# 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 |
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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 The 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 |
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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, |

Title |
## Turbulent particle-laden jets |
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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 |
## Interactions between rocks and ice |
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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 |
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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. |

Title |
## Combustion instabilities |
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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. |

Title |
## Efficient Uncertainty Quantification for PDEs with Random Data |
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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. |