2026–27 Project IV

Slow flows: obstacles, porous media and suspensions

Research area: Applied mathematics

⚠️ I am on research leave in Epiphany, so some of our meetings in this term may take place by video call.

Overview

In Fluid Mechanics III, we saw the Navier–Stokes equation,
\[\rho \left[\frac{\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u}\cdot\boldsymbol{\nabla})\boldsymbol{u}\right] = -\boldsymbol{\nabla}p + \mu \nabla^2 \boldsymbol{u} + \boldsymbol{F},\]and were able to derive some steady solutions like Poiseuille flow in a pipe.

This project explores what happens when inertia is negligible, so that flow is dominated by pressure and viscosity. This leads to a range of simplified models, depending on how much structure is present in the system.

At one end, we have the Stokes equations,
\[\mathbf{0} = -\boldsymbol{\nabla}p + \mu \nabla^2 \boldsymbol{u},\]which describe slow flow around objects or collections of particles. These can be used to study suspensions, such as clouds of sedimenting particles, interacting microswimmers, or particles driven by external forces, such as magnetic fields.

At a more macroscopic level, when the fluid moves through a porous material, an experimentally-derived model due to Darcy is often appropriate,
\[\frac{\mu}{k}\boldsymbol{u} = -\boldsymbol{\nabla}p,\]where $k$ is the permeability of the medium.

Brinkman’s equation provides a bridge between these two descriptions,
\[\frac{\mu}{k}\boldsymbol{u} = -\boldsymbol{\nabla}p + \mu \nabla^2 \boldsymbol{u},\]and can be interpreted as incorporating both viscous stresses and porous resistance.

Depending on your interests, the project could focus on:

  • Flow around one or more obstacles
  • Effective (averaged) models of flow through porous media
  • Particle-based descriptions of suspensions using Stokeslets or related methods
  • Collective behaviour such as sedimentation, clustering, or externally-driven motion (e.g. magnetic particles)

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.

How we might structure the project

At the start, you could have something like:

  • Derivation and interpretation of Stokes flow
  • Introduction to Darcy and Brinkman models (if relevant to your direction)
  • Fundamental solutions (e.g. Stokeslets) and how to build flows from them
  • Simple analytical or computational examples

Possible directions

  • An analytical project on flow around obstacles, based on Wang, 2009, comparing Stokes, Darcy and Brinkman descriptions.
  • A computational project on suspensions using Green’s functions, starting from Durlofsky & Brady, 1987, including:
    • Derivation and implementation of Stokeslets/Brinkmanlets
    • Simulation of sedimenting particle clouds (e.g. Guazelli, 2006)
    • Exploring collective effects (e.g. clustering, drafting, instabilities, jamming)
  • Regularised methods for flows around particles, based on Cortez et al., 2010, with possible extensions to Brinkman flows.
  • Flows in porous or structured domains (channels, corrugations, etc.), using Darcy/Brinkman models (e.g. Ng & Wang, 2010, Roach & Hamdan, 2023). Asymptotic expansions, inclined channels.
  • Network models using discretised Darcy models (e.g. blood flow, Obrist et al., 2010)
  • Swimming and active suspensions in Stokes or Brinkman flows (e.g. propulsive rods in Almoteri & Lushi, 2024, squirmer models in Nganguia & Pak, 2018).
  • Externally-driven suspensions, for example particles interacting via magnetic fields (e.g. Das & Saintillan, 2013, using Stokes flow as the underlying model.

Mode of operation and evidence of learning for the individual project

The project will revolve around learning through reading and programming in Python. Students will demonstrate their understanding by deriving theory, and then comparing theory to simulations and experimental literature. They will write Python code to implement core methodology, modelling a number of physically-inspired fluid flows, and then clearly communicate the material in both written and oral formats.

Relevant papers

How does fluid navigate a porous medium?