Granular gravitational collapse and chute flow

– Inelastic grains in a flow under gravitation tend to collapse into states in which the relative normal velocities of two neighboring grains is zero. If the time scale for this gravitational collapse is shorter than inverse strain rates in the flow, we propose that this collapse will lead to the formation of “granular eddies”, large scale condensed structures of particles moving coherently with one another. The scale of these eddies is determined by the gradient of the strain rate. Applying these concepts to chute flow of granular media, (gravitationally driven flow down inclined planes) we predict the existence of a bulk flow region whose rheology is determined only by flow density. This theory yields the experimental “Pouliquen flow rule”, correlating different chute flows; it also correctly accounts for the different flow regimes observed. Introduction. – Flows of hard granular systems are ubiquitous in nature and technology, yet are still poorly understood [1]. Granular systems typically have a twofold separation of energy scales: the typical energy of a particle is determined by gravity or some other body force (in a few instances by initial conditions), and is much larger than the thermal scale kBT , yet much smaller than the scale required to appreciably deform the particle. Despite the smallness of kBT on the scale of granular energies, many treatments use a pseudo-temperature connected to the random part of the kinetic energy of a particle. Such treatments often link granular phenomena to the kinetic theory of gases. The “granular gas” has an intrinsic rheology, and is driven by the external forcing. One of the pioneering treatments of this rheology was by Bagnold, who discussed chute flows, the gravitationally driven flow of a granular material down an inclined surface [2]. It is simplest to consider a flow of constant, fixed depth H , with the average velocity of the particles parallel to the free surface. The particles are spheres of monodispersed mass M and radius R. We choose axes such that the direction of flow is x̂, the direction perpendicular to the free surface of the flow is ẑ, and the direction parallel to vorticity is ŷ (see Figure 1). The shear stress σxz in such a flow is communicated by particles at slightly differing depths, whose velocities differ if ∂zvx is non-zero. We expect the momentum transfer communicated by collisions between particles at different depths to be of the order of (∗) E-mail: [email protected]