Hydrodynamic modes of a uniform granular medium

Abstract

This paper uses a fluid-mechanical model of a granular medium to calculate the hydrodynamic modes of a spatially uniform basic state. These modes are the granular analogs of the heat, sound, and shear modes of the standard fluid. Attention is focused on the possibility of an unstable mode that might result in the spontaneous development of inhomogeneities in density. Two cases are considered: the cooling medium which loses energy without replenishment, and the heated medium which reaches a steady state when an energy source balances the loss of energy through particle collisions. The spatially uniform state of the cooling granular medium is unstable. Two modes, analogous to the shear and heat conduction modes of a standard fluid, are unstable at long wavelengths. The growth of these modes is algebraic, rather than exponential, in time. The shear mode does not involve the formation of density inhomogeneities, but the heat mode does. At long wavelengths the heat mode can be visualized by imagining a converging velocity increasing the density of particles in a certain region. The increased collisional dissipation of granular thermal energy reduces the pressure, and prevents it from reversing the convergent velocities, so the condensation is not checked. The stability of the heated granular medium depends on the energy source. If the energy source selectively deposits energy in hot regions of a disturbance, the diffusion and collisional damping can be overwhelmed, and the disturbance grows exponentially. The standard fluid (completely elastic particles) can be recovered as a special case of the heated granular medium. In all cases, waves analogous to the sound and heat conduction modes are present. In some cases, a second type of sound wave is present at long wavelengths with the peculiar property of being damped more quickly for more elastic particles.