Well-posed continuum equations for granular flow with compressibility and (I)-rheology

Abstract

Continuum modelling of granular flow has been plagued with the issue of ill-posed dynamic equations for a long time. Equations for incompressible, twodimensional flow based on the Coulomb friction law are ill-posed regardless of the deformation, whereas the rate-dependent (I)-rheology is ill-posed when the non-dimensional inertial number I is too high or too low. Here, incorporating ideas from critical-state soil mechanics, we derive conditions for well-posedness of partial differential equations that combine compressibility with I-dependent rheology. When the I-dependence comes from a specific friction coefficient (I), our results show that, with compressibility, the equations are well-posed for all deformation rates provided that (I) satisfies certain minimal, physically natural, inequalities.