Research
Overview
I use mathematical modelling, asymptotic analysis, and numerical methods to explore a range of industrial and physical processes. I am particularly interested in processes involving multiple time- or length-scales, and free-boundary problems. Some of my research projects are described below.
As the water dries and the evaporation front moves further into the porous material, suspended dirt becomes concentrated, driving diffusive transport and deposition in the pores. These processes are coupled: the deposited dirt slows down the transport of water vapour out of the porous material, slowing the drying.
Drying and clogging in filters and textiles
As a filter (eg: in a washing machine) or a textile (eg: a raincoat) dries, any dirt or impurities in the water are left behind within the porous structure. It’s important to know where the dirt ends up because this impacts the future efficacy of the filter or raincoat: does it deposit uniformly throughout the porous material, or concentrate at the surface? (This is a bit like the coffee-ring problem, only in porous media.) Moreover, can the accumulating dirt ever clog the pore space entirely, and if so, how can we avoid this? Answering these questions can help to understand contaminant build-up within filters and textiles, enabling the prediction of product lifetimes and the identification of appropriate cleaning processes.
We’ve derived new models, using multi-scale techniques, that capture the coupled liquid evaporation and dirt accumulation, diffusion, and deposition within the porous material. We’ve found that the dirt deposits non-uniformly through the porous material, tending to concentrate towards the middle. Furthermore, there are two mechanisms by which the depositing dirt can clog the material before drying is complete, depending on the speed of drying relative to the transport and deposition processes.
Look out for our upcoming papers on this topic (currently under review at Journal of Fluid Mechanics and EuroPhysics Letters)!
Our work so far has assumed that there are no surface tension forces, and so no capillary flow of the drying liquid. In practice, for many porous materials these capillary flows are important during drying; future work will investigate how capillary flows affect the coupled drying and deposition of impurities.
The motion of evaporation fronts in porous media
How does a porous material dry out? In situations where an evaporation front moves into the porous material, we have used a combined homogenisation and boundary-layer approach to understand how the microstructure of a porous material affects the motion of the evaporation front. It turns out that this depends on the "chemistry" of the evaporation: we might either assume (i) that there's an evaporation rate (that depends on how humid the air is next to the evaporation front) or alternatively (ii) that the water vapour is at its saturation point adjacent to the evaporation front. In case (i) the shape of the evaporation interface within the microscale pores is crucial, and there is an effective evaporation rate which depends on the average surface area. In case (ii) however, it doesn't matter what shape the evaporating surface is in the pores: the evaporation is purely driven by the transport of water vapour out of the porous material!
Take a look at our paper for more details.
The motion of an evaporating interface through some simple , periodic, porescale geometries (solid inclusions are shown in grey, the blue lines show the position of the evaporating interface moving down through the porespace, at different times) . The microstructure of the porous material affects the shape of the evaporating interface. In the case where we impose an evaporation rate, the evaporating interface on the microscale moves with constant normal velocity: this means it bends around obstacles (like the grey triangles in the left diagram).
Efficient modelling of reactive decontamination
The decontamination of porous materials that have been contaminated with hazardous chemical agents is a difficult and important problem, with critical implications for the environment and for public health. One decontamination method employed by government agencies is to apply a cleanser at the surface of the contaminated porous material, which reacts into the contaminated porespace, neutralising the hazardous chemical. The agent and cleanser fluids are usually immiscible, so that the neutralisation reaction occurs at the interface between the fluids.
One difficulty of simulating this process is that the pore-space in the porous material forms an intricate, complicated domain through which to simulate chemical transport and reactions. Using homogenisation and boundary layer methods, we have developed averaged models for the reactive decontamination of these porous materials, which hold over the much simpler domain of the entire porous material, but rigorously take into account the effect of the microstructure. Analysis and simulation of these models is much more computationally efficient than of the original models, and can be used to understand and optimise the decontamination protocols.
See our paper for more detail, and a case study on problems in cleaning and decontamination.
You can play around with simulating these models online using VisualPDE - check out our Story on decontamination.
