A Kinetic Model for the Sedimentation of Rod-Like Particles

Abstract. We consider a kinetic multi-scale model, describing suspensions of rod-like particles, which couples a microscopic Smoluchowski equation for the distribution of rod orientations to a macroscopic Stokes or Navier-Stokes equation via an elastic stress tensor and a gravitational forcing term. A reciprocal coupling of phenomena on these different scales leads to the formation of clusters. The buoyancy force creates a macroscopic velocity gradient that causes the microscopic particles to align in such a way that their sedimentation reinforces the formation of clusters of higher particle density. We perform a linear stability analysis and show that linear theory can predict a wavelength selection mechanism for the cluster width, provided that the Reynolds number is larger than zero. An asymptotic analysis of the largest eigenvalue around Re = 0 explains this response, while numerical simulations of the nonlinear model confirm the wavelength selection. By looking at special flow configurations and at parameter scalings, we derive simpler models which describe the aggregate behavior of the suspension in various regimes. In particular, in a diffusive type of scaling, we obtain the two-dimensional Keller-Segel model.