Microscopic derivation of a one-dimensional lubrication model with roughness

Abstract

We derive a hydrodynamic model for the motion of inertial particles with a spherical hard core, interacting through lubrication forces and pairwise repulsive forces. The repulsion arises from the assumption that each particle is surrounded by a thin rough layer of reduced permeability. We prove that, as the number of particles tends to infinity (and their size tends to 0), the microscopic dynamics converges to a macroscopic hydrodynamic model in which congestion effects are encoded directly into the macroscopic interaction forces, depending on a local critical density transported by the flow. In particular, we extend the work of Lefebvre-Lepot and Maury where non-inertial particles, submitted to only a lubrication force were considered, and present the convergence proof when inertial effects and roughness are taken into account.