Development of a density inversion in driven granular gases

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