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Numerical Computation of Doubly-Periodic Stokes Flow Bounded by a Plane with Applications to Nodal Cilia

Part of: Biological fluid mechanics Incompressible viscous fluids Biology and other natural sciences

Published online by Cambridge University Press:  06 July 2017

Franz Hoffmann  and
Ricardo Cortez
Franz Hoffmann*
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
Ricardo Cortez*
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
*
*Corresponding author. Email addresses:[email protected] (F. Hoffmann), [email protected] (R. Cortez)
*Corresponding author. Email addresses:[email protected] (F. Hoffmann), [email protected] (R. Cortez)
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Abstract

A numerical method is presented for the computation of externally forced Stokes flows bounded by the plane z=0 and satisfying periodic boundary conditions in the x and y directions. The motivation for this work is the simulation of flows generated by cilia, which are hair-like structures attached to the surface of cells that generate flows through coordinated beating. Large collections of cilia on a surface can be modeled using a doubly-periodic domain. The approach presented here is to derive a regularized version of the fundamental solution of the incompressible Stokes equations 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 fluid velocity u(x,y,z) via a two-dimensional inverse fast Fourier transform for any fixed value of z. Repeating the computation for multiple values of z leads to the fluid velocity on a uniform grid in physical space. The zero-flow condition at the plane z=0 is enforced through the use of images. The performance of the method is illustrated by numerical examples of particle transport by nodal cilia, which verify optimal particle transport for parameters consistent with previous studies. The results also show that for two cilia in the periodic box, out-of-phase beating produces considerablemore particle transport than in-phase beating.

Keywords

Stokes flowEwald summationFFT

MSC classification

Secondary: 76D07: Stokes and related (Oseen, etc.) flows 76Z99: None of the above, but in this section 92-08: Computational methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 620 - 642
Copyright
Copyright © Global-Science Press 2017 

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Footnotes

Communicated by Boo-Cheong Khoo

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