Mechanics of collisional motion of granular materials .4. Expansion wave

The problem of expansion into a vacuum of a semi-infinite layer composed of chaotically moving inelastic rough spherical particles is solved analytically. A variant of the matched asymptotic expansion scheme is used to obtain a matched composite solution, which is valid for both small and large times in the wave head, wave tail and intermediate domains of the disturbed part of the layer. The effects of granular initial energy and particle collisonal properties on their hydrodynamic velocity, temperature, density and pressure are studied in the limit of low initial density of the granular gas. The total granular mass, M, within the disturbed region was found to change with time as log t. This is in contrast with the comparable classical result M similar to t obtained for conservative (molecular) gases. This logarithmic dependence stems from the influence of kinetic energy losses, which reduce the granular temperature and speed of sound in the wave head region. The ultimate escape energy and momentum (i.e. those achieved for long times by the expanding part of the layer) are shown to be finite quantities, dependent on the particle restitution coefficient, roughness and the initial granular temperature. The estimated mass of the escaping part of the layer, calculated here for dilute gases, serves as an upper bound on this quantity for all (also dense) comparable granular gases. This mass is determined by the collisional losses, as embodied within the particle restitution and roughness coefficients.