Date(s) : 02/07/2019 iCal
11 h 00 min - 12 h 00 min
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  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  in a certain averaged sense. The result holds true in the case of particles that are not so dilute as in  and for which the interactions between particles are still important.
In this paper, using the method of reflections, we extend the investigation of  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  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  and .
 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.
 M. Hillairet. On the homogenization of the Stokes problem in a perforated domain. Arch Rational Mech Anal, 230, (2018), 1179–1228.
 R. M. H ̈ofer Sedimentation of Inertialess Particles in Stokes Flows. Commun. Math. Phys. 360, (2018), 55–101.
 P. E Jabin and F. Otto Identification of the dilute regime in particle sedimentation. Communications in Mathematical Physics. 2004.