Kinetic models for granular flow

The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann-Lorentz equation is reduced to a Fokker-Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker-Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker-Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker-Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.