Computation of the singular and regularized image systems for doubly-periodic Stokes flow in the presence of a wall

Abstract

A method is presented for the efficient computation of Stokes flow due to doubly-periodic arrays of regularized forces and point forces in 3D near a wall. Our approach is based on a variation of Ewald's summation method and the method of images for regularized and singular Stokeslets. The slowly convergent series resulting from the direct summation of the doubly-periodic array of Stokeslets is recast into the sum of two series, one in real space and one in Fourier space, both having partial sums decaying in a Gaussian manner. The formulation of our method guarantees that the velocity on the wall is analytically zero independent of the number of summands used. We also introduce an empirical approach to compute the ‘optimal’ number of summands in real space and Fourier space corresponding to a given splitting parameter. Finally, we report computational run times on a CPU and a GPU, and give two numerical examples to illustrate the method.