Modelling of phase transitions in granular flows

We present in this work a system for unidimensional granular flows first mentioned in a paper of A. Lefebvre–Lepot and B. Maury (2011), which captures the transitions between compressible and incompressible phases. This model exhibits in the incompressible regions some memory effects through an additional variable called adhesion potential . We derive this system from compressible Navier–Stokes equations with singular viscosities and pressure, the singular limit between the two systems can then be seen as an analogue of the low Mach number limit for fluid with pressure dependent viscosity. It answers in a sense to the problem of transition between suspension flows and immersed granular flows identified by B. Andreotti, Y. Forterre and O. Pouliquen (2011). We illustrate the singular limit by numerical simulations.