Pressure Dependent Viscosity Model for Granular Media Obtained from Compressible Navier-Stokes Equations

摘要

The aim of this article is to justify mathematically, in the two-dimensionalperiodic setting, a generalization of a two-phase model with pressure dependentviscosity first proposed by A. Lefebvre-Lepot and B. Maury to describe a systemin one dimension of aligned spheres interacting through lubrication forces.This model involves an adhesion potential, apparent only on the congesteddomain, which keeps track of history of the flow. The solutions are constructed(through a singular limit) from a compressible Navier-Stokes system withviscosity and pressure both singular close to a maximal volume fraction.Interestingly, this study can be seen as the first mathematical connectionbetween models of granular flows and models of suspensions. As a by-product ofthis result, we also obtain global existence of weak solutions for a system ofincompressible Navier-Stokes equations with pressure dependent viscosity, theadhesion potential playing a crucial role in this result.