A depth-averaged model for granular flow consistent with the incompressible $\mu (I)$ rheology

We derive a depth-averaged model consistent with the $\mu (I)$ rheology for an incompressible granular flow down an inclined plane. The first two variables of the model are the depth and the depth-averaged velocity. The shear is also taken into account via a third variable called enstrophy. The obtained system is a hyperbolic system of conservation laws, with an additional equation for the energy. The system is derived from an asymptotic expansion of the flow variables in powers of the shallow-water parameter. This method ensures that the model is fully consistent with the rheology. The velocity profile is a Bagnold profile at leading order and the first-order correction to this profile can be calculated for flows that are not steady uniform. The first-order correction to the classical granular friction law is also consistently written. As a consequence, the instability threshold of the steady uniform flow is the same for the depth-averaged model and for the governing equations. In addition, a higher-order version that contains diffusive terms is also presented. The spatial growth rate, the phase velocity and the cutoff frequency of the version with diffusion are in good agreement with the experimental data and with the theoretical predictions for the rheology. The mathematical structure of the equations enables us to use well-known and stable numerical solvers. Numerical simulations of granular roll waves are presented. The model has the same limitations as the $\mu (I)$ rheology, in particular for the solid/ liquid and liquid/gas transitions, and needs therefore a regularisation for these transitions.