A Regularization Method for the Numerical Solution of Doubly-Periodic Stokes Flow
Karin Leiderman, Elizabeth L. Bouzarth, Hoang-Ngan Nguyen · Contemporary mathematics - American Mathematical Society · 2014
We introduce a numerical method to calculate Stokes flow driven by an infinite, doubly-periodic array of point forces in three dimensions. Flow arising from a single point force, as given by the Stokeslet, decays as 1 / r 1/r , where r r denotes the distance from the point force. Thus, in three dimensions, a direct summation over a periodic array is \color{black}only conditionally convergent. One can recast the conditionally convergent series into a sum of two rapidly decaying series, one in real space that contains the singularity, and one in Fourier space that contains only smooth terms. To obtain a smooth solution everywhere, the derivation here relies on the use of regularized Stokeslets within this summation framework. The regularized Stokeslets are ormalcolor contained wholly within the real space sum, while the sum in Fourier space remains unchanged from the expression in the singular solution. We report our convergence results for the real and Fourier space sums for various domain sizes to optimize the work split between them. Then we present an application of the method in calculating fluid flows generated by doubly-periodic, infinite arrays of airway cilia to study the effects of cilia spacing on fluid transport.