Broadly speaking, I am interested in complex, weirdly-behaving fluids and how objects and organisms move through them. In many cases, I look at fluids where particles are suspended in them, giving the fluid some interesting internal structure. Often this means writing software to simulate systems of interest, which can be very large. Some of the things I am interested in are described here.

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### Sperm swimming through mucus

Understanding the processes involved in mammalian reproduction is key to trying to diagnose and treat problems such as infertility. The female reproductive tract actively selects the sperm that will eventually fertilise the egg cell, and a key component of this selection is the mucus through which sperm swim.

Mucus is a complex fluid with lots of structure which is about the same size as the sperm. We are modelling sperm swimming through such a structured environment, looking at how different swimming patterns interact with different structures. Since experiments are currently unable to probe these interactions, these simulations hope to reveal the selection mechanism.

(Work with Dr Eric Keavney at Imperial.)

### Shear thickening in suspensions

A long-standing classification of fluids is how their viscosities change under an applied shear rate. The simplest fluids—water, air—are usually indifferent. Most common suspensions, however, have viscosities which increase or decrease with shear rate. The more exciting behaviour of these is when the viscosity increases, such as a mixture of cornflour and water. This is a famous kitchen science experiment but up until now it has not been clear *why* it behaves this way.

Current research suggests that the secret lies in contact forces between particles, yet these are usually implemented in an ad hoc way, resulting in inaccurate predictions or high computational cost. We developed a new method which allows us to investigate contact models quickly and efficiently, and suggests an important factor in models of strongly shear-thickening fluids.

(Work with Prof. Helen Wilson at UCL, as are the remaining items in this list.)

- Townsend, A. K. & Wilson, H. J., 2017 Frictional shear thickening in suspensions: The effect of rigid asperities.
*Physics of Fluids*

### Tracking a ball falling through a suspension

When a large, heavy ball is dropped through a suspension of much smaller balls, we would expect the large ball to force its way through to the bottom of the tank in a relatively boring way. However, recent experiments shows that if you spin the tank in a certain way, the ball can actually be made to fall ‘upwards’!We investigated the structure of the small particles caused by an applied oscillatory shear and the heavy ball, and offered some explanations and a mechanism for this peculiar observation.

- Townsend, A. K. & Wilson, H. J., 2017 Simulations of a heavy ball falling through a sheared suspension.
*Journal of Engineering Mathematics*

### Modelling non-Newtonian fluids as suspensions

Methods of simulating particles in viscous, Newtonian fluids are widely accepted, but when the background fluid is non-Newtonian, the standard methods begin to fail. Unfortunately, many common real-life suspensions—concretes, paints—behave like this. One solution is to ‘fake’ a non-Newtonian liquid by placing little springs in a Newtonian background fluid.

We simulate suspensions of bead-and-spring dumbbells with various force laws, placing them under standard rheological tests, to see how good a representation of real non-Newtonian fluids they are. We see how certain parameters in the models lead to the suspension exhibiting useful properties, as well as explain how to extract this information from simulations.

### Stokesian Dynamics

A lot of my computer simulations are performed using a technique called Stokesian Dynamics. It’s a good technique for simulating systems of spherical particles where you know the forces acting on the particles, and you want to know where they go. In suspension, particles can act indirectly on each other as well as directly, moving the fluid around them as they move, which in turn moves other particles. This behaviour is captured nicely in this method.

Both long-range hydrodynamics and lubrication are included in Stokesian Dynamics, and the method is suited to three-dimensional simulation in both periodic and non-periodic domains, with relatively low calculation and time penalty. We give a complete and thorough explanation of Stokesian Dynamics and provide a walkthrough of how to write your own implementation of it.

I have written a complete implementation of Stokesian Dynamics, allowing particles of different sizes, in Python (with some parts in C). It will be available for download shortly.

### Chocolate fountains

Why do chocolate fountains fall inwards? Why do you not see very many white chocolate fountains? A tasty study into the behaviour of non-Newtonian fluids in three different geometries, the project is linked to below. Since then I have given popular mathematics talks on chocolate fountains many times over all over the country!- Read more about the fluid dynamics of chocolate fountains

- Townsend, A. K. & Wilson, H. J., 2016 The fluid dynamics of the chocolate fountain.
*European Journal of Physics***37**(1)**,**015803.