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

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.