Development of a Density Inversion in Driven Granular Gases

Abstract

Granular materials fluidized by a rapidly vibrating bottom plate often develop a fascinating density inversion: a heavier layer of granulate supported by a lower-density region. We employ the Navier-Stokes granular hydrodynamics to follow a density inversion as it develops in time. Assuming a dilute low-Mach-number flow, we derive a reduced time-dependent model of the late stage of the dynamics. The model looks especially simple in the Lagrangian coordinates. The time-dependent solution describes the growth of a density peak at an intermediate height. A transient temperature minimum is predicted to develop in the region of the density peak. The temperature minimum disappears at later times, as the system approaches the steady state. At late times, the predictions of the low-Mach-number model are in good agreement with a numerical solution of the full hydrodynamic equations. At an early stage of the dynamics, pressure oscillations are predicted.