Homogeneous Free Cooling State in Binary Granular Fluids of Inelastic Rough Hard Spheres

In a recent paper [A. Santos, G. M. Kremer, and V. Garz\'o, \emph{Prog. Theor. Phys. Suppl.} \textbf{184}, 31-48 (2010)] the collisional energy production rates associated with the translational and rotational granular temperatures in a granular fluid mixture of inelastic rough hard spheres have been derived. In the present paper the energy production rates are explicitly decomposed into equipartition rates (tending to make all the temperatures equal) plus genuine cooling rates (reflecting the collisional dissipation of energy). Next the homogeneous free cooling state of a binary mixture is analyzed, with special emphasis on the quasi-smooth limit. A previously reported singular behavior (according to which a vanishingly small amount of roughness has a finite effect, with respect to the perfectly smooth case, on the asymptotic long-time translational/translational temperature ratio) is further elaborated. Moreover, the study of the time evolution of the temperature ratios shows that this dramatic influence of roughness already appears in the transient regime for times comparable to the relaxation time of perfectly smooth spheres.Comment: 6 pages; 4 figures; contributed talk at the 27th International Symposium on Rarefied Gas Dynamics (Asilomar Conference Grounds, Pacific Grove, California July 10-15, 2010

A Bhatnagar-Gross-Krook-like Model Kinetic Equation for a Granular Gas of Inelastic Rough Hard Spheres

The Boltzmann collision operator for a dilute granular gas of inelastic rough hard spheres is much more intricate than its counterpart for inelastic smooth spheres. Now the one-body distribution function depends not only on the translational velocity of the center of mass but also on the angular velocity of the particle. Moreover, the collision rules couple both velocities, involving not only the coefficient of normal restitution but also the coefficient of tangential restitution. The aim of this paper is to propose an extension to inelastic rough particles of a Bhatnagar-Gross-Krook-like kinetic model previously proposed for inelastic smooth particles. The Boltzmann collision operator is replaced by the sum of three terms representing: (i) the relaxation to a two-temperature local equilibrium distribution, (ii) the action of a nonconservative drag force proportional to the peculiar velocity, and (iii) the action of a nonconservative torque equal to a linear combination of the angular velocity and its mean value. The three coefficients in the force and torque are fixed to reproduce the Boltzmann collisional rates of change of the mean angular velocity and of the two granular temperatures (translational and rotational). A simpler version of the model is also constructed in the form of two coupled kinetic equations for the translational and rotational velocity distributions. The kinetic model is applied to the simple shear flow steady state and the combined influence of the two coefficients of restitution on the shear and normal stresses and on the translational velocity distribution function is analyzed.Comment: 8 pages; 3 figures; invited talk at the 27th International Symposium on Rarefied Gas Dynamics (Asilomar Conference Grounds, Pacific Grove, California July 10-15, 2010

Velocity Distribution and Cumulants in the Unsteady Uniform Longitudinal Flow of a Granular Gas

The uniform longitudinal flow is characterized by a linear longitudinal velocity field $u_x(x,t)=a(t)x$, where $a(t)={a_0}/({1+a_0t})$ is the strain rate, a uniform density $n(t)\propto a(t)$, and a uniform granular temperature $T(t)$. Direct simulation Monte Carlo solutions of the Boltzmann equation for inelastic hard spheres are presented for three (one positive and two negative) representative values of the initial strain rate $a_0$. Starting from different initial conditions, the temporal evolution of the reduced strain rate $a^*\propto a_0/\sqrt{T}$, the non-Newtonian viscosity, the second and third velocity cumulants, and three independent marginal distribution functions has been recorded. Elimination of time in favor of the reduced strain rate $a^*$ shows that, after a few collisions per particle, different initial states are attracted to common "hydrodynamic" curves. Strong deviations from Maxwellian properties are observed from the analysis of the cumulants and the marginal distributions.Comment: 8 pages; 4 figures; contributed paper at the 28th International Symposium on Rarefied Gas Dynamics (Zaragoza, Spain, July 9-13, 2012