A Simple Kinetic Theory for Granular Flow of Rough, Inelastic, Spherical Particles

Using statistical methods analogous to those used in the kinetic theory of dense gases, conservation equations and constitutive equations are derived for the flow of an idealized granular material consisting of uniform, rough, inelastic spherical particles. Simple forms for the singlet and pair velocity distribution functions are employed to study the effects of particle rotary inertia in the specific case of Couette flow of such materials. A constant coefficient of restitution e is used to characterize the inelasticity of the particles and a roughness coefficient β is adopted to characterize the effects of surface friction in collisions between particles. During collisions, surface friction causes particle rotational velocity fluctuations. As a result of particle rotary inertia, the stress tensor is found to be asymmetric during general deformations. However, for the special case of steady Couette flow which is studied in detail, the stress tensor remains symmetric. The partition of fluctuation kinetic energy between the translational and rotational modes is examined; equipartition is achieved only for the case of perfectly rough particles. In agreement with previous investigations, the stresses are found to decrease with decreasing e. Except for the case of almost perfectly rough particles, the effects of rotary inertia generally reduce the stresses. However, the normal stresses are reduced more than the shear stresses, and the predicted ratio of shear to normal stress (the dynamic friction angle) is higher for rough particles than for smooth ones. The inclusion of roughness in the analysis yields shear to normal stress ratios that agree more closely with experimental measurements.