Numerical Computation of Doubly-Periodic Stokes Flow Bounded by a Plane with Applications to Nodal Cilia (Communications in Computational Physics, 2017)
A numerical method is presented for the computation of externally forcedStokes flows bounded by the plane z=0 and satisfying periodic boundary conditionsin the x and y directions. The motivation for this work is the simulation of flowsgenerated by cilia, which are hair-like structures attached to the surface of cells thatgenerate flows through coordinated beating. Large collections of cilia on a surface canbe modeled using a doubly-periodic domain. The approach presented here is to derivea regularized version of the fundamental solution of the incompressible Stokesequations in Fourier space for the periodic directions and physical space for the z direction.This analytical expression for û(k,m;z) can then be used to compute the fluidvelocity u(x,y,z) via a two-dimensional inverse fast Fourier transform for any fixedvalue of z. Repeating the computation for multiple values of z leads to the fluid velocityon a uniform grid in physical space. The zero-flow condition at the plane z =0is enforced through the use of images. The performance of the method is illustratedby numerical examples of particle transport by nodal cilia, which verify optimal particletransport for parameters consistent with previous studies. The results also showthat for two cilia in the periodic box, out-of-phase beating produces considerable moreparticle transport than in-phase beating.