Velocity fluctuation, correlation and rheology in frictional granular shear flow

Abstract

The study of granular materials has received a recent upsurge of interest in physics. This has been motivated by both the relevance of such flows to a wide range of industrial and geological processes, and by the realization that granular materials provide an excellent test bed for a number of fundamental questions in the context of modern fluid dynamics and nonequlibrium statistical mechanics. Most of the theories of granular fluid are based on the Boltzmann or Enskog-Boltzmann equation of inelastic hard spheres. In this regard particle dynamics simulations make a bridge between theory and experiment. An event-driven molecular dynamics code has been developed which is based on the paper of Lubachevsky (1991) and the book by Rapaport (1995). The computational approach employed in this study mainly consists of initialization, bookkeeping and diagnostic parts. This code is very fast and can handle a large number of particles (A'^ ~ 10^) and has been generalized to three dimensions. It can be used to simulate various kind of plane granular flows (Couette flow, Poiseuille flow, Chute flow, etc.) with rough, frictional particles. For the simplest model of rough, inelastic spheres, two material parameters are needed to characterize the collision process: the normal coefficient of restitution, e, and the tangential coefficient of restitution, /?. For a more realistic collision model of rough particles, we have taken into account the effect of Coulomb friction which helps to distinguish between the sliding and the rolling contacts, resulting in a contact-angle (7) dependent tangential restitution coefficient (3{'y). With the above code, we have probed various microscopic and macroscopic propreties of unbounded granular shear flow for which a steady linear velocity profile is applied with a constant velocity gradient via shifting the replica images of top and bottom boundaries without deforming the box (Lees-Edwards boundary condition). This code has been tested for bounded shear flow too for which physical boundaries arc required: wall-particle interactions are modeled using the same collision dynamics of particle-particle interaction. For either case, the system is allowed to reach a 'non-equilibrium' steady-state condition by monitoring the temporal evolution of system's kinetic energy. In this work we have thoroughly examined the effects of particle roughness and rotation on the probability distributions of fluctuating translational and rotational velocities as well as density and spatial velocity correlations in the Boltzmann limit (for which the particle volume fraction, 0, is close to zero) of unbounded shear flow. We found that both translational and rotational velocity distribution functions (VDF) arc non-Gaussian with stretched exponential tails, except in the limiting cases of perfectly smooth (/? = —1) and rough (0—1) particles with elastic collisions (e = 1). One important finding is that the translational and rotational velocities are correlated in direction. Particle roughness has important effects on oricntational and velocity correlations even when the collisions are perfectly elastic (e = 1) and the system is homogeneous. Oricntational and spatial velociy correlation are responsible for non-Gaussian distributions of translational and rotational velocities. With increasing system density, the dissipation-induced density inhomogeneity is observed over the whole domain. A pronounced asymmetry about the mean value is observed for the probability distributions of local density, local shear rate and local spanwise rotational velocity. Therefore the calculation of "local" VDF is a proper way to study such inhomogeneous systems. For a moderately dense system {(() — 0.3), an interesting phenomenon is observed for the local VDF of streamwise translational velocity, its tails undergo a transition from an stretched exponential to an under-populated Gaussian distribution with decreasing dissipation and finally to a Gaussian for no dissipation. The VDF of spanwise rotational velocity makes a transition from stretched exponential tails to a Gaussian with decreasing dissipation. For the dense system {4> = 0.5) with dissipation, the VDF for streamwise translational velocity is a Gaussian with under-populated tails. The effect of Coulomb friction on VDFs has been studied for different values of friction coefficient ^ ior (p = 0.3 with the critical roughness being set to Po — 0. With the incorporation of Coulomb friction, a pronounced asymmetry of the tails of the VDFs of rotational velocities is seen and the skewness of the distribution increases with increasing dissipation.Lastly, we have calculated some rheologoical properties of unbounded shear flow of rough, frictional particles. Our simulations results on pressure and shear viscosity compare well with the predictions of Lun's (1991) rheological model at small dissipations. The model predictions deteriorate with increasing dissipation which is tied to the inherent assumptions of the underlying model which is valid for quasi-elastic (e ~ 1) particles in the prefectly smooth and rough (|/3| ~ 1) limits. Our results on normal stress differences (A/i and A/'2) suggest that a non-Newtonian constitutive model is required for moderately dissipative rough, frictional particles.