Sedimentation of Inertialess Particles in Stokes Flows

Abstract

We investigate the sedimentation of a cloud of rigid, spherical particles of identical radii under gravity in a Stokes fluid. Both inertia and rotation of particles are neglected. We consider the homogenization limit of many small particles in the case of a dilute system in which interactions between particles are still important. In the relevant time scale, we rigorously prove convergence of the dynamics to the solution of a macroscopic equation. This macroscopic equation resembles the Stokes equations for a fluid of variable density subject to gravitation.We consider a sequence of initial particle configurations indexed by ε and we assume N ε → ∞ and R ε → 0 in the limit ε → 0. Moreover, we assume lim ε→0 ξ ε = ξ * ∈ [0, ∞).Furthermore, we consider the mass density of the particleswhere the particle positions depend on ε. The dynamics (1) implies that the particles are transported by the velocity fieldv ε , i.e, ∂ tρε +v ε · ∇ρ ε = 0.

Sedimentation of particles in Stokes flow

“…The assumption on the initial density ρ 0 is the one introduced by Höfer in [10] which is ρ 0 , ∇ρ 0 ∈ X β , for some β > 2. See Section 5.1 for the definition of X β .…”

“…Moreover, in the case where the minimal distance between particles is much larger than 1/N 1/3 the result in [12] shows that particles do not interact and sink like single particles. We refer finally to [10] where the author considers a particle system with minimal distance of order 1/N 1/3 and proves that, under a relevant time scale, the spatial density of the cloud converges in a certain averaged sense to the solution of the Vlasov-Stokes equation (58). In this paper, we continue the investigation of [10] by looking for a more general set of particle configurations that is conserved in time and prove the convergence to the Vlasov-Stokes equation (58).…”

“…We refer finally to [10] where the author considers a particle system with minimal distance of order 1/N 1/3 and proves that, under a relevant time scale, the spatial density of the cloud converges in a certain averaged sense to the solution of the Vlasov-Stokes equation (58). In this paper, we continue the investigation of [10] by looking for a more general set of particle configurations that is conserved in time and prove the convergence to the Vlasov-Stokes equation (58). Also, we include particle rotation in the modeling.…”

A model for suspension of clusters of particle pairs

“…Remark 0.2. Analogously to the model (1), global existence a uniqueness result can be shown for the former model following the result of [10].…”

“…Theorem 0.2. We consider the additional assumption (10). Assume that there exists a function F 0 ∈ W 1,∞ such that ξ i (0) = F 0 (x i + (0)) for all 1 ≤ i ≤ N. There exists T > 0 independent of N and unique F N ∈ L ∞ (0, T ; W 1,∞ ) such that for all t ∈ [0, T ] we have:…”