PhD student
The University of Oxford
Role Overview
I completed my DPhil (PhD) in 2019 at the Mathematical Institute at the University of Oxford as part of the Industrially Focussed Mathematical Modelling (InFoMM) program. I was supervised by Prof. Ian Griffiths and Prof. Jim Oliver at Oxford, and Dr Ritchie Parker at Nestlé. My project was titled The Mathematical Modelling of Cereal Extrusion, and involved using asymptotic and numerical methods to interogate the complicated multi-phase flow equations that govern extruded mixtures.
For more information on this work see the project page HERE.
Through the InFoMM program I took part in a range of additional programs designed to strengthen my ability to bridge industry and mathematics. For example, I completed short term placements with Schlumberger and Nestlé, and participated in numerous training courses focussed on technical and non-technical communication skills, informing decision making, problem design.
Key projects
Bubble growth (microscale modelling)
The evolution of bubbles that form in extruded cereals is one of the key determinants of the products final shape and texture. The first project we undertook was to construct a suitable bubble evolution model, and then develop methods to efficiently simulate these models. This required accounting for the mechanics of bubble growth and the diffusion of moisture and heat to fuel the growth. We used asymptotic and numerical methods to generate a suite of simulator options depending on the parameter regime, ensuring that we were simulating all cases accurately and efficiently. The resultant product was a simulator that takes pressure as an input and returns the change in bubble size.
Extruded fluids (macroscale modelling, approximation)
The overall extruded fluid evolution depends on the bubbles, and the bubbles depend on the pressure (and flow rate for bubble transport) of the fluid. So the second component of this work was to develop methods to simulate the evolution of a compressible fluid when coupled to a micrsocale bubble model. A key challenge is that the viscous compressible fluid equations are already complicated and because the fluid is unconstrained there is a free boundary (we don’t know our solution domain a priori, this is determined as part of the solution). We found that in some, operationally relevant parameter regimes, asymptotic methods could drastically simplify the system of equations, giving accurate solutions in a fraction of the time with very simple numerical solvers. We exploited this fact to explore the dynamics of the mixtures, and factored in additional physics, such as temperature dependent viscosity, allowing us to easily explore the impact of the control parameters (e.g. feed rate, temperature, die size) on the final product. Moreover, we were able to build on past work to model the evolution of the cross-sectional shape as well (see below), something that would be incredibly difficult using three-dimensional numerical solvers.
Extruded fluids (macroscale modelling, numerical)
When these simple models aren’t appropriate, full numerical solutions are required. In these cases the only way forward was to develop a finite-element solver to simulate the evolution of the product. This solver was only able to simulate two-dimensional or axiallly symmetric extrusions, and did take much longer to generate solutions than the approximate model. The benefit of approaching this problem in two ways was that we could now run both methods in the area of overlap (when, theoretically, the should both recover the same answer) and confirm that both were consistent.
Fig.1 A 3D illustration of the evolution of an extruded mixture. The transition from red to blue colouring indicates the drop in temperature as the product expands.