Selection of velocity profile and flow depth in granular flows

The dynamics of a two-dimensional pile constituted by spherical grains organized in parallel layers is investigated theoretically. Only three effects are taken into account in the model: driving by gravity, nonlocal dissipation due to shocks, and trapping of grains by the bumps of the underneath layer. This is sufficient to recover the basic properties of granular avalanches: the transition between static and flowing state is hysteretic; the pile does not flow on the whole height but only in a layer at the surface; the velocity profile inside the flowing layer is approximately linear and is followed by a creep motion in the (quasi) static part. The flow height increases as a function of the pile angle and tends to infinity for a critical angle ${\phi }_{\infty }.$ The dependence of this critical angle with the static angle ${\phi }_s,$ the restitution coefficient ρ, and the moment of inertia J, is investigated.