Propagating fronts on sandpile surfaces

– The flow of granular matter such as sand is often characterized by the motion of a thin superficial layer near the free surface, while the bulk of the solid remains immobile. A pair of equations called the BCRE equations (Bouchaud J-P., Cates M. E., Ravi Prakash J. and Edwards S. F. J. Phys. 4 (1994) 1383) have been proposed to model these flows and account for the dynamic exchange of mass between moving and stationary grains using the simplest kinematic considerations. We uncover a new conservation law for the BCRE equations and its variants that unifies a variety of recent special solutions and show that these equations support simple waves, and are capable of finite time singularities that correspond to propagating erosion fronts. The flow of sand in a hourglass, the ripples on a beach and the clogging of a grain hopper are commonplace examples of our experience with particulate matter. The study of these materials cuts across the traditional boundaries of solids, fluids and gases; the finite angle of repose of a mound of sand is like that of a solid that preserves its shape, a snow avalanche is reminiscent of a flowing fluid, and the motion of a sediment suspension is like that of a dilute gas. Theoretical approaches to these problems at a macroscopic level use a combination of ideas from continuum mechanics and phenomenology, and at a microscopic level use inelastic molecular dynamics and statistical/kinetic approaches [1]. Of the phenomenological approaches, one model that accounts for the flow of granular materials in a thin superficial layer in such phenomena as avalanches [1] was first explicitly laid out in [2], although variants of these equations had been proposed earlier [3]. In its simplest form, the model characterizes the dynamics of these free-surface flows in terms of two dependent variables corresponding to the height Ĥ(x, t) of a solid-like immobile phase and the effective height R(x, t) of a liquid-like mobile phase that is akin to a free-surface shear band sliding on top of the solid-like phase. It is most convenient to think of the height H(x, t) of immobile grains relative to a sandpile resting at the angle of repose so that H(x, t) = Ĥ(x, t)−x tan θ, where θ is the angle of repose. Although the original equations also () Permanent address: 1-310, MIT, Cambridge, MA 02139-4307, USA. E-mail: l [email protected]