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

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

Andrés Santos
###### Abstract

In a recent paper [A. Santos, G. M. Kremer, and V. Garzó, Prog. Theor. Phys. Suppl. 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.

Granular gases, Rough spheres, Free cooling state
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45.70.Mg, 05.20.Dd, 51.10.+y
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8x11single

## 1 Introduction

The simplest model of a granular fluid consists of a system of identical, inelastic smooth hard spheres with a constant coefficient of normal restitution [1]. Obviously, the model can be made closer to reality by introducing extra ingredients. In particular, polydispersity and roughness are especially relevant because they unveil an inherent breakdown of energy equipartition in granular fluids, even in homogeneous and isotropic states [2, 3, 4, 5, 6, 7].

In this work a fluid mixture of inelastic rough hard spheres characterized by mutual coefficients of normal restitution and tangential restitution is considered. First, a recent derivation by kinetic-theory arguments [8] of the collisional energy production rates ( and ) associated with the translational and rotational temperatures ( and ) is recalled and the energy production rates are decomposed into equipartition and cooling rates. Next, the results are applied to a binary mixture and the asymptotic long-time values and the time evolution of the three independent temperature ratios , , and and are analyzed in the homogeneous and isotropic free cooling state Special attention is paid to a paradoxical effect: the asymptotic long-time values of the temperature ratio in the nearly-smooth limit () differ from those obtained for purely smooth spheres ().

## 2 Collisional energy production rates

Let us consider a multi-component granular gas made of hard spheres of masses , diameters , and moments of inertia . The value of the dimensionless parameter depends on the mass distribution within the sphere and runs from the extreme values (mass concentrated on the center) to (mass concentrated on the surface). Collisions between a sphere of component and a sphere of component are characterized by a coefficient of normal restitution and a coefficient of tangential restitution . The former coefficient ranges from (perfectly inelastic particles) to (perfectly elastic particles), while the latter runs from (perfectly smooth particles) to (perfectly rough particles). The total (translational plus rotational) kinetic energy is dissipated upon collisions unless and . The particular case of one-component systems has been widely studied in the literature (see, for instance, Refs. [5, 6, 7]).

Let and be the one-body and two-body distribution functions, respectively, where is the velocity of the center of mass and is the angular velocity. By starting from the Liouville equation and following standard steps, one can derive the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy. The first equation of the hierarchy reads [8]

 ∂tfi(vi,ωi)+vi⋅∇fi(vi,ωi)=∑jJij[vi,ωi|fij], (1)
 Jij[vi,ωi|fij]=σ2ij∫dvj∫dωj∫dˆσΘ(vij⋅ˆσ)(vij⋅ˆσ)[1α2ijβ2ijfij(ri,v′′i,ω′′i;r−i,v′′j,ω′′j)−fij(ri,vi,ωi;r+i,vj,ωj)]. (2)

Here, is the collision operator, , , , and the double primes denote pre-collisional quantities giving rise to unprimed quantities as post-collisional values. The explicit form of the restituting collision rules can be found in Ref. [8].

Given an arbitrary one-body function , we define its average as

 ⟨ψi(vi,ωi)⟩≡1ni∫dvi∫dωiψi(vi,ωi)fi(vi,ωi),ni=∫dvi∫dωifi(vi,ωi), (3)

where is the number density of component . The rate of change of the quantity due to collisions with particles of component is

 Jij[ψi(vi,ωi)]≡1ni∫dvi∫dωiψi(vi,ωi)Jij[vi,ωi|fij]. (4)

In particular, the partial temperatures associated with the translational and rotational degrees of freedom are

 Ttri=mi3⟨(vi−u)2⟩,Troti=Ii3⟨ω2i⟩, (5)

where is the flow velocity. The corresponding collisional rates of change define the (partial) collisional energy production rates as

 ξtri=∑jξtrij,ξroti=∑jξrotij,ξtrij≡−mi3TtriJij[(vi−u)2],ξrotij≡−Ii3TrotiJij[ω2i]. (6)

The global temperature of the gas and its associated net cooling rate are

 T=∑ini2n(Ttri+Troti),ζ=∑i,jni2nT(Ttriξ% trij+Trotiξrotij), (7)

where is the total number density. Note that, as usually done in the literature on granular gases, the Boltzmann constant has been absorbed in the definitions of , , and , so that these quantities have dimensions of energy.

The energy production rates and defined in Eq. (6) do not have a definite sign. They can be decomposed into two classes of terms: equipartition rates and cooling rates (see Fig. 1). The equipartition terms, which exist even when energy is conserved by collisions ( and ), tend to make temperatures equal [10]. Therefore, they can be positive or negative depending on the signs of the differences , , and . On the other hand, the cooling terms reflect the collisional energy dissipation and thus they are positive if and/or , vanishing otherwise. Only the cooling terms in and contribute to the net cooling rate .

