Inelastic and frictional hard spheres as a model of granular gases
By: Andres Santos
From: University of Badajoz
At: Instituto de Investigação Interdisciplinar, Anfiteatro
[2012-02-28]
A granular fluid is usually modeled as a system of identical, inelastic smooth hard spheres with a constant coefficient of normal restitution. Despite its simplicity, this model has been useful to capture the basic properties of granular flows. On the other hand, the model can be made closer to reality by introducing more ingredients, such as coefficients of normal restitution depending on the impact velocity, presence of an interstitial fluid, non-spherical shapes, polydispersity, or roughness (or friction), to name just a few. Roughness is especially relevant, not only because friction is unavoidable in beads and grains, but also because this  ingredient unveils an inherent breakdown of energy equipartition in granular fluids, even in homogeneous and isotropic states. In this talk I will focus on roughness and consider a dilute granular gas of hard spheres colliding inelastically with constant coefficients of normal and tangential restitution. The basic quantities characterizing the distribution function  of linear  and angular  velocities are the second-degree moments defining the  translational  and rotational  temperatures. The deviation of the velocity distribution function from the Maxwellian  parameterized by both temperatures can be measured by the cumulants and velocity correlation functions associated with the fourth-degree moments. Two of those parameters measure the kurtosis of the translational and rotational distribution functions, while the translational-rotational correlations are measured by other two parameters. The collisional rates of change  of these second- and fourth-degree moments are evaluated by means of a Sonine approximation where the velocity distribution function is approximated by the two-temperature Maxwellian times a truncated polynomial expansion. The results are subsequently applied to  a paradigmatic state, namely the so-called homogeneous cooling state. It is found that the Maxwellian approximation for the temperature ratio does not deviate much from  the Sonine prediction. On the other hand, non-Maxwellian properties measured by the cumulants and velocity correlation functions cannot be ignored,  especially for medium and small roughness. Moreover, some quantities differ in the quasi-smooth limit  from those of pure smooth spheres. This singular behavior is directly related to the unsteady character of the homogeneous cooling state and thus it is absent in steady states.