# Final year projects

I am offering a final-year project for fourth-year MMath students at Durham for the upcoming 2020–21 year.

I would be very happy to talk you about the project and possible directions if you would like to get in touch.

### MATH4072 (Project IV) 2020–21

#### Project title

Particle suspensions in confined spaces

#### Overview

Fluids move differently to solids, in part because of their ability to be funnelled. So while a liquid can move through a funnel quite happily, any solid particles suspended in the fluid may be rearranged in order to pass through.

Fluids with solid particles suspended in them are common in nature and industry: ceramics, paint, blood, and concrete, to name a few, can all be characterised as viscous fluids in which small particles are distributed. As part of their application, these particle suspensions often find themselves being transported through pipes or channels, which can have varying widths.

This project looks at concentrated particle suspensions as they slowly pass through funnels. If the funnel angle is very shallow, the particles will pass through almost undisturbed. Too steep, and the particles will simply clog the outlet.

Experimental data shows, for certain funnel angles and concentrations, shockwaves passing backwards through the oncoming suspension. We will use computer models to try to reproduce this behaviour and perhaps go on to try more interesting funnel shapes.

#### 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
• Coding up the simplest model of particle interactions in a dilute viscous fluid

From here, we can use an existing code which uses a more sophisticated particle model to examine the behaviour of concentrated suspensions.

There are lots of questions we could ask! For example:

• Is there an optimum funnel angle, shape, or gap size? Does that optimum change for different particle sizes or particle concentrations?
• What about particles which attract each other in some way?
• Does the surface roughness of the funnel walls make a difference?

After the initial derivation of the governing equations, this project is primarily computational and this project also provides an opportunity to learn about, and take part in, scientific computing.

#### Prerequisites

None, although useful modules to have taken are Continuum Mechanics III for the fluid dynamics, and PDEs 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.

And from this point, primarily the project will be computational. We will use code written in Python, and some comfort with programming is therefore required.