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.


Acoustic properties of nanofibre composites

Group Industrial and Applied Mathematics

Noise pollution is a serious problem in many aspects of modern society. Although many materials exist that can provide mechanisms for sound absorption, particularly in the higher frequency ranges, compact low-frequency noise attenuation and absorption remains a significant challenge. It is therefore becoming increasingly important to design improved materials that can be employed in low-frequency noise control scenarios.

Recently, a new class of materials known as nano-fibre composites have been shown experimentally to offer excellent sound absorption characteristics at low frequencies. However the mechanisms by which they provide this enhanced acoustic absorption are not clearly understood and existing models fail to adequately describe this behaviour.

This project will therefore develop mathematical models of acoustic propagation in nano-fibre composites with the objective of improving the understanding of the mechanisms of sound absorption in such media.

This project is a collaboration between the School of Mathematics at the University of Manchester and Dyson Technology Ltd. The student will be expected to spend time at the industrial collaborator and work with them to validate theoretical results experimentally.

Candidates with a strong background in applied mathematics and/or physics, with excellent theoretical and technical ability and a strong motivation and enthusiasm for interdisciplinary scientific research are encouraged to apply. The successful applicant should have a high first class honours degree and ideally a related Masters degree and be available to start in September 2018 or shortly after. Applications should include a cover letter (two pages maximum) describing background and motivation and a complete CV (two pages maximum). These will be considered upon receipt and the position will remain open until filled.  
Informal queries should be emailed to Prof. William J. Parnell (

Start date: 17th September 2018


Funding covers all tuition fees and annual maintenance payments of the Research Council minimum (£14,777 for academic year 2018/19) for eligible UK and EU applicants as well as a CASE top-up of at least £3K per annum on average over the 4 year PhD.



Fluid-structure interaction effects in the sedimentation of thin elastic sheets

Group Industrial and Applied Mathematics

There is much current interest in so-called two-dimensional materials 
because of their unusual and attractive mechanical and electrical properties. 
Much of their processing is performed in a fluid environment, e.g., during
the size selection of dispersed graphene flakes by centrifugation, or
their deposition by ink-jet printing. The flakes' large aspect ratio 
implies that despite their impressive in-plane stiffness they have a
very small bending stiffness and are therefore easily deformed by the 
traction that the surrounding fluid exerts on them. The resulting
strong fluid-structure interaction affects not only the dynamics of 
individual flakes but also their collective behaviour.

The aim of this project is to perform a systematic study of 
the behaviour of thin elastic sheets in a viscous fluid. 
Specifically, we wish to establish how the flow-induced 
deformation affects the sedimentation of such sheets, paying
particular attention to

-- the effect of the sheets' aspect ratio; long narrow sheets are
likely to behave in a manner similar to elastic rods: at which point
does their finite aspect ratio become significant?

-- the effect of wrinkling instabilities and the development of
symmetry-breaking frustrated patterns: how do they arise in sheets of 
canonical shapes (circular, rectangular, polygonal,...) and how do
they affect the sheets' sedimentation?

The focus of this specific project is on computational/semi-analytical
approaches and would suit a student with a good background in Applied
Mathematics (especially fluid and solid mechanics) and Scientific
Computing. There is an opportunity for hands-on involvement
in an associated experimental study in the School of Physics
and Astronomy.



FUNDING: Funding is available and would provide fees and maintenance
at RCUK level for home/EU students, or a fees-only bursary
for overseas students.

DEADLINE: Applications are accepted at any time until the position
is filled.



Thermo-visco-acoustic metamaterials for underwater applications

Group Industrial and Applied Mathematics

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.


Fluid Mechanics of Cleaning and Decontamination

Group Industrial and Applied Mathematics

Cleaning and decontamination processes are important in many applications: from the daily chores of doing the dishes (with or without a dishwasher), to ensuring clean hygiene in hospitals, the food industry, or pharmaceutical companies. Although a lot of research has been done in chemistry and chemical engineering to improve detergents and cleaning devices, much less work has been done on the modelling of the underlying physical and chemical processes. In some cleaning applications, such as the neutralisation of toxic chemicals after a spill, it is crucial to avoid using strong mechanical forces in order to prevent the dispersion of the toxic material in the environment. Instead, a localised dissolution process, aided by chemical reactions neutralising the material, is used. This PhD project will investigate the advection, diffusion and reaction processes involved in this scenario.

Through a combination of experiments and modelling work the student will study the influence of flow properties: such as the Reynolds number and the Péclet number; geometry: whether the material is attached to a permeable or impermeable surface; and chemical properties such as solubility, reactivity and diffusivity.

This project is directly motivated by industrial applications and will suit candidates interested in using mathematical approaches to solve real challenges. Suitable candidates should have experience in the lab or a keen interest to support theoretical work in fluid dynamics by experimental evidence.

Reference: Landel, Thomas, McEvoy & Dalziel (2016). Convective mass transfer from a submerged drop in a thin falling film, Journal of Fluid Mechanics, 789: 630.


Turbulent particle-laden jets

Group Industrial and Applied Mathematics

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.


Interactions between rocks and ice

Group Industrial and Applied Mathematics

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

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.


Mathematical theory of diffraction

Group Industrial and Applied Mathematics

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.

-- 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.


Combustion instabilities

Group Industrial and Applied Mathematics

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.

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


Efficient Uncertainty Quantification for PDEs with Random Data

Group Industrial and Applied Mathematics

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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