Here is an archive of final year projects I ran at Durham in previous years.

Contents

### MATH3382 (Project III) 2022–23

#### Project title

Theme park physics

#### Overview

What is your favourite theme park ride? The tower of terror? The space mountain rollercoaster? The teacups?

Theme park rides offer customers an exciting and exhilarating illusion of danger in a controlled environment. Making rides as thrilling as possible while keeping members of the public safe is the challenge of theme park ride designers.

In this project, we will mathematically model the 3D motion of these rides and consider their effects on the human body.

#### How we might structure the the project

At the start, we could have something like:

- Using mechanics to model the motion of the tower of terror
- Determining how fast you can spin teacups while keeping control
- Researching what sort of rollercoaster designs currently exist and determining if there are other designs that could work
- Deciding how to represent 3D motion along a path

#### Prerequisites

This is an applied maths project with its basis in Newtonian mechanics.

You may also have to do a bit of practical coding. You should therefore have some familiarity with Python, but you do not have to be a numerical analysis expert.

#### Resources

### MATH3382 (Project III) 2021–22

#### Project title

Sheepdog trials: collective motion in 2D

#### Overview

Fans of BBC Alba’s Farpaisean Chon-Chaorach will know that sheep herding is a competitive business. It will not surprise you, then, to learn that the motion of individual sheep as a moving herd — collective motion known as flocking or swarming — is mathematically quite well described.

At the local level, each sheep behaves as an individual, but if we zoom out, the herd of sheep behaves as a continuum. And at the continuum level, we can describe the motion of sheep using partial differential equations. How do herds decide where to go? Do they ever end up just going round in circles?

This project offers you to chance to model this interesting behaviour. We will derive the governing PDEs of some common models, and then numerically solve these equations. At the same time, we can simulate a bunch of individual sheep operating using simple rules. Will these two approaches converge? We will see!

This dichotomy of microscale vs macroscale, and how we move between them, is a key concept in applied mathematics.

#### How we might structure the the project

At the start, we could have something like:

- What is flocking? Where else do we see this behaviour?
- Looking up different models for how sheep behave around other sheep
- Coding up a few sheep in Python and seeing what they do
- Investigating the Vicsek model
- Deriving PDEs for our sheep and beginning to look for ways to numerically solve them

#### Prerequisites

This is an applied maths project and you will have to do a bit of practical coding. You should therefore have some familiarity with Python, but you do not have to be a numerical analysis expert at all! We will learn some numerical differential equation solving techniques together. There will be plenty of support.

#### Courses to take at the same time

Nothing essential. You might enjoy comparing these models to those you see in Mathematical Biology III and Fluid Mechanics III, but this is optional.

#### Resources

### MATH4072 (Project IV) 2022–23

#### Project title

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.

### MATH4072 (Project IV) 2021–22

#### Project title

How do microorganisms swim?

#### Overview

If you were a microorganism — say, a bacterium or a sperm cell — the world would feel very different to you. Not just because you are 1µm tall, but also because the physics you experience at this scale is quite different to what we experience as metre-sized humans.

As humans, we are familiar with the notion that everything moves according to Newton’s second law: $F = ma$. But as a microorganism, your mass is so small, that the acceleration term becomes neglible. Instead, you live in an inertialess world, where all forces just balance out, all the time.

Many microorganisms swim, and if you have taken a fluid mechanics course (although this is not required), you will know that fluids move according to the Navier–Stokes equation. A big part of what makes this equation hard is the acceleration term. Not a problem for a microorganism!

So really our governing equations become a lot simpler, and we are able to use techniques you will have learnt over the last three years to start looking at how these microorganisms really swim.

#### How we might structure the the project

At the start, we could have something like:

- Introduction to Stokes flow; derivation as a limit of the Navier–Stokes equations
- Solutions to Stokes flow using Green’s functions
- Looking up different methods of solving the Stokes equations on the shape of a microorganism
- Coding up the simplest model of a microorganism (if you like Python)

#### Prerequisites

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

There are no prerequisite models for this project. Possibly useful modules to have taken are Fluid Mechanics III and Mathematical Biology III for the Green’s functions. However, **these are not essential** because, together, we will derive the governing equations and learn an intuition for the physics which is completely new.

#### Resources

- Life at low Reynolds number, , 1976.
- Small particles in a viscous fluid, course notes, , 2014.
- A Physical Introduction to Suspension Dynamics, textbook, , 2012.
- Stokesian Dynamics online game, to get a feel for the physics, .