Volume 79, Number 3, August 2007
Article Number 34001
Number of page(s) 6
Section Electromagnetism, Optics, Acoustics, Heat Transfer, Classical Mechanics, and Fluid Dynamics
Published online 13 July 2007
EPL, 79 (2007) 34001
DOI: 10.1209/0295-5075/79/34001

A mean-field model for the rheology and the dynamical phase transitions in the flow of granular matter

B. Andreotti

Laboratoire de Physique et Mécanique des Milieux Hétérogènes, UMR 7636 CNRS, Université Paris 7 10 rue Vauquelin, 75231 Paris Cedex 05, France

received 8 November 2006; accepted in final form 14 June 2007; published August 2007
published online 13 July 2007

Based on a large set of experiments and numerical simulations, it has been recently shown (GDR MiDi (collective work), Eur. Phys. J. E, 14 (2004) 341) that dense granular flows are well described by a local rheology: the ratio of the shear stress $\tau $ to the normal stress P is an increasing function of the properly rescaled shear rate I. We propose a mean field model for this quasi-local constitutive relation and the phase diagram of granular matter, based on the motion of single spherical grain on an array of identical grains. The model recovers a "solid-liquid" transition that is controlled by potential trapping, as well as a subcritical "liquid-gas" transition governed by the restitution coefficient. The system presents a "triple point" above which the grain directly leaves the static equilibrium to enter the gaseous regime. In the liquid regime, the relation between force and velocity is found to be almost independent of the microscopic parameters, friction and restitution coefficient. The dynamics is dominated by potential trapping and leads to a constitutive relation of the form $\tau/P=\mu_d+\delta \mu \; \exp\left( -I_d/I \right) $, in close agreement with experimental and numerical results. This rheology is only quasi-local as the inertial number is redefined, introducing the effective number of grains $\mathcal{N} $ involved in a collision: $I=\dot \gamma \sqrt{\mathcal{N} m/Pd} $.

45.70.Mg - Granular flow: mixing, segregation and stratification.
83.50.Ax - Steady shear flows, viscometric flow.

© Europhysics Letters Association 2007