The quantities are functionals of the two-body distribution function and therefore they are generally rather intricate. Now, let us imagine that, instead of the full knowledge of , we only have at our disposal the local values of the two densities ( and ) and the four partial temperatures (, , , and ). Then, we can get reasonable estimates of from the replacement

 fij(vi,ωi;vj,ωj)→ninjχij⎛⎜⎝mimj4π2TtriT% trj⎞⎟⎠3/2exp⎡⎢⎣−mi(vi−u)22Ttri−mj(vj−u)22Ttrj⎤⎥⎦froti(ωi)frotj(ωj), (8)

where is the contact value of the pair correlation function. Equation (8) can be justified by maximum-entropy arguments, except that here a Maxwellian form for the rotational probability densities and is not needed. The expressions for and derived from the replacement (8) and the assumption can be found in Ref. [8]. Here, those expressions are recast into sums of equipartition and cooling rates as follows:

 ξtrij=1+αij2ξ(1)ij+1+βij2⎡⎣ξ(2)ij+κiTrotiTtriξ(3)ij⎤⎦+ζtr,1ij+κiT% rotiTtriζrotij,ξrotij=1+βij2ξ(3)ij+ζrotij, (9)
 ξ(1)ij=103νijmimj(mi+mj)2Ttri−TtrjTtri,ξ(2)ij=103νijmjmi+mjκij1+κijTtri−TrotiTtri, (10)
 ξ(3)ij=103νijTrotimimjκiκj(mi+mj)2(κij1+κij)2[Troti−Trotj+κj(Ttri−Ttrj)+κj(1+mj/mi)(Troti−Ttri)], (11)
 ζtr,1ij=56νij1−α2ijTtrimim2j(mi+mj)2⎛⎜⎝Ttr% imi+Ttrjmj⎞⎟⎠, (12)
 ζrotij=56νij1−β2ijTrotimim2jκi(mi+mj)2(κij1+κij)2⎛⎜⎝Ttrimi+Ttrjmj+Trotimiκi+Trotjmjκj⎞⎟⎠. (13)

In these equations, ,

 νij≡8√2π5χijnjσ2ij ⎷Ttrimi+Ttrjmj (14)

is an effective collision frequency, and we have assumed . The quantities represent equipartition rates. They do not have a definite sign and vanish if all the temperatures are equal. The equipartition rate is always present (even for perfectly elastic spheres, ) and tends to equilibrate both translational temperatures. The rates and do not contribute in the case of smooth spheres (). The former tends to equilibrate the translational and rotational temperatures of component , while the latter tends to equilibrate both rotational temperatures but is also affected by the other temperature differences. The quantities and , on the other hand, are positive definite and represent cooling rates. The former only vanishes if the spheres are elastic, whilst the latter only vanishes if the spheres are either perfectly smooth () or perfectly rough (). It is straightforward to check that and . Therefore, as expected, the equipartition rates do not contribute to the net cooling rate. The latter is given by

 ζ = ∑ijni2nT[Ttriζtr,1ij+(1+κi)Trotiζrotij] (15) = 524∑ijninTνijmimjmi+mj⎡⎢⎣(1−α2ij)⎛⎜⎝Ttrimi+Ttrjmj⎞⎟⎠+κij1+κij(1−β2ij)⎛⎜⎝Ttrimi+Ttrjmj+Trotimiκi+Trotjmjκj⎞⎟⎠⎤⎥⎦.

## 3 Homogeneous free cooling state

In the homogeneous and isotropic free cooling state the evolution equations for the total and partial temperatures are

 ∂tT=−ζT,∂tlnTtriT=−(ξtri−ζ),∂tlnTrotiT=−(ξroti−ζ). (16)

The first equation describes the monotonic decrease of temperature (unless and for all the pairs). The remaining equations show that the temperature ratios and decrease or increase, depending on the sign of the differences and , respectively. After a certain transient stage, those temperature ratios eventually reach constant values, while the total temperature keeps decreasing following Haff’s law . In this asymptotic regime one has equal production rates, i.e., , where is the number of components. Use of expressions (9)–(13) allows one to get an algebraic set of equations whose solution gives the asymptotic long-time values of the temperature ratios.

In the particular case of a binary mixture (), it is convenient to choose the three relevant temperature ratios as , , and . The parameter space is twelve-dimensional: the three coefficients of normal restitution (, , ), the three coefficients of tangential restitution (, , ), the two moment-of-inertia parameters ( and ), the mass ratio (), the size ratio (), the concentration ratio () and the total packing fraction .

