Using theory and experiments of spheres moving near boundaries to optimize the method of images for regularized Stokeslets

Abstract

The general system of images for regularized Stokeslets (GSIRS) developed by Cortez and Varela (2015) is used extensively to model Stokes flow phenomena such as microorganisms swimming near a boundary. Our collaborative team uses dynamically similar scaled macroscopic experiments to test theories for forces and torques on spheres moving near a boundary and use these data and the method of regularized Stokeslets (MRS) created by Cortez et al. (2015) to calibrate the GSIRS. We find excellent agreement between theory and experiments, which provides the first experimental validation of such series solutions for spheres moving near an infinite plane boundary. We test two surface discretization methods commonly used in the literature: the six-patch method and the spherical centroidal Voronoi tessellation (SCVT) method. Our data show that the SCVT method provides the most accurate results when the motional symmetry is broken by the presence of a boundary. We use theory and the MRS to find optimal values for the regularization parameter in free space and show that the optimal regularization parameter values can be fit with simple formulae when using the SCVT method, so that other researchers have an easy reference. We also present a regularization function with higher order accuracy when compared with the regularization function previously introduced by Cortez et al. (2005). The simulated force and torque values compare very well with experiments and theory for a wide range of boundary distances. But, we find the simulations lose accuracy when the gap between the edge of the sphere and the wall is smaller than the average distance between grid points using the SCVT method. Our computational parameters and MATLAB and PYTHON implementations of the Lee and Leal (1980) theory provide researchers with important resources to optimize the numerical simulations of spheres moving near boundaries.