We consider the sedimentation of N identical spherical particles in a uniform gravitational field. Particle rotation is included in the model while inertia is neglected.
In the dilute case, the result in [5] shows that the particles do not get closer in finite time. The rigorous convergence of the dynamics to the solution of a Vlasov-Stokes equation is proven in [4] in a certain averaged sense. The result holds true in the case of particles that are not so dilute as in [5] and for which the interactions between particles are still important.
In this paper, using the method of reflections, we extend the investigation of [4] by discussing the optimal particle distance which is conserved in finite time. The set of particle configurations considered herein is the one introduced in [3] for the analysis of the homogenization of the Stokes equation. 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 [1] and [2].
Key-words: Suspension flows, Interacting particle systems, Stokes equations, Vlasov-like equations
References
[1] M. Hauray, Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci. 19 (2009) pp. 1357-1384.
[2] M. Hauray and P.-E. Jabin. Particle approximation of Vlasov equations with singular forces: propagation of chaos, Ann. Sci. Ec. Norm. Super. 48-4 (2015), pp. 891-940.
[3] M. Hillairet, On the homogenization of the Stokes problem in a perforated domain. Arch. Rational Mech. Anal. (2018), https://doi.org/10.1007/s00205-018-1268-7
[4] R.-M. Höfer, Sedimentation of Inertialess Particles in Stokes Flows, Commun. Math. Phys. 360-1 (2018), pp. 55-101.
[5] P.-E. Jabin and F. Otto, Identification of the dilute regime in particle sedimentation, Commun. Math. Phys. 250 (2004), pp. 415-432.