Hydrodynamic equations for nearly smooth granular gases are derived from the pertinent Boltzmann equation. The angular velocity distribution field needs to be included in the set of hydrodynamic fields. The angular velocity distribution is strongly non-Maxwellian for the homogeneous cooling state and any homogeneous steady state. In the case of steady wall-bounded shear flows the average spin (created at the boundaries) has a finite penetration length into the bulk.

A novel computer-aided method for solving kinetic equations has been developed and implemented in a study of the Boltzmann equation corresponding to elastic and inelastic hard spheres. Accurate results are obtained for the linear transport coefficients for all physical values of the coefficient of normal restitution, α. These coefficients are bounded and nonsingular even in the limit of vanishing α. Using the new method we also calculated the full homogeneous cooling state (HCS) distribution function (after replacing the standard divergent expansion by a convergent one) and confirmed the conjecture that it possesses an exponential tail. Further implications and applications of these results are outlined.

Hydrodynamic equations of motion for a monodisperse collection of nearly smooth homogeneous spheres have been derived from the corresponding Boltzmann equation, using a Chapman-Enskog expansion around the elastic smooth spheres limit. Because in the smooth limit the rotational degrees of freedom are uncoupled from the translational ones, it turns out that the required hydrodynamic fields include (in addition to the standard density, velocity, and translational granular temperature fields) the (infinite) set of number densities, n(s,r, t), corresponding to the continuum of values of the angular velocities. The Chapman-Enskog expansion was carried out to high (up to 10th) order in a Sonine polynomial expansion by using a novel computer-aided method. One of the consequences of these equations is that the asymptotic spin distribution in the homogeneous cooling state for nearly smooth, nearly elastic spheres, is highly non-Maxwellian. The simple sheared flow possesses a highly non-Maxwellian distribution as well. In the case of wall-bounded shear, it is shown that the angular velocity injected at the boundaries has a finite penetration length.

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One of the goals of the present paper is to present a brief, and admittedly somewhat biased, review of some recent theoretical advances in the field of granular gases. Another goal is to highlight some challenges facing this field. A third goal is to present some new results concerning the Chapman-Enskog expansion. These include an extension to weakly frictional granular gases, and a study of a stationary granular gas subject to a heat flux (an earlier study describing a stationary sheared granular gas is presented as well). In addition, a computer-aided method for inverting the linearized Boltzmann operator, which should be useful for kinetic theorists, is presented in an appendix. A further goal is to briefly discuss possible extensions beyond Boltzmann kinetics, as well as beyond hydrodynamics, i.e., the moderately dense regime on one hand, and the Knudsen regime on the other hand. An attempt is made throughout this paper to put granular gases in a general context, by distinguishing between those features that are common to granular and molecular gases and those that are not. Also, stress is put on the physics (and methods) rather than phenomena. While this is not a review article, parts of it are intended for the uninitiated in the field.Theory of granular gases: some recent results and some open problems S2593