Diphasic non - local model for granular surface flows

– Considering recent results revealing the existence of multi-scale rigid clusters of grains embedded in granular surface flows, i.e. flows down an erodible bed, we describe here the surface flows rheology through a non-local constitutive law. The predictions of the resulting model are compared quantitatively to experimental results: The model succeeds to account for the counter-intuitive shape of the velocity profile observed in experiments, i.e. a velocity profile decreasing exponentially with depth in the static phase and remaining linear in the flowing layer with a velocity gradient independent of both the flowing layer thickness, the angle between the flow and the horizontal, and the coefficient of restitution of the grains. Moreover, the scalings observed in rotating drums are recovered, at least for small rotating speed. Introduction. – Granular media share properties with both usual liquids and solids: They can sustain an inclined free surface without flowing but, when the angle exceeds a critical value identified since Coulomb [1] with some effective macroscopic friction angle Φ M , an avalanche occurs. The motion has the peculiarity of being a surface flow. Two phases can be observed: A " solid " phase experiencing a creep motion where the averaged streamwise velocity decreases exponentially with the depth [2,3], and a flowing phase exhibiting a linear velocity profile with a velocity gradient independent of the flowing layer thickness, the angle between the mean flow and the horizontal, and the coefficient of restitution of the beads [4,3]. Such profiles cannot be described using any conventional local and univocal constitutive laws of Continuum Mechanics for two main reasons: (i) The velocity gradient is found to be constant in the cascading layer whereas momentum balance implies that the shear stress increases linearly with depth and (ii) the velocity profile in the flowing layer down an erodible bed differs significantly from the Bagnold-like velocity profile observed in dense flows down rough inclines [5]. Several models have been recently proposed to describe dense granular flows [6], but, to our knowledge, none of them succeeds to account for the velocity profile measured experimentally in surface flows.