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 doable 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 
Adaptive Numerical Algorithms for the Forward Propagation of Uncertainty in TimeDependent CFD 

Group  Industrial and Applied Mathematics 
Supervisor  
Description 
Fullyfunded iCASE PhD project supported by IBM Research UK
Uncertainty quantification (UQ) is a rapidlyevolving field, incorporating several traditional mathematical disciplines. This project will develop new adaptive numerical algorithms for the forward propagation of uncertainty in largescale timedependent CFD (computational fluid dynamics) simulations and is a collaboration between the School of Mathematics at the University of Manchester and IBM Research UK. The project will be jointly supervised by Dr. Catherine Powell and Professor David Silvester from the School of Mathematics, and Dr. Malgorzata Zimon from IBM Research UK. The PhD student will be based in the School of Mathematics but will also have the opportunity to spend a minimum of 3 months working alongside researchers at IBM Research UK's premises in Daresbury.
Project Outline: In realworld applications, when using mathematical models to simulate realworld processes (such as fluid flows) we frequently encounter situations where we are uncertain about one or more of the inputs (viscosity, material parameters, initial conditions, geometry etc). In forward UQ, the main aim is to assess the impact of uncertainty in the model inputs on quantities of interest associated with the model's outputs. For this, we require computationally efficient numerical methods that can take in a probability distribution for the model's inputs and deliver accurate approximations of statistical quantities of interest related to the model's outputs. For timedependent problems, and especially those with nonsmooth solutions, the approximation space often needs to be adapted in time to maintain accuracy. How to design adaptive numerical algorithms with guaranteed error control is highly challenging.
Requirements: Candidates with a strong background in applied mathematics and numerical analysis with a passion for solving realworld problems efficiently on computers are encouraged to apply. Some prior experience in scientific computing (Python, MATLAB, C or Fortran etc) is desirable but not essential. Applicants should have (or be on track to to be awarded) either (i) a first class honours MMath degree or (ii) a first class honours BSc degree in Mathematics and a oneyear MSc degree in a relevant mathematical discipline. Ideally, applicants should be available to start in September 2019 or shortly after.
How to apply: Informal email queries should be directed to Dr. Catherine Powell and/or Professor David Silvester in the first instance. Formal applications can then be submitted online. As well as transcripts and references, applicants should supply a cover letter describing their academic background and motivation for the project, as well as a complete CV (two pages maximum). These will be considered upon receipt and the PhD position will remain open until it is filled.
Funding: For eligible UK applicants, funding covers all tuition fees and annual maintenance payments at the standard EPSRC rate (£15,009 for the academic year 2019/20), plus a CASE topup (amount TBC). For eligible EU applicants, funding is only available to cover tuition fees. 
Title 
Mathematics & Meteorites 

Group  Industrial and Applied Mathematics 
Supervisor  
Description 
Mankind has long been fascinated by the falling of material from space. This material gives us insight into the composition of the early solar system and the remaining planets within. Huge efforts across the globe are placed into the analysis of such material and their parent bodies, plus how it is transported and delivered around the solar system. However the use of applied mathematics (very broadly defined), presents a prism for studying this material from a novel vantage an example being the 'The Lost Meteorites of Antarctica Project' (www.ukantarcticmeteorites.com). Other applications for mathematics include fireball camera network analysis (e.g. predicting what an observed fireball actually was), crater counting studies within the solar system, and collection statistics from Antarctica. Having now established ourselves as a leader in such studies, we seek applications from PhD students will a strong background in mathematics and firm interdisciplinary mindset, who wish to push boundaries in this exciting area of space and planetary science. 
Title 
Marangoni driven film dynamics for diffusive mixtures 

