The hydrodynamic equations for a gas of hard spheres with dissipative dynamics are derived from the Boltzmann equation. The heat and momentum fluxes are calculated to Navier-Stokes order and the transport coefficients are determined as explicit functions of the coefficient of restitution. The dispersion relations for the corresponding linearized equations are obtained and the stability of this linear description is discussed. This requires consideration of the linear Burnett contributions to the energy balance equation from the energy sink term. Finally, it is shown how these results can be imbedded in simpler kinetic model equations with the potential for analysis of more complex states.

The homogeneous cooling state of a granular flow of smooth spherical particles described by the Boltzmann equation is investigated by means of the direct simulation Monte Carlo method. The velocity moments and also the velocity distribution function are obtained and compared with approximate analytical results derived recently. The accuracy of a Maxwell-Boltzmann approximation with a time-dependent temperature is discussed. Besides, the simulations show that the state of uniform density is unstable to long enough wavelength perturbations so that clusters and voids spontaneously form throughout the system. The instability has the characteristic features of the clustering instability which has been observed in molecular dynamics simulations of dense fluids and predicted by hydrodynamic models of granular flows.This research was partially supported by Grant No. PB96- 0534 from the Dirección General de Investigación Científica y Técnica (Spain

A kinetic model for a dilute multicomponent gas system is proposed. It is constructed by replacing the Boltzmann collision operator with a relaxation-time term, in the same manner as in the Bhatnagar–Gross–Krook (BGK) model for a single gas. The model contains several parameters that are determined by keeping some of the main properties of the Boltzmann description. In contrast to previous works, the BGK equation is recovered when mechanically identical particles are considered. Thus the model can be expected to apply to systems in which masses are comparable. The transport properties to the Navier–Stokes level are studied and Onsager’s reciprocal relations are found to hold.

A self-diffusion equation for a freely evolving gas of inelastic hard disks or spheres is derived starting from the Boltzmann–Lorentz equation, by means of a Chapman–Enskog expansion in the density gradient of the tagged particles. The self-diffusion coefficient depends on the restitution coefficient explicitly, and also implicitly through the temperature of the system. This latter introduces also a time dependence of the coefficient. As in the elastic case, the results are trivially extended to the Enskog equation. The theoretical predictions are compared with numerical solutions of the kinetic equation obtained by the direct simulation Monte Carlo method, and also with molecular dynamics simulations. An excellent agreement is found, providing mutual support to the different approaches.