Model kinetic equation for low-density granular flow

Abstract

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, \ensuremath{\epsilon}. The kinetic equation can be solved exactly for the homogeneous cooling state, providing explicit expressions for both the time-dependent ``temperature'' and the velocity distribution function. In contrast to the Maxwellian for fluids with energy conservation, this distribution exhibits algebraic decay for large velocities. Hydrodynamic equations are derived by expanding in the gradients of the hydrodynamic fields around the homogeneous cooling state, without the limitation to \ensuremath{\epsilon} asymptotically close to unity. The equation for the energy density contains, in addition to a source term describing the energy lost in collisions, a contribution to the heat flux which is proportional to the gradient of the density. The linear stability of the homogeneous cooling state is investigated by analyzing the hydrodynamic modes of the system. The shear modes are found to decay slowly at long wavelengths, in the sense that spatial perturbations of the macroscopic flow field decay slower than the cooling rate for the thermal velocity of the reference homogeneous state. On the other hand, the heat mode is always stable. \textcopyright{} 1996 The American Physical Society.