Shear Flows Of Inelastic Spheres Research Articles

Dynamics of dense sheared granular flows. Part II. The relative velocity distributions

The distribution of relative velocities between colliding particles in shear flows of inelastic spheres is analysed in the volume fraction range 0.4–0.64. Particle interactions are considered to be due to instantaneous binary collisions, and the collision model has a normal coefficient of restitution en (negative of the ratio of the post- and pre-collisional relative velocities of the particles along the line joining the centres) and a tangential coefficient of restitution et (negative of the ratio of post- and pre-collisional velocities perpendicular to line joining the centres).The distribution of pre-collisional normal relative velocities (along the line joining the centres of the particles) is found to be an exponential distribution for particles with low normal coefficient of restitution in the range 0.6–0.7. This is in contrast to the Gaussian distribution for the normal relative velocity in an elastic fluid in the absence of shear. A composite distribution function, which consists of an exponential and a Gaussian component, is proposed to span the range of inelasticities considered here. In the case of rough particles, the relative velocity tangential to the surfaces at contact is also evaluated, and it is found to be close to a Gaussian distribution even for highly inelastic particles.Empirical relations are formulated for the relative velocity distribution. These are used to calculate the collisional contributions to the pressure, shear stress and the energy dissipation rate in a shear flow. The results of the calculation were found to be in quantitative agreement with simulation results, even for low coefficients of restitution for which the predictions obtained using the Enskog approximation are in error by an order of magnitude. The results are also applied to the flow down an inclined plane, to predict the angle of repose and the variation of the volume fraction with angle of inclination. These results are also found to be in quantitative agreement with previous simulations.

Dynamics of dense sheared granular flows. Part 1. Structure and diffusion

Shear flows of inelastic spheres in three dimensions in the volume fraction range 0.4–0.64 are analysed using event-driven simulations. Particle interactions are considered to be due to instantaneous binary collisions, and the collision model has a normal coefficient of restitution en (negative of the ratio of the post- and pre-collisional relative velocities of the particles along the line joining the centres) and a tangential coefficient of restitution et (negative of the ratio of post- and pre-collisional velocities perpendicular to the line joining the centres). Here, we have considered both et = +1 and et = en (rough particles) and et = −1 (smooth particles), and the normal coefficient of restitution en was varied in the range 0.6–0.98. Care was taken to avoid inelastic collapse and ensure there are no particle overlaps during the simulation. First, we studied the ordering in the system by examining the icosahedral order parameter Q6 in three dimensions and the planar order parameter q6 in the plane perpendicular to the gradient direction. It was found that for shear flows of sufficiently large size, the system continues to be in the random state, with Q6 and q6 close to 0, even for volume fractions between φ = 0.5 and φ = 0.6; in contrast, for a system of elastic particles in the absence of shear, the system orders (crystallizes) at φ = 0.49. This indicates that the shear flow prevents ordering in a system of sufficiently large size. In a shear flow of inelastic particles, the strain rate and the temperature are related through the energy balance equation, and all time scales can be non-dimensionalized by the inverse of the strain rate. Therefore, the dynamics of the system are determined only by the volume fraction and the coefficients of restitution. The variation of the collision frequency with volume fraction and coefficient of restitution was examined. It was found, by plotting the inverse of the collision frequency as a function of volume fraction, that the collision frequency at constant strain rate diverges at a volume fraction φad (volume fraction for arrested dynamics) which is lower than the random close-packing volume fraction 0.64 in the absence of shear. The volume fraction φad decreases as the coefficient of restitution is decreased from en = 1; φad has a minimum of about 0.585 for coefficient of restitution en in the range 0.6–0.8 for rough particles and is slightly larger for smooth particles. It is found that the dissipation rate and all components of the stress diverge proportional to the collision frequency in the close-packing limit. The qualitative behaviour of the increase in the stress and dissipation rate are well captured by results derived from kinetic theory, but the quantitative agreement is lacking even if the collision frequency obtained from simulations is used to calculate the pair correlation function used in the theory.

Hydrodynamic modes of a sheared granular flow from the Boltzmann and Navier–Stokes equations

The initial growth rates for the hydrodynamic modes of the shear flow of a three-dimensional collection of inelastic spheres is analyzed using two models. The first is the generalized Navier–Stokes equations, derived for the shear flow of inelastic spheres using the Chapman–Enskog procedure, where the energy equation has an additional dissipation term due to inelastic collisions. The second is the solution of the linearized Boltzmann equation, where the distribution function in the base state is determined using a Hermite polynomial expansion in the velocity moments. For perturbations with variations in the velocity and gradient directions, it is found that the solutions obtained by two procedures are qualitatively similar, though there are quantitative differences. For perturbations with variations in the vorticity direction, it is found that there are qualitative differences in the predictions for the initial growth rate of the perturbations.

Collisional source of the second moment and pressure tensor for inhomogeneous rapid granular flows of highly inelastic spheres

By extending the constitutive theories for homogeneous granular flows of highly inelastic spheres by Richman (J. Rheol 33 (1989) 1293), Chou (J. CSME 16-6 (1995) 577), and Chou and Richman (Physica A 259 (1998) 430), the collisional source of the second moment of fluctuation velocity and pressure tensor were developed in this study for inhomogeneous rapid granular flows of identical smooth highly inelastic spheres. The important mean fields in this flow are the solid fraction, mean velocity, and full second moment of fluctuation velocity. The collisional source of second moment and the collisional flux of momentum are based upon an anisotropic Maxwellian velocity distribution function. The constitutive theory was combined with the experimental results measured by Hsiau and Jang (Exp. Thermal Fluid Sci. 17 (1998) 202) so as to determine the profiles of pressure tensor and collision source of second moment in the inhomogeneous rapid granular shear flows of inelastic spheres. The normal pressure discrepancies were also observed.