Sedimentation of particles in Stokes flow

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 [10] 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 [7] and [8].

A model for suspension of clusters of particle pairs

“…when the minimal distance between particles is larger than 1 N 1/3 . In [10,17] the justification has been extended to regimes that are not so dilute but where the minimal distance between particles is still large compared to the particles radii. The coupled equations derived are:…”

“…As in [17], the idea of proof of Theorem 0.1 and 0.2 is to provide a derivation of the kinetic equation satisfied weakly by µ N . This is done by computing the first order terms of the velocities of each pair:…”

“…The interaction force Φ is the Oseen tensor, see formula (17). This development is a corollary of the method of reflections which consists in approaching the solution u N of 2N separated particles by the superposition of fields produced by the isolated 2N particle solutions.…”

“…We also refer to [11] where a converging method of reflections is developed and is used in [10]. In this paper we reproduce the same method of reflections developed in [17,Section 3]. However this method is no longer valid in the case where the minimal distance is comparable to the particle radii.…”

“…This requires an analysis of the solution of the Stokes equation past a pair of particles. In particular, we need to show that these special solutions have the same decay rate as the Stokeslets, see [17,Section 2.1]. The convergence of the method of reflections is ensured under the condition that the minimal distance d min between the centers x i + satisfies…”