MODEL KINETIC-EQUATION FOR LOW-DENSITY GRANULAR FLOW

A model kinetic equation is proposed to describe the time evolution of a gas composed of particles which collide inelastically. Dissipation in collisions is described by means of a parameter which is related to the coefficient of restitution, epsilon. The kinetic equation can be solved exactly for the homogeneous cooling state, providing explicit e xpressions for both the time-dependent ''temperature'' and the velocit y distribution function. In contrast to the Maxwellian for fluids with energy conservation, this distribution exhibits algebraic decay for l arge velocities. Hydrodynamic equations are derived by expanding in th e gradients of the hydrodynamic fields around the homogeneous cooling state, without the limitation to epsilon asymptotically close to unity . The equation for the energy density contains, in addition to a sourc e term describing the energy lost in collisions, a contribution to the heat flux which is proportional to the gradient of the density. The l inear stability of the homogeneous cooling state is investigated by an alyzing the hydrodynamic modes of the system. The shear modes are foun d to decay slowly at long wavelengths, in the sense that spatial pertu rbations of the macroscopic flow field decay slower than the cooling r ate for the thermal velocity of the reference homogeneous state. On th e other hand, the heat mode is always stable.