Sedimentation of particles with very small inertia I: Convergence to the transport-Stokes equation

We consider the sedimentation of N spherical particles with identical radii R in a Stokes flow in R3. The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton’s law but with very small particle inertia as N tends to infinity and R to zero. In a mean-field scaling, we show that the particle evolution is well approximated by the transport-Stokes system which has been derived previously as the mean-field limit of inertialess particles. In particular, this justifies neglecting the particle inertia in the microscopic system, which is a typical modeling assumption in this and related contexts. The proof is based on a modulated energy argument that exploits the coercivity of the particle forces with respect to the particle velocities in a Stokes flow. We combine this with an adaptation of Hauray’s method for mean-field limits to 2-Wasserstein distances. Moreover, in order to control the minimal distance between particles, we prove a representation of the particle forces. This representation makes the heuristic “Stokes law” rigorous that the force on each particle is proportional to the difference of the velocity of the individual particle and the mean-field fluid velocity generated by the other particles.