Energy considerations in accelerating rapid shear granular flows

Abstract

We present a complete expression for the total energy associated with a rapid frictional granular shear flow down an inclined surface. This expression reduces to the often used energy for a non-accelerating flow of an isotropic, ideal fluid in a horizontal channel, or to the energy for a vertically falling mass. We utilize thickness-averaged mass and momentum conservation laws written in a slope-defined coordinate system. Both the enhanced gravity and friction are taken into account in addition to the bulk motion and deformation. The total energy of the flow at a given spatial position and time is defined as the sum of four energy components: the kinetic energy, gravity, pressure and the friction energy. Total energy is conserved for stationary flow, but for non-stationary flow the non-conservative force induced by the free-surface gradient means that energy is not conserved. Simulations and experimental results are used to sketch the total energy of non-stationary flows. Comparison between the total energy and the sum of the kinetic and pressure energy shows that the contribution due to gravity acceleration and frictional resistance can be of the same order of magnitude, and that the geometric deformation plays an important role in the total energy budget of the cascading mass. Relative importance of the different constituents in the total energy expression is explored. We also introduce an extended Froude number that takes into account the apparent potential energy induced by gravity and pressure.