2024–25 Project IV

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.

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 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

Possible directions

  • Wang, 2009 could form the basis of a relatively analytical project which looks at flows around a single obstacle in all three regimes:
    • General separable solution to the Brinkman equations in spherical and cylindrical coordinates (Appendix A and B)
    • Stokes flow over a sphere?
    • Darcy flow over a sphere
    • Why you can’t get Stokes flow around a cylinder (the Stokes paradox)
    • Darcy flow over a cylinder
    • Solution to the Brinkman equations over a sphere
    • Solution to the Brinkman equations around a cylinder
    • Solution to the Darcy/Brinkman equations in parallel flow
  • Durlofsky & Brady, 1987 could form the start of a computational project which models spherical objects moving through a low-Reynolds number fluid. It could go like
    • Derivation of Green’s function for Stokes flow
    • Faxén’s laws
    • Simulation of particle clouds under sedimentation in Stokes (see Guazzelli 2006)
    • Green’s function for Brinkman
    • Simulation of particle clouds under sedimentation in Brinkman
  • Cortez et al., 2010 could form the basis of a perhaps more challenging computational project, but along the same lines as the Durlofsky & Brady paper. It again would model objects moving through a Brinkman fluid. It could go like
    • Derivation of Green’s function for Stokes flow
    • Green’s function for Brinkman
    • Regularising the delta functions (i.e. making them smooth)
    • Using these to simulate the flow around a stationary sphere in Stokes and in Brinkman
    • Possibly looking at how to turn this into a model for a swimming organism in Brinkman. The details are in this paper but you would have to be happy with the coding.
  • Ng & Wang, 2010 and Roach & Hamdan, 2023 could form the basis of a mostly analytic take on Brinkman flows through channels: either inclined or corrugated (or both). It could go like:
    • Stokes flow in a channel
    • Brinkman flow in a channel
    • Asymptotic expansion of the solutions near walls which are sine curves (following the corrugated paper)
    • Discussing how the phase difference of the top and bottom walls affects the flow (again following the corrugated paper)
    • Solving the Brinkman equations in an inclined channel (following the second paper)
    • Maybe possible to combine?
  • Almoteri & Lushi, 2024 could start you off looking at swimming in Stokes flow and Brinkman flows. This 2024 paper creates a zoomed out ‘continuum’ model for when you have a lot of microswimmers swimming around in a Brinkman fluid. Solving these equations numerically is probably too awkward for a 4H project, but the steps along the way are interesting in their own way. The presentation is also a bit ‘following paper X’, which could leave space for us to fill in the gaps as there are lots of techniques used:
    • Green’s function for Stokes flow and Brinkman flow
    • The ‘propulsive rod’ model for swimming in Stokes and Brinkman (we can do simulations with eqs 7 and 8 alone, I think)
    • Turning that into a continuum model
  • Nganguia & Pak, 2018 is another take on swimming in Stokes and Brinkman flows: the squirmer model. This paper is mostly analytic and in places it looks challenging, but I think it is followable, and we could do a good job of filling in gaps. I think you would want to let Mathematica do a good chunk of the algebra.
    • Green’s function for Stokes and Brinkman
    • Squirmer models for swimming
    • How the squirmer swims: deriving some results and plotting streamlines

Relevant papers