Sedimentation of particles in Stokes flow

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Date(s) - 02/07/2019
11 h 00 min - 12 h 00 min

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We consider the sedimentation of N identical spherical particles in a uniform gravitational field. Particle rotation is included in the model while fluid and particle 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 dis- cussing 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].

[1] M. Hauray. Wasserstein distances for vortices approximation of Euler-type equations. Math. Models Methods Appl. Sci. 19, [1357,1384],2009.
[2] M. Hauray and P. E. Jabin. Particle approximation of Vlasov equations with singular forces: propa- gation of chaos. Ann. Sci. E ́c. Norm. Sup ́er. (4), 48(4):[891-940], 2015.
[3] M. Hillairet. On the homogenization of the Stokes problem in a perforated domain. Arch Rational Mech Anal, 230, (2018), 1179–1228.
[4] R. M. H ̈ofer Sedimentation of Inertialess Particles in Stokes Flows. Commun. Math. Phys. 360, (2018), 55–101.
[5] P. E Jabin and F. Otto Identification of the dilute regime in particle sedimentation. Communications in Mathematical Physics. 2004.

http://www.theses.fr/s174812

Olivier CHABROL
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