Identification of the Dilute Regime in Particle Sedimentation

We investigate the dynamics of rigid, spherical particles of radius R sinking in a viscous fluid. Both the inertia of the particles and the fluid are neglected. We are interested in a large number N of particles with average distance d≫R. We investigate in which regime (in terms of N and R/d) the particles do not significantly interact and approximately sink like single particles. We rigorously establish the lower bound for the critical number Ncrit of particles. This lower bound agrees with the heuristically expected Ncrit in terms of its scaling in R/d. The main difficulty lies in showing that the particles cannot get significantly closer over a relevant time scale. We use the method of reflection for the Stokes operator to bound the strength of the particle interaction.

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