For the sake of illustration, here we particularize to an equimolar mixture where all the spheres are uniformly solid and are made of the same material, the size of the spheres of one component being twice that of the other component. More specifically, , , , , , and . Moreover, we consider a dilute granular gas (), so that . Thus only the parameters and remain free.

In Ref. [8] it was observed that the asymptotic value of the translational/translational temperature ratio exhibits a peculiar behavior in the smooth-sphere limit : it tends to a finite value different from (in fact higher than) the value directly obtained in the case of perfectly smooth spheres [2]. Thus, a tiny amount of roughness has dramatic effects on the temperature ratio , producing an enhancement of non-equipartition. This singular behavior of the case in the free cooling state is further elaborated in this paper.

The physical origin of the singularity is the following one. If one strictly has , the rotational degrees of freedom are “frozen”, so that the rotational temperatures are finite while the translational temperatures keep decreasing. The rotational/translational temperature ratios diverge but, since the spheres are perfectly smooth, there is no mechanism transferring energy from the rotational to the translational degrees of freedom. This situation is represented by the left diagram of Fig. 2. On the other hand, if , where , the huge rotational/translational temperature ratios are eventually able to “feed” the weak energy channels connecting the rotational and translational temperatures (see the right diagram of Fig. 2), thus producing a non-negligible effect (“ghost” effect) on the ratio .

Let us see this in more detail. In the limit one can see that the rotational/translational temperature ratios diverge as and . It can be found that the energy production rates behave as , , , and , where are explicit functions of the indicated arguments. Since both rotational production rates vanish in the limit , the condition of equal production rates implies that , , and ). The first two conditions yield and . Once obtained, the third condition gives . In the pure smooth-sphere problem, only the translational degrees of freedom matter and therefore the temperature ratio is obtained from the condition with , which becomes . Obviously, the obtained solution differs from that given by .

In the preceding discussion we have implicitly assumed that the spheres are inelastic (). In the perfectly elastic case (), the scalings for the temperature ratios are quite different and turn out to be , , and . The production rates behave as , , , and , where are linear functions. The coefficient is simply obtained from . Once is known, and follow from and , respectively.

Figure 3 shows the quantities , , and as functions of for several values of , including the limit . It is clearly observed that the translational/translational temperature ratio in the limit differs from the values obtained in the perfectly smooth case [2]. In the latter case, moreover, the rotational temperatures and maintain their initial values and thus the scaled ratio vanishes, while is indetermined.

Since the scaled energy production rates go to zero in the quasi-smooth limit , it is reasonable to wonder whether the paradoxical singular effect, , is apparent only for times much longer than the characteristic relaxation time of . In order to investigate this possible scenario, the set of three coupled equations , , and has been numerically solved for several values of , the solution being compared with that of at . Here, is a measure of time in units of the accumulated number of 1–1 collisions. Figure 4 shows the time evolution of the three temperature ratios for and some representative values of . We can observe that in the perfectly smooth case () the translational/translational temperature ratio monotonically increases until reaching a plateau value at , the rotational/rotational ratio keeps its initial value, and the rotational/translational ratio increases exponentially with (Haff’s law). The important feature in the cases is that, while the relaxation time needed to reach the asymptotic long-time values actually increases as decreases, the curves for and markedly differ from the case of perfectly smooth spheres just after . Therefore, the dramatic difference between the quasi-smooth limit () and the perfectly smooth case () manifests itself not only in the asymptotic long-time regime but also in the transient regime for times comparable to the relaxation time of perfectly smooth spheres.

## 4 Conclusion

In this paper the homogeneous free cooling state of a binary granular gas made of inelastic rough hard spheres has been analyzed. Special attention has been paid to the quasi-smooth limit (), where an interesting singular phenomenon appears. It turns out that a vanishingly small amount of roughness has a significant effect on the translational/translational temperature ratio, with respect to the strict perfectly smooth case (). This paradoxical (“ghost”) phenomenon is essentially due to the fact that, even if is close to , the transfer of energy from the rotational to the translational degrees of freedom eventually becomes activated when the rotational temperatures are much larger than the translational.

It is important to remark that this singular behavior of the smooth case is closely tied to the non-stationary character of the free cooling state and thus disappears in homogeneous steady states. For instance, in a granular gas driven by a white-noise thermostat, the stationarity conditions are and . It is straightforward to check that in this case the ratio in the limit coincides with that of [4].

###### Acknowledgements.
Insightful discussions with G. M. Kremer, V. Garzó, and J. W. Dufty are gratefully acknowledged. This work has been supported by the Ministerio de Educación y Ciencia (Spain) through Grant No. FIS2007–60977 (partially financed by FEDER funds) and by the Junta de Extremadura (Spain) through Grant No. GR10158.

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