Dynamics of inelastically colliding rough spheres: Relaxation of translational and rotational energy

Abstract

We study the exchange of kinetic energy between translational and rotational degrees of freedom for inelastic collisions of rough spheres. Even if equipartition holds in the initial state it is immediately destroyed by collisions. The simplest generalisation of the homogeneous cooling state allows for two temperatures, characterizing translational and rotational degrees of freedom separately. For times larger than a crossover frequency, which is determined by the Enskog frequency and the initial temperature, both energies decay algebraically like t −2 with a fixed ratio of amplitudes, different from one.Kinetic theory of inelastically colliding particles has become a subject of growing research activity, motivated partly by renewed interest in granular materials. A Boltzmann equation and the Enskog variant of it have been formulated for inelastically colliding particles with normal and tangential restitution [1][2][3][4][5][6]. Whereas the derivation of the kinetic equation is relatively straightforward, the methods of solution which were developped for elastic collisions cannot be taken over to the inelastic case because there is no simple stationary or local equilibrium distribution, around which one could expand the nonlinear kinetic equation. One simple distribution is the homogeneous cooling state (HCS) [7,8], which depends on time only implicitly via the average kinetic energy T (t). The latter is predicted to decay like t −2 for large times. The homogeneity assumption is certainly violated when clustering occurs and, in particular, if inelastic collapse happens. Nevertheless we have found recently [9] in a model of inelastically colliding rods that the kinetic energy follows on average a t −2 behaviour, even if clustering occurs. This suggests that the above scaling law may be a useful approximate description, even when the assumptions of HCS break down.In this note we show that rotational and translational energy relax with different rates in general. Once friction is included, HCS with a single time dependent temperature is no longer consistent with the time evolution of translational and rotational energy separately. Instead one has to introduce two temperatures which characterize translational and rotational degress of freedom separately. Both are found to fall off like t −2 with the ratio approaching a constant value, which is determined by the coefficients of normal and tangential restitution.We briefly recall the collision dynamics of hard spheres with normal and tangential restitution. These results can for example be found in Cerginani [6]. We consider two spheres of equal diameter a, mass M and moment of inertia I. The unit-vector from the center of the first sphere to the center of the second is denoted byn and velocities and angular velocities before collision by v 1 , v 2 , ω 1 and ω 2 . The relative velocity of the contactpoint before collision is given by V = v 2 + a 2n × ω 2 − v 1 + a 2n × ω 1 . Normal and tangential restitution determine the relative velocity after collision according ton...

Transport coefficients of a granular gas of inelastic rough hard spheres

“…The stationary solution (3.13) represents the value of the temperature ratio in the HCS [28,[31][32][33]38]. In such a state the whole time dependence of the distribution function only occurs through the granular temperature T (t), so that the Boltzmann equation (2.12a) becomes 17) where, according to Eq.…”

A driven two-dimensional granular gas with Coulomb friction

“…In the following we present different approximations for frictional particles, referred to as models A-E. Model A is the well known model using constant coefficients of normal and tangential restitution, cf., e.g., [5,10]. Model E implements Coulomb friction as introduced by Walton [17].…”

“…In the simplest model, one describes inelastic collisions by a normal restitution coefficient r only. However, surface roughness and friction are important [10,13,20,21,22,29], since they allow for an exchange of translational and rotational energy and influence the overall dissipation. In the standard approach [5,10,22], surface roughness is accounted for by a constant tangential restitution coefficient r t , which is defined in analogy to r in the tangential direction.…”

“…However, surface roughness and friction are important [10,13,20,21,22,29], since they allow for an exchange of translational and rotational energy and influence the overall dissipation. In the standard approach [5,10,22], surface roughness is accounted for by a constant tangential restitution coefficient r t , which is defined in analogy to r in the tangential direction. A more realistic friction law involves the Coulomb friction coefficient µ [17,30,31,32], so that the tangential restitution r t (γ) depends on the impact angle γ, i.e.…”

“…Many phenomenona are well reproduced by model granular media, where spheres are used instead of other, possibly more realistic shapes. In order to study such model systems, kinetic theories [4,5,6,7,8,9,10,11,12,13,14,15,16] and numerical simulations [4,15,17,18,19,20,21,22,23] have been applied for special boundary conditions and a variety of interesting experiments have been performed, see for example [24,25,26,27,28]. The dynamics of the system is usually assumed to be dominated by instantaneous two-particle collisions.…”