### Fluid flow through obstacles: Stokes, Darcy and Brinkman flows

#### Overview

In Fluid Mechanics III, we (well, you and Peter) saw the Navier–Stokes equation,

\[\rho \left(\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u}\cdot\boldsymbol{\nabla u}\right) = -\boldsymbol{\nabla}p + \mu \nabla^2 \boldsymbol{u} + \boldsymbol{F},\]and were able to derive some steady solutions like Poiseuille flow in a pipe.

This project looks at what happens when fluid travels not through a nice open pipe, but through something partially blocked. This could be a small number of obstacles (a few particles clogging the pipe), or something more substantial, like trying to soak through a sponge or some porous rock.

An early attempt to model this flow was an experimentally-derived simplification of Navier–Stokes due to Darcy,

\[\frac{\mu}{k}\boldsymbol{u} = -\boldsymbol{\nabla}p,\]where $k$ is the permeability of the medium. Here, you can imagine the left-hand side as a friction-like term, because it’s proportional to the velocity (remember that from Dynamics I or A-level mechanics?).

Meanwhile, a mathematical simplification of Navier–Stokes when the Reynolds number is small (so the flow is very slow, or the fluid is very viscous), leads us to the Stokes equations,

\[\mathbf{0} = -\boldsymbol{\nabla}p + \mu \nabla^2 \boldsymbol{u}.\]

This project looks at both of these models, and a third — Brinkman flows — which combines the two,

\[\frac{\mu}{k}\boldsymbol{u} = -\boldsymbol{\nabla}p + \mu \nabla^2 \boldsymbol{u}.\]

We’ll take a look at these models and apply them to interesting shapes and objects, with the target ultimately directed by your interests.

#### How we might structure the the project

At the start, you could have something like:

- Mathematical derivation of Darcy and Stokes flows
- Simple solutions of these flows
- Matching between microscopic and macroscopic models

#### Relevant papers

Darcy–Brinkman flow with solid inclusions .

Swimming in a two-dimensional Brinkman fluid: Computational modelling and regularised solutions
.

#### Prerequisites

You need to have taken Fluid Mechanics III to do this project.

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