The μ(I)-rheology has been recently proposed as a potential candidate to model the flow of frictional grains in a dense inertial regime. However, this rheology was shown to be ill-posed in the mathematical sense for a large range of parameters, notably in the slow and fast flow limits \citep{Barker2015}. In this rapid communication, we extend the stability analysis to compressible flows. We show that compressibility regularizes mostly the equations, making them well-posed for all parameters, at the condition that sufficient dissipation is associated with volume changes. In addition to the usual Coulomb shear friction coefficient μ, we introduce a bulk friction coefficient μb, associated to volume changes and show that the equations are well-posed in two dimensions if μb>2−2μ (μb>3−7μ/2 in three dimensions). Moreover, we show that the ill-posed domain defined in \citep{Barker2015} transforms into a domain where the equations are unstable but stay well-posed when compressibility is taken into account. These results suggest thus the importance of compressibility in dense granular flows.
@article{arxiv.1609.01502,
title = {Compressibility regularizes the "$\mu(I)$" rheology for granular flows},
author = {J. Heyman and R. Delannay and H. Tabuteau and A. Valance},
journal= {arXiv preprint arXiv:1609.01502},
year = {2017}
}