Sedimentation of particles in Stokes flow

TL;DR

This paper analyzes the sedimentation behavior of identical spherical particles in Stokes flow, including particle rotation, and establishes the optimal particle distance, interaction forces, and mean field approximation validity.

Contribution

It extends previous work by discussing the optimal particle distance, proves the singular interaction force via the Oseen tensor, and justifies the mean field approximation in sedimentation.

Findings

Optimal particle distance conserved in finite time

Particles interact via singular Oseen tensor force

Mean field approximation of Vlasov-Stokes equations justified

Abstract

In this paper, we consider $N$ identical spherical particles sedimenting in a uniform gravitational field. Particle rotation is included in the model while inertia is neglected. Using the method of reflections, we extend the investigation of [R. M. H\"ofer, Sedimentation of inertialess particles in Stokes flows, arXiv:1610.03748, (2016)] by discussing the optimal particle distance which is conserved in finite time. We also prove that the particles interact with a singular interaction force given by the Oseen tensor and justify the mean field approximation of Vlasov-Stokes equations in the spirit of [M. Hauray and P. E. Jabin, Particle approximation of Vlasov equations with singular forces : propagation of chaos, Ann. Sci. Ec. Norm. Super. (4), (2015)] and [M. Hauray, Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci. 19, (2009), pp. [1357,1384]].