Collision statistics in sheared inelastic hard spheres

The dynamics of sheared inelastic-hard-sphere systems is studied using nonequilibrium molecular-dynamics simulations and direct simulation Monte Carlo. In the molecular-dynamics simulations Lees-Edwards boundary conditions are used to impose the shear. The dimensions of the simulation box are chosen to ensure that the systems are homogeneous and that the shear is applied uniformly. Various system properties are monitored, including the one-particle velocity distribution, granular temperature, stress tensor, collision rates, and time between collisions. The one-particle velocity distribution is found to agree reasonably well with an anisotropic Gaussian distribution, with only a slight overpopulation of the high-velocity tails. The velocity distribution is strongly anisotropic, especially at lower densities and lower values of the coefficient of restitution, with the largest variance in the direction of shear. The density dependence of the compressibility factor of the sheared inelastic-hard-sphere system is quite similar to that of elastic-hard-sphere fluids. As the systems become more inelastic, the glancing collisions begin to dominate over more direct, head-on collisions. Examination of the distribution of the times between collisions indicates that the collisions experienced by the particles are strongly correlated in the highly inelastic systems. A comparison of the simulation data is made with direct Monte Carlo simulation of the Enskog equation. Results of the kinetic model of Montanero et al. [J. Fluid Mech. 389, 391 (1999)] based on the Enskog equation are also included. In general, good agreement is found for high-density, weakly inelastic systems.

Figure 1

The system size dependence of the mean free time between collisions, $t_{\text{avg}}$, for simulations of a sheared inelastic-hard-sphere system with $\alpha =0.4$ in a cubic box with sides of length $L_{\text{box}}$. The number of particles is $N=256$ in the smallest system, and $N=10976$ in the largest.

Figure 2

Variations in the mean kinetic energy per particle with density for (i) $\alpha =0.4$ (circles), (ii) $\alpha =0.5$ (squares), (iii) $\alpha =0.6$ (diamonds), (iv) $\alpha =0.7$ (up triangles), (v) $\alpha =0.8$ (left triangles), and (vi) $\alpha =0.9$ (down triangles). The dotted lines are the suggested expressions of Ref. [15] and the solid lines are DSMC results.

Figure 3

The ratios of the mean-square velocity (a) in the $y$ and $x$ directions and (b) in the $z$ and $x$ directions for sheared inelastic-hard-sphere systems with (i) $\alpha =0.4$ (circles), (ii) $\alpha =0.5$ (squares), (iii) $\alpha =0.6$ (diamonds), (iv) $\alpha =0.7$ (up triangles), (v) $\alpha =0.8$ (left triangles), and (vi) $\alpha =0.9$ (down triangles). The dotted lines are the suggested expressions of Ref. [15], and the solid lines are the DSMC results.

Figure 4

The peculiar velocity distributions in the (a) $x$ direction, (b) $y$ direction, and (c) $z$ direction in sheared inelastic-hard-sphere systems with (i) $\rho {\sigma }^3=0.4$ and $\alpha =0.4$ (circles), (ii) $\rho {\sigma }^3=0.9$ and $\alpha =0.4$ (squares), (iii) $\rho {\sigma }^3=0.4$ and $\alpha =0.9$ (diamonds), and (iv) $\rho {\sigma }^3=0.9$ and $\alpha =0.9$ (triangles). The solid line is a Gaussian distribution.

Figure 5

(a) The dimensionless pressures $p\sigma /m{\dot{\gamma }}^2$ and (b) the compressibility factors $Z$ for inelastic-hard-sphere systems with (i) $\alpha =0.4$ (circles), (ii) $\alpha =0.5$ (squares), (iii) $\alpha =0.6$ (diamonds), (iv) $\alpha =0.7$ (up triangles), (v) $\alpha =0.8$ (left triangles), and (vi) $\alpha =0.9$ (down triangles). The uncertainty is smaller than the symbol size. The dotted lines are the suggested expressions of Ref. [15], and the solid lines are DSMC results. The solid line in (b) is the Carnahan-Starling equation of state for elastic-hard-sphere fluids, and the dotted lines are DSMC results for various $\alpha$.

Figure 6

The viscosities for a homogeneously sheared inelastic-hard-sphere system with (i) $\alpha =0.4$ (circles), (ii) $\alpha =0.5$ (squares), (iii) $\alpha =0.6$ (diamonds), (iv) $\alpha =0.7$ (up triangles), (v) $\alpha =0.8$ (left triangles), and (vi) $\alpha =0.9$ (down triangles). The dotted lines are the suggested expressions of Ref. [15], and the solid lines are DSMC results.

Figure 7

Mean times between collisions for sheared inelastic hard spheres with (i) $\alpha =0.4$ (circles), (ii) $\alpha =0.5$ (squares), (iii) $\alpha =0.6$ (diamonds), (iv) $\alpha =0.7$ (up triangles), (v) $\alpha =0.8$ (left triangles), and (vi) $\alpha =0.9$ (down triangles). Inset: variations in the mean time between collisions with the coefficient of restitution for (i) $\rho {\sigma }^3=1.0$ (circles), (ii) $\rho {\sigma }^3=0.9$ (squares), (iii) $\rho {\sigma }^3=0.8$ (diamonds), (iv) $\rho {\sigma }^3=0.7$ (up triangles), (v) $\rho {\sigma }^3=0.6$ (left triangles), (vi) $\rho {\sigma }^3=0.5$ (down triangles), and (vii) $\rho {\sigma }^3=0.4$ (right triangles). The solid lines are the DSMC simulation results.

Figure 8

The distributions of collisional angles for (i) $\rho {\sigma }^3=0.4$ and $\alpha =0.4$ (circles), (ii) $\rho {\sigma }^3=0.8$ and $\alpha =0.4$ (left triangles), (ii) $\rho {\sigma }^3=0.9$ and $\alpha =0.4$ (squares), (iii) $\rho {\sigma }^3=0.4$ and $\alpha =0.9$ (diamonds), and (iv) $\rho {\sigma }^3=0.9$ and $\alpha =0.9$ (up triangles). The dashed line is for elastic hard spheres and the solid line is from a DSMC simulation of $\rho {\sigma }^3=0.9$ and $\alpha =0.4$.

Figure 9

The changes in kinetic energy on collision for sheared inelastic-hard-sphere systems with (i) $\rho {\sigma }^3=0.4$ and $\alpha =0.4$ (circles), (ii) $\rho {\sigma }^3=0.9$ and $\alpha =0.4$ (squares), (iii) $\rho {\sigma }^3=0.4$ and $\alpha =0.9$ (diamonds), and (iv) $\rho {\sigma }^3=0.9$ and $\alpha =0.9$ (triangles). The dashed line represents the kinetic-energy loss on collision if the velocity were given by the Maxwell-Boltzmann distribution, and the solid line is from a DSMC simulation at $\rho {\sigma }^3=0.9$ and $\alpha =0.4$. The inset highlights the frequency of collisions that result in low-energy losses.

Figure 10

Distributions of time between (a) one collision, (b) five collisions, and (c) ten collisions in sheared inelastic hard spheres with (i) $\rho {\sigma }^3=0.4$ and $\alpha =0.4$ (circles), (ii) $\rho {\sigma }^3=0.9$ and $\alpha =0.4$ (squares), (iii) $\rho {\sigma }^3=0.4$ and $\alpha =0.9$ (diamonds), and (iv) $\rho {\sigma }^3=0.9$ and $\alpha =0.9$ (triangles). The solid line is for a Poisson process.