Abstract

This paper presents quite general bidimensional gas-dynamic equations-derived from kinetic theory-which include the fourth cumulant ?(r?,t) as a dynamic field. The dynamics describes a low-density system of inelastic hard spheres (disks) with normal restitution coefficient r. Two illustrative examples are given and the role of ? in them is discussed. Our general gas-dynamic equations would deal with 9 hydrodynamic fields (which corresponds to 14 in three-dimension). These fields are the standard hydrodynamic fields plus the components pij of the traceless part of the pressure tensor, the energy flux vector Q? and the fourth cumulant ?. The present formulation requires no constitutive equations. The two examples are: the well-known homogeneous cooling state and a system, with and without gravity, steadily heated by two parallel walls. In the first case, the dynamics yield a description of the homogeneous cooling state consistent with known results adding extra details mainly about the transient time behavior. The steadily heated system kept in a static state gives rise to quite simple but nontrivial equations. In the case with gravity, it is shown that when ? is included as a dynamic field, the formalism leads to a non-Fourier law already to first order in dissipation. Setting gravity g=0 a perturbative solution is shown and favorably compared with observations obtained from molecular dynamics (MD). In both cases, with and without gravity, ? is not homogeneous. An analytic extension suggests a divergent situation for a small negative value of q, which originates in the unavoidable extension of the formalism to exothermic collisions associated with a restitution coefficient larger than one. This divergent behavior is observed in MD. ©2002 The American Physical Society.