Group  Industrial and Applied Mathematics 
Supervisor  
Description 
Marangonidriven flows develop due to local surface tension gradients at the interface between two liquids. These flows have many implications for biochemical systems, microfluidic systems or chemical reactors. When a drop of a liquid substance such as ethanol is deposited at the surface of a liquid film layer, e.g. water, the surface tension gradient expands the drop rapidly to form a very thin film. Competition between the Marangonidriven film flow and the diffusion of some or all of the film components into the other phase can lead to complex mixing dynamics as well as interfacial instabilities. In this project, we will explore experimentally and theoretically the competition between convection and diffusion mechanisms and their impact on the dynamics and stability of these rapidly expanding thin films. Experimental techniques such as highspeed particle image velocimetry and dye attenuation will be used to understand the various flow phenomena and the mixing properties. Pure or mixed substances for the drop will be tested varying the miscibility and surface tension properties with the liquid film layer onto which it is deposited. We will also investigate the impact of the results of these studies for practical applications in biochemical systems and microfluidic systems. The project is suitable for an enthusiastic and creative candidate who has some experience in laboratory experimentation and good knowledge in fluid mechanics. Some knowledge in imaging analysis technique is desired but not necessary. Funding Notes Funding is available and would provide fees and maintenance at RCUK level for home/EU students, or a feesonly bursary for overseas students. Competitive bursaries are also available for overseas students to fully cover both fees and maintenance at RCUK level. Anticipated start date: September 2019. 
Title 
Instabilities of buoyancy driven flows in a confined environment 

Group  Industrial and Applied Mathematics 
Supervisor  
Description 
Buoyancy driven flows are important in many geophysical and environmental applications: from natural ventilation in buildings to the evolution of cloud particles after a volcanic eruption or an explosion. Much work has been done on these flows in both turbulent and laminar flow regimes. However, the impact of lateral confinement on the flow has not received as much attention. One of the distinctive features is the development of an instability due to lateral shear. This effect can have important consequences for the large scale dynamics as well as the small scale mixing and dispersion properties of these flows when carrying particles or other tracers. In this project, we will explore experimentally and theoretically the origin of the instability for various fundamental buoyancydriven flows such as fountains and thermals. New experimental techniques, based on novel experimental design and imaging techniques, recently developed in our laboratory have allowed to probe further into the complex dynamics of these flows. The goal of this project is to study the phenomenology of the flow in order to determine the source of the instability. We will also analyse how the instability affects mixing and dispersion of active or passive tracers. The project is suitable for an enthusiastic and creative candidate who has some experience in laboratory experimentation and good knowledge in fluid mechanics. Some knowledge in imaging analysis technique is desired but not necessary. Funding Notes Funding is available and would provide fees and maintenance at RCUK level for home/EU students, or a feesonly bursary for overseas students. Competitive bursaries are also available for overseas students to fully cover both fees and maintenance at RCUK level. Anticipated start date: September 2019. 
Title 
Acoustic properties of nanofibre composites 

Group  Industrial and Applied Mathematics 
Supervisor  
Description 
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 lowfrequency noise attenuation and absorption remains a significant challenge. It is therefore becoming increasingly important to design improved materials that can be employed in lowfrequency noise control scenarios. Recently, a new class of materials known as nanofibre 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 nanofibre 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. Start date: 16^{th} September 2019 Funding: 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 topup of at least £3K per annum on average over the 4 year PhD.

Title 
Fluidstructure interaction effects in the sedimentation of thin elastic sheets 

Group  Industrial and Applied Mathematics 
Supervisors 

Description 
There is much current interest in socalled twodimensional materials The aim of this project is to perform a systematic study of  the effect of the sheets' aspect ratio; long narrow sheets are  the effect of wrinkling instabilities and the development of The focus of this specific project is on computational/semianalytical
MAIN SUPERVISOR: Professor Matthias Heil (School of Mathematics) COSUPERVISORS: Professor Anne Juel (School of Physics and Astronomy) START DATE: September 2019 (or as soon as possible thereafter) OTHER ASSOCIATED PROJECT AREAS: Physics FUNDING: Funding is available and would provide fees and maintenance DEADLINE: Applications are accepted at any time until the position 
Title 
Thermoviscoacoustic 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 “inair” 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 semiinfinite 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 
Fluid Mechanics of Cleaning and Decontamination 

Group  Industrial and Applied Mathematics 
Supervisor  
Description 
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. 
Title 
Turbulent particleladen jets 

Group  Industrial and Applied Mathematics 
Supervisors  
Description 
Turbulent particleladen 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 twophase 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 particleladen 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 
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 objectoriented multiphysics finiteelement library oomphlib, developed by M. Heil and A.L. Hazel and their collaborators, and available as open source software at http://www.oomphlib.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 halfplane. Since, some very ingenious mathematical methods have been developed to tackle such problems. One of the most famous being the WienerHopf technique. 
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 multiscale, multiphysics 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. 
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. 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. 