arXiv:1609.01502v2 [cond-mat.dis-nn] 14 Sep 2017 This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1 Compressibility regularizes the μ(I )-rheology for dense granular flows J. Heyman 1 †, R. Delannay 1 , H. Tabuteau 1 and A. Valance 1 1 Institut de Physique de Rennes, UMR CNRS 6251, Universit de Rennes 1, Campus de Beaulieu, Btiment 11A, 263 Avenue Gnral Leclerc, 35042 Rennes CEDEX, France (Received xx; revised xx; accepted xx) The µ(I )-rheology was recently proposed as a potential candidate to model the incom- pressible flow of frictional grains in the 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 low and large inertial number limits (Barker et al. 2015). In this rapid communication, we extend the stability analysis of Barker et al. (2015) to compressible flows. We show that compressibility regularizes mostly the equations, making the problem well-posed for all parameters, with the condition that sufficient dissipation be associated with volume changes. In addition to the usual Coulomb shear friction coefficient µ, we introduce a bulk friction coefficient µ b , associated with volume changes and show that the problem is well-posed if µ b > 1 − 7µ/6. Moreover, we show that the ill-posed domain defined by Barker et al. (2015) transforms into a domain where the flow is unstable but remains well-posed when compressibility is taken into account. These results suggest the importance of taking into account dynamic compressibility for the modelling of dense granular flows and open new perspectives to investigate the emission and propagation of acoustic waves inside these flows. Key words: acoustics, complex fluids, granular media 1. Introduction The so called µ(I )-rheology was recently proposed to model granular flows in the dense inertial regime (GDR MiDi 2004; da Cruz et al. 2005). This rheology rests on the fact that unidirectional granular shear flows are fairly well described using a single friction coefficient µ—ratio of the shear stress τ to the confinement pressure p—that varies with an inertial (dimensionless) number I , defined as the ratio of a microscopic grain rearrangement time scale to a macroscopic flow time scale. This rheology may be thought as a generalization of the basic Coulomb friction model τ/p = µ, with a friction coefficient that varies according to the local shear rate and confinement pressure. This simple scaling was shown to break at low inertial numbers close to the jamming limit, where non-local effects become important (Kamrin & Koval 2012). On the other hand, at very high inertial numbers, granular flows enter the collisional regime and are best described by kinetic theory. Regarding two-dimensional incompressible flows, the µ(I )-rheology was also recently shown to lead to ill-posed problems for a large range of parameters, notably for high and low inertial numbers (Barker et al. 2015). For this range of parameters, infinitely small wavelengths are indeed amplified at an † Email address for correspondence: joris.heyman@univ-rennes1.fr