrogen via the formation and growth of discrete uranium hydride blisters. This project is closely related to another industrially-sponsored project on mathematical modelling of diffusion-driven oxidation in metals. The proposed PhD will study the blister-formation problem at a fundamental level using continuum mechanical models that couple the transport and reaction of chemicals, transport and conversion of heat and deformation of the material.

Typically, the timescales of each process are well separated so that dynamic deformation effects (oscillations) on the order of seconds are faster than the transport of heat which is faster than the diffusion of chemicals.

The project will contain a strong modelling component and will pursue analytical and numerical approaches for the solution of the resulting model equations. The initial model development will be performed in a low-dimensional framework to allow rapid assessment of the influence of different assumptions. The later stages of the project will extend the framework to a realistic three-dimensional geometry. The project is suitable for any student with a strong applied mathematics background.

Title

Interactions between rocks and ice

Group Industrial and Applied Mathematics
Supervisor
Description

Many glaciers are covered by a debris layer whose presence has multiple, competing effects on the glacier's melt rate. The debris layer shields the ice from incoming solar radiation and thus reduces its melt rate. However, since the albedo of the debris layer is much smaller than that of the ice, the debris layer is heated up very rapidly by the solar radiation, an effect that is likely to increase the melt rate.

The project aims to develop theoretical/computational models to study how solid objects (rocks) which are placed on (or embedded in) an ice layer affect the ice's melt rate. The work will employ (and contribute to) the object-oriented multi-physics finite-element library oomph-lib, developed by M. Heil and A.L. Hazel and their collaborators, and available as open source software at http://www.oomph-lib.org.

The project would suit students with an interest in mathematical modelling, continuum mechanics and scientific computing and will be performed in close collaborations with Glaciologists at the University of Sheffield and the Bavarian Academy of Science.

Title

Mathematical theory of diffraction

Group Industrial and Applied Mathematics
Supervisor
Description

There is a long history of mathematicians working on canonical diffraction (or scattering) problems. The mathematical theory of diffraction probably started with the work of Sommerfeld at the end of the 19th century and his famous solution to the diffraction of acoustic waves by a solid half-plane. Since, some very ingenious mathematical methods have been developed to tackle such problems. One of the most famous being the Wiener-Hopf technique.

However, despite tremendous efforts in this field, some canonical problems remain open, in the sense that no clear analytical solution is available for them.

One of this problem is the quarter-plane problem, the problem of diffraction of acoustic waves by a solid quarter-plane. Thus far, it has not been possible to apply classical methods such as the Wiener-Hopf method successfully in that case, and hence some new mathematical methods need to be developed in order to tackle this problem. This makes it very interesting as it implies that many different types of mathematics can be used and it makes the subject intrinsically multidisciplinary.

Many industrial problems can be linked to the theory of diffraction, for example the noise generated by a jet engine (acoustic waves) or radar detection (electro-magnetic waves) and defect detection in materials (elastic waves).

PhD projects are available in this field.

References:
-- R. C. Assier and N. Peake. On the diffraction of acoustic waves by a quarter-plane. Wave Motion, 49(1):64-82, 2012
-- R. C. Assier and N. Peake. Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane. IMA J. Appl. Math., 77(5):605-625, 2012.

Title

Combustion instabilities

Group Industrial and Applied Mathematics
Supervisor
Description

Combustion is essential to energy generation and transport needs, and will remain so for the foreseeable future. Mitigating its impact on the climate and human health, by reducing its associated emissions, is thus a priority. One suggested strategy to reduce NOx is to operate combustors at lean conditions. Unfortunately, combustion instability is more likely to occur in the lean regime, and may have catastrophic consequences on the components of combustion chambers, such as vibrations and structural fatigue.

Ramjet engines, rocket engines and in general any type of gas turbine engines may be subject to this detrimental instability. The ability to predict and control the instability is crucial for implementing the lean burn strategy. Combustion instability involves an intricate interplay of several key physical processes, which take place in regions of different length scales. Due to this multi-scale, multi-physics nature of the problem, direct numerical simulations of realistic combustors are extremely challenging. For this reason, simplified mathematical models capturing qualitatively and quantitatively the main characteristics of combustion instability are essential. In particular, by exploiting the scale disparity, systematic asymptotic analyses may be carried out to derive relevant models on first principles, and to provide guidance for developing reliable and efficient numerical algorithms.

Recent progress have been made in the mathematical modelling of such instabilities, using refining and implementing this model would make a good PhD project.

Reference:
R. C. Assier and X. Wu. Linear and weakly nonlinear instability of a premixed curved flame under the influence of its spontaneous acoustic field. J. Fluid Mech. (2014), vol. 758, pp. 180-220

Title

Efficient Uncertainty Quantification for PDEs with Random Data

Group Industrial and Applied Mathematics
Supervisor
Description

Uncertainty Quantification (UQ) is the science of accounting for uncertainty in mathematical models. Research in this area has undergone rapid growth in the last few years and is currently considered a 'hot topic'. This growth has been driven by the need for scientists in today's world to provide decision makers with ever more accurate and reliable predictions that are based on results obtained from mathematical models.

Many physical processes such as fluid flows are governed by partial differential equations (PDEs). In practical applications in the real world, it is unlikely that all the inputs (boundary conditions, geometry, coefficients) for the chosen PDE model will be known. One possibility is to model the quantities that we don't know as random variables. Solving these problems is not always hard in theory but solving them efficiently in practice is a massive challenge.

I am interested in working with students who want to develop numerical analysis and numerical methods (e.g. solvers, error estimators) for solving partial differential equations with uncertain inputs (stochastic PDEs). I welcome any enquiries to work in this area. Specific projects could be theoretical or computational, according to the strengths of the student.

Projects on this topic would suit students who have taken undergraduate courses in numerical analysis and applied mathematics who have a keen interest in computational mathematics and developing practical algorithms. Some prior programming experience is essential.

Background reference:

An Introduction to Computational Stochastic PDEs (Cambridge Texts in Applied Mathematics), G. J. Lord, C.E. Powell and T. Shardlow, 2014.

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