Boundary integral equation analysis for suspension of spheres in Stokes flow

TL;DR

This paper analyzes boundary integral operators for Stokes flow around spheres, providing spectral formulas, efficient evaluation methods, and numerical verification for applications in porous media, active matter, and magneto-hydrodynamics.

Contribution

It introduces spectral diagonalization of boundary integral operators on spheres and develops efficient computational schemes for hydrodynamic interactions.

Findings

Operators diagonalize on vector spherical harmonics

Spectrally accurate evaluation schemes are validated

Fast multipole method accelerates computations

Abstract

We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical expressions for evaluating the operators away from the boundary. When two particle are located close to each other, we use a truncated series expansion to compute the hydrodynamic interaction. On the other hand, we use the standard spectrally accurate quadrature scheme to evaluate smooth integrals on the far-field, and accelerate the resulting discrete sums using the fast multipole method (FMM). We employ this discretization scheme to analyze several boundary integral formulations of interest including those arising in porous media flow, active matter and magneto-hydrodynamics of rigid particles. We provide numerical results verifying the accuracy and scaling of their evaluation.