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

TL;DR

This paper derives conditions ensuring well-posed continuum equations for granular flow by combining compressibility with $$-dependent rheology, resolving longstanding issues of ill-posedness in granular flow modeling.

Contribution

It introduces a framework that guarantees well-posed PDEs for granular flow with compressibility and $$-rheology, based on natural inequalities for the friction coefficient.

Findings

Equations are well-posed for all deformation rates with appropriate $$-dependent friction laws.

Incompressibility alone leads to ill-posed equations regardless of flow conditions.

Conditions derived are physically natural and applicable to a range of granular flow scenarios.

Abstract

Continuum modelling of granular flow has been plagued with the issue of ill-posed equations for a long time. Equations for incompressible, two-dimensional flow based on the Coulomb friction law are ill-posed regardless of the deformation, whereas the rate-dependent $\mu(I)$-rheology is ill-posed when the non-dimensional strain-rate $I$ is too high or too low. Here, incorporating ideas from Critical-State Soil Mechanics, we derive conditions for well-posedness of PDEs that combine compressibility with $I$-dependent rheology. When the $I$-dependence comes from a specific friction coefficient $\mu(I)$, our results show that, with compressibility, the equations are well-posed for all deformation rates provided that $\mu(I)$ satisfies certain minimal, physically natural, inequalities.