Hydrodynamics of granular gases and granular gas mixtures

Abstract

It is shown that a vertically vibrated binary granular gas mixture of light and heavy particles can segregate (in the presence of gravity) in such a way that the bottom and top layers are composed mostly of light particles, even if all other parameters (including size) are the same for both species. The corresponding concentration profile possesses certain universal properties. It is also shown that such mixtures can segregate when the only difference between the species is the value of the coefficient of restitution. These findings follow from a set of hydrodynamic equations for granular gas mixtures which we derived from the pertinent Boltzmann equation. The above results comprise the second part of this article, the first part of which is devoted to a brief and somewhat biased review of the main physical properties of granular gases. This includes their (generic) tendency to coagulate into clusters and other micro- and macrostructures. A fundamental property of granular materials in general, and granular gases in particular, is the lack of scale separation; an explanation and some consequences are presented. The answer to the basic question of whether the dynamics of granular gases lends itself to description by (appropriate) hydrodynamic equations seems to be positive, though some restrictions apply.

Impurity in a granular gas under nonlinear Couette flow

“…In particular, if the spatial gradients present in the system are weak, the Navier-Stokes (NS) constitutive equations for the fluxes of mass, momentum, and energy have been derived (with explicit expressions for the transport coefficients) for the model of inelastic hard spheres characterized by constant coefficients of normal restitution α ij . Most of the early derivations were restricted to the quasielastic limit (α ij ≈ 1), thus assuming an expansion around Maxwellians at the same temperature [3,4,5,6,7,8]. However, the nonequipartition of energy becomes significant beyond the quasi-elastic limit, as confirmed by kinetic theory [9,10,11,12], computer simulations [11,13,14,15,16,17,18,19,20,21], and real experiments [21,22,23].…”

Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport

“…Another key difference between existing polydisperse theories is the base state used in the Chapman-Enskog (CE) expansion. Some theories [8,9,10,11,12,13,14,37,39] assume an expansion about a perfectly elastic (molecular equilibrium) base state, and thus are restricted to nearly-elastic systems. However, in the CE method the base state must not be chosen a priori, but rather it is determined as the solution to the kinetic equation to zeroth order in the gradient expansion.…”

Theory of Dilute Binary Granular Gas Mixtures

“…Theoretical studies, most of which focus on the monodisperse case, as well as numerical simulations, reveal that after the initial decrease in the value of the granular temperature as a function of height above the floor, the temperature profile possesses a minimum above which the temperature increases as a function of height [3,49,43]. Therefore thermophoresis would push large or massive intruders to this minimum rather than the top of the system.…”

“…The problem of a single intruder in a vibrated granular gas was treated in [4], and a comparison between theory, experiments and simulations for slightly inelastic binary mixtures was given in [50]. Recently, it was shown [43] that a dilute vertically vibrated binary mixture of particles of the same size with different masses arranges itself in a sandwich-like configuration in which a layer of heavy particles is trapped between two layer of light particles. This is a result of the competition between buoyancy and thermal diffusion: the former pushes the light particles to the top and the latter pushes the heavy particles to the region of minimal temperature, leaving room for the light particles (also) at the bottom of the system.…”

“…While in most of these studies the segregation mechanisms involved are directly or indirectly related to the inelastic nature of the granular constituents, few studies have concentrated on the direct influence of inelasticity alone on segregation, which would be the purest manifestation of the effect of inelasticity. In reference [43] it was shown that for near-elastic collisions particles with identical mass and size may segregate on the basis of differences in their inelastic properties alone, when subject to a temperature gradient. This was corroborated in molecular dynamic (MD) simulations of vertically vibrated granular gas mixtures [43,6].…”