# 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 to be approximately proportional to , where is the diameter of the disk and is the mean diameter.

## I Introduction

The minimal model to describe the dynamical properties of a granular fluid consists of a collection of identical, smooth hard disks (in two-dimensional geometry) or spheres (in the three-dimensional case). Particles dissipate kinetic energy via binary collisions and this is characterized in the minimal model by means of a constant coefficient of normal restitution. While this simple model captures most of the basic properties of granular flows Dufty (2000); Ottino and Khakhar (2000); Pöschel and Luding (2001); Goldhirsch (2003); Kudrolli (2004); Brilliantov and Pöschel (2004); Aranson and Tsimring (2006); Rao and Nott (2008), it can be made more realistic, for instance, by assuming that the coefficient of normal restitution depends on the impact velocity Brilliantov and Pöschel (2004); Brilliantov et al. (2004); Duan and Feng (2017), taking into account the presence of an interstitial fluid Xu et al. (2009), considering non-spherical particles Hidalgo et al. (2009), introducing the effect of surface friction in collisions, or accounting for a multicomponent character of the granular fluid.

In particular, there exists a vast literature about polydisperse systems of smooth disks or spheres Jenkins and Mancini (1989); Garzó and Dufty (1999); Hong et al. (2001); Jenkins and Yoon (2002); Montanero and Garzó (2002); Barrat and Trizac (2002); Dahl et al. (2002); Garzó and Dufty (2002); Brey et al. (2005); Serero et al. (2006); Garzó et al. (2007a, b); Garzó (2008a, b); Uecker et al. (2009), as well as about friction (or roughness) in monodisperse systems Jenkins and Richman (1985); Lun and Savage (1987); Campbell (1989); Lun (1991); Lun and Bent (1994); Goldshtein and Shapiro (1995); Luding (1995); Lun (1996); Zamankhan et al. (1998); Huthmann and Zippelius (1997); McNamara and Luding (1998); Luding et al. (1998); Herbst et al. (2000); Aspelmeier et al. (2001); Mitarai et al. (2002); Cafiero et al. (2002); Jenkins and Zhang (2002); Polashenski et al. (2002); Moon et al. (2004); Herbst et al. (2005); Goldhirsch et al. (2005); Zippelius (2006); Brilliantov et al. (2007); Gayen and Alam (2008); Kranz et al. (2009); Kremer (2010); Santos (2011a); Santos et al. (2011); Santos and Kremer (2012); Mitrano et al. (2013); Vega Reyes et al. (2014a, b); Kremer et al. (2014); Rongali and Alam (2014); Vega Reyes and Santos (2015); Fullmer and Hrenya (2017); Scholz and Pöschel (2017); Duan and Feng (2017); Garzó et al. (2018). On the other hand, much fewer works have dealt with multicomponent gases of rough spheres Viot and Talbot (2004); Piasecki et al. (2007); Cornu and Piasecki (2008); Santos et al. (2010); Santos (2011b); Vega Reyes et al. (2017a, b). This class of systems is especially relevant because of an inherent breakdown of energy equipartition, even in homogeneous and isotropic states (driven or undriven), as characterized by independent translational () and rotational () temperatures associated with each component . The rate of change of the translational mean kinetic energy of particles of component due to collisions with particles of component defines the energy production rate . A similar energy production rate measures the rate of change of the rotational mean kinetic energy.

By means of kinetic-theory tools, the energy production rates and for (three-dimensional) hard spheres were obtained in Ref. Santos et al. (2010) as functions of , , , , and of the mechanical parameters (masses, diameters, moments of inertia, and coefficients of normal and tangential restitution) of each pair . Those expressions were derived by assuming collisional molecular chaos, statistical independence between the translational and angular velocities, and a Maxwellian form for the translational velocity distribution function. The application of the results to the homogeneous cooling state (HCS) of a tracer particle immersed in a granular gas of inelastic and rough hard spheres shows a very good agreement with computer simulations Vega Reyes et al. (2017a, b).

From the experimental point of view, however, most of the setup geometries are two-dimensional Olafsen and Urbach (1998); Rouyer and Menon (2000); Feitosa and Menon (2002); Schmick and Markus (2008); Daniels et al. (2009); Grasselli et al. (2009); Tatsumi et al. (2009); Nichol and Daniels (2012); Altshuler et al. (2013); Grasselli et al. (2015); Scholz et al. (2016); Scholz and Pöschel (2017). Moreover, while capturing most of the physics of the problems at hand, two-dimensional computer simulations are much easier to carry out and interpret than three-dimensional ones. Hence, the extension of the analysis carried out in Ref. Santos et al. (2010) to multicomponent hard disks has undoubtedly a practical interest beyond its added academic value. In contrast to what happens for smooth, spinless particles, where an unambiguous kinetic-theory treatment of -dimensional hard spheres is possible Vega Reyes et al. (2007), the existence of angular motion due to surface friction or roughness establishes a neat separation between the cases of spheres and disks. Whereas both classes of particles are embedded in a common three-dimensional space, spinning spheres have three translational plus three rotational degrees of freedom, but spinning disks on a plane have two translational and only one rotational degrees of freedom.

