Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield

TLDR

Granular flow of uniform smooth inelastic spherical particles is examined using kinetic‑theory statistical methods. The study develops two theories: one for Couette flow of arbitrarily inelastic particles and another for general flow of nearly elastic particles, the latter employing a Chapman‑Enskog–based approach. The first theory applies Savage & Jeffrey (1981) with an ad hoc collision distribution, while the second uses a rational approximation to determine collisional distribution functions, applicable to general flowfields and incorporating kinetic and collisional contributions to stress and energy flux for dilute to dense solids. The theories agree moderately with experiments, and the Jenkins–Savage asymptotic analysis yields similar results even for highly inelastic particles; when collisional effects dominate, stresses predicted by the second theory match those of the first in simple shear.

Abstract

The flow of an idealized granular material consisting of uniform smooth, but nelastic, spherical particles is studied using statistical methods analogous to those used in the kinetic theory of gases. Two theories are developed: one for the Couette flow of particles having arbitrary coefficients of restitution (inelastic particles) and a second for the general flow of particles with coefficients of restitution near 1 (slightly inelastic particles). The study of inelastic particles in Couette flow follows the method of Savage & Jeffrey (1981) and uses an ad hoc distribution function to describe the collisions between particles. The results of this first analysis are compared with other theories of granular flow, with the Chapman-Enskog dense-gas theory, and with experiments. The theory agrees moderately well with experimental data and it is found that the asymptotic analysis of Jenkins & Savage (1983), which was developed for slightly inelastic particles, surprisingly gives results similar to the first theory even for highly inelastic particles. Therefore the ‘nearly elastic’ approximation is pursued as a second theory using an approach that is closer to the established methods of Chapman-Enskog gas theory. The new approach which determines the collisional distribution functions by a rational approximation scheme, is applicable to general flowfields, not just simple shear. It incorporates kinetic as well as collisional contributions to the constitutive equations for stress and energy flux and is thus appropriate for dilute as well as dense concentrations of solids. When the collisional contributions are dominant, it predicts stresses similar to the first analysis for the simple shear case.