An integral equation formulation for rigid bodies in Stokes flow in three dimensions

TL;DR

This paper introduces a new boundary integral equation method for simulating the three-dimensional motion of arbitrarily-shaped rigid particles in Stokes flow, using a single-layer representation, spectral quadrature, and fast multipole acceleration.

Contribution

A novel second-kind integral equation formulation for 3D rigid particles in Stokes flow that avoids auxiliary sources and enhances computational efficiency.

Findings

High accuracy and good conditioning demonstrated in numerical tests.

Linear scaling of computations with the number of particles using FMM.

Effective high-order time-stepping for particle dynamics.

Abstract

We present a new derivation of a boundary integral equation (BIE) for simulating the three-dimensional dynamics of arbitrarily-shaped rigid particles of genus zero immersed in a Stokes fluid, on which are prescribed forces and torques. Our method is based on a single-layer representation and leads to a simple second-kind integral equation. It avoids the use of auxiliary sources within each particle that play a role in some classical formulations. We use a spectrally accurate quadrature scheme to evaluate the corresponding layer potentials, so that only a small number of spatial discretization points per particle are required. The resulting discrete sums are computed in $\mathcal{O}(n)$ time, where $n$ denotes the number of particles, using the fast multipole method (FMM). The particle positions and orientations are updated by a high-order time-stepping scheme. We illustrate the accuracy, conditioning and scaling of our solvers with several numerical examples.