By following steps similar to those followed in Ref. Santos et al. (2010), the energy production rates and are derived in this paper for a multicomponent gas made of inelastic and rough disks. The results are subsequently applied to the HCS and illustrated for monodisperse and bidisperse gases. An interesting mimicry effect is also analyzed. According to this effect, the HCS of a polydisperse gas of disks having common values of the coefficient of restitution and of the reduced moment of inertia can be indistinguishable from that of a monodisperse gas in what concerns the rotational-translational temperature ratio. It is shown here that the condition for this mimicry effect is that the mass of each component must be approximately proportional to , where is the diameter of a disk of component and is the mean diameter.

The organization of this paper is as follows. Section II describes the collision rules, which are then used in Sec. III to express the collisional rates of change in terms of two-body averages. Next, those averages are estimated by assuming molecular chaos, statistical independence between the translational and angular velocities, and a Maxwellian translational velocity distribution function. The energy production rates and are defined in Sec. IV, their explicit expressions being displayed in Table 3. Those results are applied to the HCS of monodisperse and bidisperse systems in Sec. V. Section VI deals with the mimicry effect described above. Finally, the paper ends with some concluding remarks in Sec. VII.

## Ii Binary collisions. Coefficients of restitution

### ii.1 Collisional rules

Let us consider an -component granular gas of hard disks (lying on the plane). Disks of component have a mass , a diameters , and a moment of inertia , where the value of the dimensionless quantity depends on the mass distribution within the disk, running from the extreme values (mass concentrated on the center) to (mass concentrated on the perimeter). If the mass is uniformly distributed, then .

Figure 1 sketches a binary collision between two disks of components and . Let us denote by the pre-collisional relative velocity of the center of mass of both disks, by and the respective pre-collisional angular velocities, by the unit vector pointing from the center of to the center of , and by its perpendicular unit vector. The velocities of the points of the disks which are in contact at the collision are

(1) |

so that the corresponding relative velocity is

(2) |

Post-collisional velocities will be denoted by primes. The conservation of linear and angular momenta yields

(3a) | |||

(3b) | |||

(3c) |

Angular momentum (with respect to the point of contact) is conserved for each particle separately because during a collision the forces act only at the point of contact and hence there is no torque with respect to that point Zippelius (2006). Equations (3) imply that

(4a) | |||

(4b) |

where the (so-far) undetermined quantity is the impulse exerted by particle on particle . Therefore, the post-collisional relative velocities are

(5a) | |||

(5b) |

where

(6) |

are the reduced mass and a sort of reduced inertia-moment parameter, respectively.

The collisional rules can be closed by relating the normal (i.e., parallel to ) and tangential (i.e., parallel to ) components of the relative velocities and :

(7) |

Here, and are the constant coefficients of normal and tangential restitution, respectively. While ranges from (perfectly inelastic particles) to (perfectly elastic particles), the coefficient runs from (perfectly smooth particles, i.e., no change in the tangential component of the relative velocity) to (perfectly rough particles, i.e., reversal of the tangential component). The insertion of Eq. (5b) into Eq. (7) yields

(8) |

with the introduction of the parameters

(9) |

Therefore, with the help of Eqs. (2) and (8), the impulse is expressed in terms of the pre-collisional velocities and the unit vector as

(10) |

This, together with Eqs. (4), closes the collision rules . Note that one has in the special case of perfectly smooth disks (), so that in that case and, according to Eq. (4b), the angular velocities of the two colliding disks are unaffected by the collision, as expected.

### ii.2 Energy dissipation

While linear and angular momenta are conserved by collisions, kinetic energy is not. Let us see this point in more detail. From Eqs. (4) and (10), it follows that the collisional changes of , , , and are

(11a) | ||||

(11b) | ||||

(11c) | ||||

(11d) |

Similar expressions are obtained for particle by exchanging , , and . The total kinetic energy before collision is

(12) |

Combining Eqs. (11c) and (11d), plus their counterparts for particle , one obtains

(13) |

The right-hand side is a negative definite quantity. Thus, energy is conserved only if the disks are elastic () and either perfectly smooth () or perfectly rough (). Otherwise, and kinetic energy is dissipated upon collisions.

### ii.3 Restituting collisions

By inverting the direct collisional rules given by Eq. (4) and (10), one can find the restituting collisional rules as

(14a) | |||

(14b) |

where

(15) |

Here, the double primes denote pre-collisional quantities giving rise to unprimed quantities as post-collisional values.

