Velocity and pressure fields for Stokes flow due to a force singularity of arbitrary orientation and arbitrary distance between two parallel plates are found, using the image technique and a Fourier transform. Two alternative expressions for the solution, one in terms of infinite integrals and the other in terms of infinite series, are given. The infinite series solution is especially suitable for computation purposes being an exponentially decreasing series. From the series the “far field” behaviour is extracted. It is found that a force singularity parallel to the two planes has a far field behaviour of source and image for the parallel components (a two dimensional source doublet of height-dependent strength) whereas the normal component, and all fields due to a force singularity normal to the planes, die out exponentially. Velocity fields are compared with those of the one plane case. An estimate of the influence of the second wall and when its effect can be disregarded is obtained.
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Liron, N., Mochon, S. Stokes flow for a stokeslet between two parallel flat plates. J Eng Math 10, 287–303 (1976). https://doi.org/10.1007/BF01535565