# Solving Stokes flow with walls using Oseen tensor

Introduction

I've developed a code to solve for generalised, incompressible 2D Stokes flow

$\eta \nabla^2 \mathbf{v} - \nabla p + \mathbf{S} = 0$

$\nabla . \mathbf{v} = 0$

where $\mathbf{S}$ can include forces or gradients of an additional stress tensor $\Pi$,

$\mathbf{S} = \mathbf{f} + \nabla . \mathbf{\Pi} + \cdots$.

In this code, I solve the above in Fourier space using the following hydrodynamic propagator,

$\tilde{\mathbf{v}} = \frac{1}{\eta k^2}\left(\mathbf{\delta} - \hat{\mathbf{k}}\hat{\mathbf{k}}\right) . \tilde{\mathbf{S}}$

I numerically do this using FFTW, with periodic boundary conditions in the x and y directions. This code has been verified, for example, on the benchmark problem of flow past a periodic array of cylinders, with forces generated by an immersed boundary code.

Question

If I want to extend this code to include walls at $y = 0, L_y$, with no slip and no permeation, can I still solve this with a modified version of the above?

One idea might be possibly to only use $sin$ modes in the y-direction for the Fourier transform? Would this amount to zero gradient BC on $\Pi$?

Thank you - any help is greatly appreciated!

• I think there's a sign error in your Oseen tensor. – AJK May 29 '14 at 19:22
• You're right - fixed! – Hemmer May 29 '14 at 19:38

If you want to use nonperiodic boundary conditions, you could change the Fourier basis in the $y$-direction for your spectral method to a Chebyshev basis. Fast Chebyshev Transforms make use of Fast Fourier Transforms; both are linear transformations on function spaces, so switching between the two amounts to a change of basis, and you can reuse significant parts of your implementation.

• Many thanks! I'm less familiar with Chebyshev transforms, but they seem potentially very useful. I've tracked down a copy of Boyd's "Chebyshev + Spectral Methods", any other resources you would recommend? Also would this method, in principle, be capable of a boundary condition where one of the walls is being sheared? – Hemmer May 29 '14 at 19:47
• In principle, I'd think so, although it's been a while since I used spectral methods. I learned the basics out of the first volume of the series by Canuto, Hussaini, Quarteroni, and Zang. – Geoff Oxberry May 30 '14 at 22:44

I think the answer may be obvious once you develop an Oseen tensor that is appropriate for the boundary conditions. (This assumes you want to continue using an exact Oseen method to do the hydrodynamics, and not solve the Stokes equation explicitly.)

The Oseen tensor you give above is appropriate for a response to a point force in free space. You need to find the Oseen tensor for the response to a point force with your boundary conditions - v to zero at the channel walls. I think you may be able to find this using the method of images, or possibly just factorizing. I'll take a shot at it if I get a chance.

The benefit of the FFT method in your application is that the Oseen tensor is diagonal - you may need to choose a different basis for this to remain true, and it may or may not be efficient to transform into this basis.

(I am not sure how well this will work if you have forces on the boundaries - so I'm not sure what you mean by a boundary condition on $\Pi$.)

• I have seen a preprint doing something related, though it is much messier and I have not been able to extract a great answer out of it: math.ucsb.edu/~atzberg/pmwiki_intranet/uploads/… – AJK May 29 '14 at 19:41
• Thanks for your answer. I might eventually be looking to implement shearing forces at the boundaries, though static walls is a good start. My comment on $\Pi$ BCs was just that if v were to be represented with only sin modes then gradients of stress must also be expressed only in sin modes, though I'm less confident that last part makes sense! – Hemmer May 29 '14 at 19:42
• This paper also seems related: journals.cambridge.org/action/… – Hemmer May 29 '14 at 20:03

I would just add a comment to the answer by AJK, but I don't have enough rep. There is a classic paper by Liron and Mochon (http://link.springer.com/article/10.1007/BF01535565) from the late 1970's which gives a series solution for the Green's function for Stokes flow between two parallel plates.