### Non-traditional constitutive laws in fluid mechanics and biology

#### Overview

If you did Fluid Mechanics III this year, you will remember the general form of the Navier–Stokes equation is

$$\frac{\mathrm{D}u}{\mathrm{D}t} = \boldsymbol{\nabla}\cdot\boldsymbol{\sigma} + \boldsymbol{f}(\boldsymbol{u}).$$

If you did Mathematical Biology III this year, you will remember that many spatial and time-dependent population laws can be written as advection–diffusion equations,

$$\frac{\partial u}{\partial t} = \boldsymbol{\nabla}\cdot\boldsymbol{J} + f(u).$$

In fluid mechanics, the assumption that the fluid is Newtonian led the stress tensor term to become $\mu \nabla^2 \boldsymbol{u}$.

In mathematical biology, the assumption of Fick’s law led the flux term to become $D \nabla^2 u$.

These assumptions are called constitutive laws, and in this project, we’re going to pick some more interesting ones!

For fluid dynamics, this will mean entering the world of non-Newtonian, or viscoelastic fluids: fluids like melted cheese and chocolate with fun elasticity and viscosity properties.

For biology, this will mean entering the world of cross-diffusion in 2D systems and ambush predators—continuing on from the end of the course.

This project can go either way depending on your interests and will serve as an introduction to these interesting, and mathematically related, areas.

This project will enjoy some informal joint supervision with Andrew Krause.

#### How we might structure the the project

At the start, you could have something like:

- Introduction to fluids and biological systems with different constitutive laws
- For the fluids: Introduction to different methods of measuring viscosity and elasticity (rheometry)
- For the fluids: Solving flow in a simple geometry (e.g. a pipe) for a non-Newtonian fluid
- For the biology: Understanding how porous media changes the structure of travelling waves and pattern formation
- For the biology: Studying how models of directed motion give more realistic (but more difficult!) representations of animal migration

#### Prerequisites

You need to have taken Fluid Mechanics III (for the fluids part) or Mathematical Biology III (for the maths bio part) to do this project. Ideally you will have taken both.

This is an applied maths project and there will be some computational work as well as some analytical work. You should be competent with Python, but you do not have to be a numerical analysis expert.