## Shocks in rapid granular flows

##### Abstract

The speed of a pressure wave (the speed of sound) in rapid granular flows is typically only a few centimeters per second while the collective streaming motion of the particles is on the order of meters per second. In this supersonic regime, shocks form when a granular flow encounters an obstacles. This work examines the shocks formed in three geometries: the surface wake behind a cylinder, the oblique shock formed at a wedge and a normal shock propagating through a funnel. In each case we evaluate the applicability of a hydrodynamic description to shocks in rapid granular flows.
We study the V-shaped wake formed by a cylindrical rod moving through a vertically vibrated granular layer. The wake appears for rod velocities vR greater than a critical velocity c. We measure the half-angle θ of the wake as a function of vR and layer depth h. We find that c and θ can be described by a hydrodynamic description applied to shallow fluids, where c = √gh is the speed of a gravitational wave on a shallow fluid and sinθ = c/vR is the Mach
relation. We find the decrease in the height of the wake hmax as it propagates away from the rod agrees with Landau’s theory for the decay of shock waves far from their origin.
We measure the time-averaged velocity, density and temperature fields for a gravity driven granular flow past a wedge. The flow is supersonic with a sound speed less than 10% of the flow speed. We find the shock formed at the wedge tip is nearly identical to oblique shocks found in a supersonic, elastic gas. Molecular dynamics simulations of Newton’s laws yield fields in quantitative agreement with experiment. A numerical solution of granular hydrodynamic equations is only in qualitatively accord with experiment. We show that hydrodynamic theory fails because it does not include friction. We use molecular dynamics simulations to examine the effect of friction on the dissipation of energy and scattering angles in collisions.
We examine the propagation of a normal shock formed in a quasi-two dimensional funnel. For shocks propagating without change in a fluid, one can use the Rankine-Hugoniot approximation to predict the velocity of the shock and the difference in flow values across the shock. We show that inelastic collisions between particles cause the shock to continuously evolve, hence the Rankine-Hugoniot predictions are inadequate for describing the evolution of granular shocks.

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