Interplay between polydispersity, inelasticity, and roughness in the freely cooling regime of hard-disk granular gases

Abstract

A polydisperse granular gas made of inelastic and rough hard disks is considered. Focus is laid on the kinetic-theory derivation of the partial energy production rates and the total cooling rate as functions of the partial densities and temperatures (both translational and rotational) and of the parameters of the mixture (masses, diameters, moments of inertia, and mutual coefficients of normal and tangential restitution). The results are applied to the homogeneous cooling state of the system and the associated nonequipartition of energy among the different components and degrees of freedom. It is found that disks typically present a stronger rotational-translational nonequipartition but a weaker component-component nonequipartition than spheres. A noteworthy "mimicry" effect is unveiled, according to which a polydisperse gas of disks having common values of the coefficient of restitution and of the reduced moment of inertia can be made indistinguishable from a monodisperse gas in what concerns the degree of rotational-translational energy nonequipartition. This effect requires the mass of a disk of component i to be approximately proportional to 2σ_{i}+〈σ〉, where σ_{i} is the diameter of the disk and 〈σ〉 is the mean diameter.

Intruders in disguise: Mimicry effect in granular gases

“…17,20 We can think of this phenomenon by imagining that a number of intruder spheres are added to a one-component granular gas and their partial temperatures mimic the corresponding values of the host gas. 42,43 Our results show that in fact there are regions in the parameter space of an scomponent system displaying this mimicry effect. More specifically, for given values of the s − 1 concentration parameters and the s − 1 diameter ratios, there are s − 1 conditions whose solution gives the s− 1 mass ratios such that partial equipartition (in the sense described before) exists.…”

Driven and undriven states of multicomponent granular gases of inelastic and rough hard disks or spheres

“…where ζ Ω i j are spin production rates, and ξ tr i j and ξ rot i j are energy production rates. While the exact determination of those quantities is not possible, a kinetic-theory approach (namely the Boltzmann equation) supplemented by a multitemperature Maxwellian approximation [38,47,48] allows one to express them in terms of the partial densities (n i , n j ), temperatures (T tr i , T rot i , T tr j , T rot j ), mean angular velocities (Ω Ω Ω i , Ω Ω Ω j ), and the mechanical parameters. The unified expressions for disks (d tr = 2, d rot = 1) and spheres (d tr = d rot = 3) are [38] ζ…”

“…In a recent work [38], we have presented a unified kinetictheory derivation (in terms of the number of degrees of freedom d tr and d rot ) of the collisional rates of energy production in multicomponent granular gases, so that previous results for disks [47] and spheres [48] are obtained by taking (d tr , d rot ) = (2, 1) and (d tr , d rot ) = (3, 3), respectively. Those unified expressions will be applied here to study the granular temperature ratios in monodisperse and bidisperse gases of rough disks or spheres in homogeneous states, both undriven and driven.…”

Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. I. Transport coefficients

“…In the HS case, the translational velocity v = v x i+v y j+v z k and the angular velocity ω = ω x i + ω y j+ ω z k have d t = 3 and d r = 3 nontrivial components, respectively. On the other hand, in the HD case, v = v x i + v y j and ω = ω z k have d t = 2 and d r = 1 nontrivial components, respectively, what simplifies the collision rules [50]. An important consequence of the distinction between spheres and disks is that the Jacobian of the transformation between pre-and postcollisional velocities turns out to depend on d r , namely…”

“…The results for HS and HD gases are given in the fourth column of Table II. In the case of λ, P N (κ 39,24,2,11,12,21) and (37,151,50,12,75,101,102) for HD and HS, respectively. It must be noted that, when setting α = β = 1 in the expressions of Table I, we took the licence of using d r = 1 2 d t (d t − 1) (which applies to both HS and HD) in order to simplify the final results for η * and λ * .…”