Depth and minimal slope for surface flows of cohesive granular materials on inclined channels

We present analytical predictions of the depth and onset slope of the steady surface flow of a cohesive granular material in an inclined channel The rheology of Jop, Forterre & Pouliquen (Nature, vol. 441, 2006, pp. 727-730) is used, assuming co-axiality between the stress and strain-rate tensors, and a coefficient of friction dependent on the strain rate through the dimensionless inertial number I. This theological law is augmented by a constant stress representing cohesion. Our analysis does not rely on a precise mu (I) functional, but only on its asymptotic power law in the limit of vanishing strain rates. Assuming a unidirectional flow, the Navier Stokes equations can be solved explicitly to yield parametric equations of the iso-velocity lines in the plane perpendicular to the flow. Two types of channel walls are considered: rough and smooth, depicting walls whose friction coefficient is respectively larger or smaller than that of the flowing material. The steady flow starts above a critical onset angle and consists of a sheared zone confined between a surface plug flow and a deep dead zone. The details of the flow are discussed, depending on dimensionless parameters relating the static friction coefficient, cohesion strength of the material, incline angle, wall friction, and channel width. The depths of the flow at the centre of the channel and at the walls are calculated by a force balance on the flowing material. The critical angle for the onset of the flow is also calculated, and is found to be strongly dependent on the channel width, in agreement with experimental results on heap stability and in rotating drums. Our results predict the important conclusion that a cohesive material always starts to flow for an incline angle lower than 90 between smooth walls, whereas in a narrow enough channel with rough walls, it may not flow, even if the channel is inclined vertically.