Exponential velocity profile of granular flows down a confined heap

Thick granular flows are essential to many natural and industrial phenomena. Experimentally, it has been well established that the grain velocity profile is linear from the free surface to a certain depth, after which it decreases exponentially in the so-called “creep region”. In this paper we obtain an exponential velocity profile based on the force balance of a grain near a wall, where the Janssen effect and the non-locality of interactions between grains are considered. When experimental parameters such as flow angles and friction coefficients are introduced in our model, it is able to reproduce experimental creep velocity profiles previously reported in the literature.

Figure 1

Picture illustrating the different granular layers in a thick flow of silicon sand confined into a Hele-Shaw cell. The picture was recorded during $0.1$ s, so the mobile grains look blurred. The origin of coordinates helps illustrating the region where the creep zone stars (corresponding to $y\ge 0$). Above it, there is a transition zone and a layer with a linear velocity profile. We define ${\theta }_c$ as the angle of the flow.

Figure 2

Force diagram of a generic grain. (a) The gravity force, the weight of the upper grains, the reaction of the lower grains, and the friction forces are represented. We define $\overline{AB}=\overline{DC}=l$ and $\overline{AD}=\overline{BC}=\delta h$. (b) The other tangential and normal forces acting on the grain are shown; the shadow region represents the wall nearest to the right-hand side of the grain.

Figure 3

Force diagram of the grain that helps establishing the relation between the arms of the normal forces exerted at points $A$ and $C$. We assume that the grain cannot rotate around the vertical axis.

Figure 4

Experimental velocity profiles for datasets shown in Table 1 plotted in semi-log scale. The velocity is normalized to the velocity of the grain where the creep region starts, and the depth, to the characteristic length of the grain in the $y$ direction. The theoretical fits are showed with lines.