About the Project
In fluid mechanics, lubrication theory elegantly simplifies the Navier-Stokes equations into Darcy or Darcy-like laws for incompressible, isothermal thin-film flows. While this physical transition is well understood for simple fluids, it remains highly undeveloped for flows involving non-isothermal conditions, fluid mixing, and chemical reactions. This theoretical gap represents a significant frontier for mathematical research and clean-energy applications.
This fully funded PhD project focuses on developing the mathematical theory of non-isothermal and reactive lubrication flows under confinement. You will explore how physical boundaries transform the fundamental physics of fluid flow, heat transport, and interface deformation.
A central focus of the project will be tracking the emergence and evolution of hydrodynamic instabilities. Using narrow channels, you will investigate how confinement alters the coupled influence of Saffman-Taylor, Darrieus-Landau, and Rayleigh-Taylor instabilities under Darcy-like laws compared to standard unconfined Navier-Stokes equations. You will apply this framework to iconic configurations, such as flow around a heated cylinder, to discover how altering local mixing and controlling these instabilities can anchor and stabilise lean hydrogen flames. You will also investigate other classical setups, including 2D stagnation flows, to map out structural transitions and mixing limits.
Using a combination of analytical mathematics, such as asymptotic analysis and perturbation theory, alongside numerical simulations using software like MATLAB or COMSOL, you will solve systems of non-linear partial differential equations to track stability branches. Your research will help address a vital challenge in designing safe and predictable zero-emission micro-combustion technologies.
As a member of our research team in the Department of Mathematics, you will receive dedicated guidance and comprehensive training in applied mathematics, fluid dynamics, and numerical programming. There is flexibility for the project to be adapted to your mathematical interests. We welcome applications from quantitative candidates with a strong background in mathematics, physics, or engineering.
This project is expected to start in September 2027.
Before you apply: We strongly recommend that you contact the supervisors for this project before you apply.
How to apply: To be considered for this project you must complete a formal application through our online application portal. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships. If you don't already have an applicant account, please follow the instructions here.
When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications. Please note late or incomplete applications will not be considered.
Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status.
We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder.
Eligibility: The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in applied mathematics, physics, mechanical/chemical engineering, or a highly quantitative discipline OR any upper-second class (2:1) honours degree and a Master's degree at merit (or international equivalent) in applied mathematics, physics, mechanical/chemical engineering, or a highly quantitative discipline. A strong foundational interest in fluid mechanics, differential equations, and transport phenomena is highly desirable. Practical familiarity with basic scientific programming and software packages, such as MATLAB or COMSOL, will be advantageous to conduct the numerical simulation components of the research.
This project will remain open until filled.
If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026.
If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027.
Self or externally funded students can also be considered for this project.
Funding Notes
Funding covers tuition fees, an annual stipend at the UKRI minimum rate (currently £21,805 per year), and a research training support grant of up to £2,500 per year for the full 3.5-year programme.