A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles

TLDR

The study examines an idealized granular material of identical, smooth, nearly elastic spherical particles flowing at high density and deformation rate where interactions are limited to binary collisions. Using probability distribution functions for particle velocity and collision likelihood, the authors derive local balance equations and, with simple parameterized PDFs, express stress, energy flux, and dissipation solely in terms of mean density, velocity, and kinetic energy. They solve the balance laws to predict field behavior, exemplified by shear flow between two parallel plates in relative motion.

Abstract

We focus attention on an idealized granular material comprised of identical, smooth, imperfectly elastic, spherical particles which is flowing at such a density and is being deformed at such a rate that particles interact only through binary collisions with their neighbours. Using general forms of the probability distribution functions for the velocity of a single particle and for the likelihood of binary collisions, we derive local expressions for the balance of mass, linear momentum and fluctuation kinetic energy, and integral expressions for the stress, energy flux and energy dissipation that appear in them. We next introduce simple, physically plausible, forms for the probability densities which contain as parameters the mean density, the mean velocity and the mean specific kinetic energy of the velocity fluctuations. This allows us to carry out the integrations for the stress, energy flux and energy dissipation and to express these in terms of the mean fields. Finally, we determine the behaviour of these fields as solutions to the balance laws. As an illustration of this we consider the shear flow maintained between two parallel horizontal plates in relative motion.