Convergence of the method of reflections for particle suspensions in Stokes flows

TL;DR

This paper proves the convergence of the method of reflections for Stokes flows around many particles, providing optimal rates under certain separation conditions, advancing understanding of suspension modeling.

Contribution

It establishes convergence and optimal rates of the method of reflections for particle suspensions in Stokes flows, under specific separation assumptions.

Findings

Method of reflections converges in ot H^1 with mild separation.

Optimal convergence rates in ot W^{1,q} and L^ty are achieved.

Convergence depends on particle volume fraction and separation conditions.

Abstract

We study the convergence of the method of reflections for the Stokes equations in domains perforated by countably many spherical particles with boundary conditions typical for the suspension of rigid particles. We prove that a relaxed version of the method is always convergent in $\dot H^1$ under a mild separation condition on the particles. Moreover, we prove optimal convergence rates of the method in $\dot W^{1,q}$, $1 < q < \infty$ and in $L^\infty$ in terms of the particle volume fraction under a stronger separation condition of the particles.