It is interesting to note that the modulus of the Jacobian of the transformation between pre- and post-collisional velocities is

(16) |

Interestingly, this differs from the case of spheres, for which the Jacobian is Santos et al. (2010).

## Iii Collisional rates of change

### iii.1 One- and two-body distribution functions

By starting from the Liouville equation, making use of the collisional rules, and following standard steps, one can derive the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy Brey et al. (1997), whose first equation reads

(17) |

where the short-hand notation has been introduced, is the two-body distribution function, and

(18) |

is the one-body distribution function, normalized as . Here, is the number of disks of component and . Finally, the collision operator is

(19) |

where , , and , being the Heaviside step function.

### iii.2 Balance equations

Given a one-body function , its average value is

(20) |

where is the number density of component and, for the sake of brevity, the spatial and temporal arguments have been omitted. In particular, one can define partial temperatures associated with the translational and rotational degrees of freedom of each component as

(21) |

where

(22) |

is the flow velocity. Note that in the definition of the angular velocities are not referred to any average value because of the lack of invariance of the collision rules under the addition of a common value to every angular velocity. Also, Eq. (21) takes into account that the number of translational and rotational degrees of freedom are and , respectively. The global temperature is

(23) |

where is the total number density.

In general, the balance equation for can be obtained by multiplying both sides of Eq. (17) by and integrating over :

(24) |

where the collisional integral is

(25) |

Therefore, is the rate of change of the quantity due to collisions with particles of component . This rate of change is a functional of the two-body distribution function , as indicated by the notation. The most basic cases are . The corresponding rates of change are obtained by inserting Eqs. (11) into Eq. (25). Note that so far all the results are formally exact.

### iii.3 Collisional integrals as two-body averages

To proceed, let us make the approximation

(26) |

where

(27) |

is the orientational average of the pre-collisional distribution . Equation (26) replaces the formally exact collisional integral (25) by a simpler one where the angular integral

(28) |

can be evaluated independently of . As a consequence,

(29) |

where

(30) |

is a two-body average.

It is important to bear in mind that the approximation (26) refers to pre-collisional quantities inside integrals over , , and . Thus, it is much weaker than the bare approximation . On the other hand, it must be pointed out that the equality holds if (i) the gas is in the Boltzmann limit (, ), in which case one can formally take in the contact value of , or (ii) the system is homogeneous and isotropic (regardless of the reduced densities and ), in which case only depends on . Thus, the approximation (26) is justified if the density of the granular gas and/or its heterogeneities are small enough to make the value of at contact hardly dependent on the relative orientation of the two colliding disks.

Let us now particularize to . The needed angular integrals are

(31a) | |||

(31b) | |||

(31c) |

were is an arbitrary unit vector and is its orthogonal unit vector. After some algebra, one can find the expressions displayed in Table 1, where is a vector orthogonal to .

### iii.4 Estimates of two-body averages

Table 1 expresses the collisional rates of change of the main quantities as linear combinations of two-body averages of the form (30). They are local functions of space and time and functionals of the orientation-averaged pre-collisional distribution . While, thanks to the approximation (26), the expressions in Table 1 are much more explicit than the formally exact results stemming from Eq. (25), they still require the full knowledge of .

Quantity | Expression |
---|---|

Suppose, for simplicity, that . Now, let us imagine that, instead of the full knowledge of , we only know the common flow velocity () and the two translational temperatures ( and ). One can resort to information-theory (i.e., maximum-entropy) arguments to make the approximation

(32) |

where is the contact value of the pair correlation function and

(33) |

is the marginal distribution function associated with the rotational degrees of freedom. Similarly, the translational marginal distribution function is

(34) |

Equation (32) is the least biased ansatz consistent with the input quantities , , and . It implies (a) molecular chaos (i.e., ), (b) statistical independence between the translational and angular velocities (i.e., ), and (c) a Maxwellian form for the distribution of translational velocities. The generalization to can be carried out following similar steps as done in Ref. Vega Reyes et al. (2007) for smooth spheres. Since the angular velocities only appear linearly or quadratically in Table 1, a Maxwellian form for does not need to be assumed, so that the local densities ( and ), the average angular velocities ( and ), and the rotational temperatures ( and ) do not appear explicitly in Eq. (32).

It must be stressed that, while small deviations from the three assumptions (a), (b), and (c) behind Eq. (32) have been documented in the literature Soto and Mareschal (2001); Soto et al. (2001); Brilliantov et al. (2007); Santos et al. (2011); Vega Reyes et al. (2014a), the expectation is that the two-body averages can be estimated reasonably well by performing the replacement (32). This expectation has been confirmed in the hard-sphere case Vega Reyes et al. (2014a, 2017a, 2017b).