Schematic of the reacting counter-current flow in the furnace material bed. Gas (yellow) flows left-to-right, heating up the solid (grey) flowing right-to-left, so that it is hot enough to react.
Silicon furnaces
Silicon is made by reducing quartz rock (containing silicon dioxide) with carbon in a furnace. The energy for the highly-endothermic chemical reactions is provided by an electric arc (see below). During my DPhil I studied the interacting thermal, chemical and electrical processes inside a silicon furnace.
Within the bed of granular material in the furnace, the solid raw materials are consumed in an endothermic reaction, producing gases (from which silicon later forms). The flow of the gas out of the furnace is in the opposite direction to the flow of the raw materials down into the hearth of the furnace. The heat for the reaction is provided by both the flow of hot gases, and by heat radiation onto the free-boundary of the domain. The resulting system is an endothermic, reacting counter-current flow. Asymptotic analysis of this system reveals intricate boundary- and transition-layer structures in a variety of different parameter regimes. This analysis identifies the industrially relevant parameter regime, and highlights how the heating in the majority of the furnace is due to heat transfer from the gas, rather than heat conduction.
See our paper for more details on counter-current, endothermic, reacting flows.
The electrical, thermal, and chemical processes in the furnace are tightly coupled but occur over a variety of timescales, from the timescale of the alternating electric current (milliseconds) to the slow evolution of the material bed furnace (hours). We used a multiple-timescales homogenisation to incorporate processes on both these timescales into an averaged model for the slow furnace evolution. The averaged model is able to predict the slow growth of the crater region of the furnace towards a stable steady state, unless the furnace is stoked, as seen in practise by furnace operators.
See our paper and my thesis for more on this averaged furnace model.
Electric arcs
Electric arcs are regions of hot plasma which carry an electric current between two electrodes. They occur in circuit breakers, and are used to convert electrical energy to heat in several types of furnace, including silicon furnaces. The flow of current within the arc generates heat by Ohmic heating, and the ionisation of the plasma (and so its electrical conductivity) increases with temperature. The region of plasma that is hot enough to conduct electricity is therefore self-determined by the coupled thermal and electromagnetic problem. The large electric current also generates a fluid flow due to the Lorentz force exerted on the charged particles, and convection is an important heat transfer mechanism, so that the fluid flow is also tightly coupled with the electrical and thermal problems.
This fluid-flow, electromagnetism, and heat-transfer problem can be modelled with a magnetohydrodynamics (MHD) framework, and simulations of a radially-symmetric arc are shown in the figure to the right. MHD models are complicated and expensive to simulate. We have used dimensional analysis to motivate a much simpler arc model consisting of a single ODE for the arc temperature, with heat lost from the arc solely by radiation. This simple radiation model essentially characterises the arc as a non-linear resistor, and it can be coupled into an equivalent circuit model for the overall electrical system, comparing favourably with empirical arc models currently used in practice.
A (non-technical) case study can be found here. See our paper for more details.
A simulation of a radially-symmetric electric arc. Colour shows electrical conductivity (S/m), and streamlines show the flow of current.
Quantum Physics
Bose-Einstein condensates (BECs) are formed of a dilute gas of bosons, cooled to nearly absolute zero, so that the particles condense down into their lowest energy state. BECs are studied because they can be made to exhibit quantum phenomena on a macroscopic scale, showing behaviours such as interference and diffraction, which we would only normally expect to see in much smaller particles such as photons and electrons. Because (almost) all the particles in the BEC are in their lowest energy state, we can model the entire system using a single Schrödinger equation, rather than a system of coupled Schrödinger equations (one for each particle in the system). The inter-particle interaction enters this single Schrödinger equation in the form of a self-interaction term, which is usually cubic in the wavefunction. In certain specific situations a quintic or cubic-quintic self-interaction is more appropriate; we have studied solutions of the Schrödinger equation in this case, showing how the behaviour deviates from the purely-cubic model under a variety of constraining potentials.
See our paper for more details.