The mean-field limit for solid particles in a Navier-Stokes flow

We propose a mathematical derivation of Brinkman's force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Ω³ for the velocity field u of an incompressible fluid with kinematic viscosity ν and density 1. Brinkman's force consists of a source term 6π ν j where j is the current density of the particles, and of a friction term 6π ν ρ u where ρ is the number density of particles. These additional terms in the motion equation for the fluid are obtained from the Stokes or steady Navier-Stokes equations set in Ω minus the disjoint union of N balls of radius ε=1/N in the large N limit with no-slip boundary condition. The number density ρ and current density j are obtained from the limiting phase space empirical measure, where x_k is the center of the k-th ball and v_k its instantaneous velocity. This can be seen as a generalization of Allaire's result in [Arch. Ration. Mech. Anal. 113:209-259, [1991]] who considered the case of periodically distributed x_k s with v _k =0, and our proof is based on slightly simpler though similar homogenization arguments. Similar equations are used for describing the fluid phase in various models for sprays.