On the sedimentation of particle suspensions in Stokes flows

Small particles moving in a fluid are encountered in various situations in nature and technology. In many cases, gravitation is the driving force for the dynamics of the particles. If the particles are not too small, the system can be microscopically modeled by the Navier-Stokes equations coupled with a system of ODEs for the particles according to Newton's laws. Although the force acting on each particle due to the gravity is directly proportional to its mass, and we do not include direct (e.g. electromagnetic) interaction between the particles themselves, the motion of the particles will be quite complex in many situations. The complexity arises from the interaction of the particles through the fluid. Indeed, the presence of each particle induces a disturbance in the fluid flow which again influences all the other particles.

Assuming that the fluid inertia is negligible, we focus on three closely related microscopic models for spherical and non-spherical particles with and without inertia. We will then discuss corresponding macroscopic models which consist of systems that couple a Vlasov equation to Stokes equations. In the case of inertialess spherical particles, the macroscopic system can be rigorously derived from the microscopic dynamics in the limit of many small particles.

This talk is based on joint work with Juan Velázquez and Arianna Giunti.