Continuum representation of a continuous size distribution of particles engaged in rapid granular flow

Abstract

Natural and industrial granular flows often consist of several particle sizes, approximately forming a continuous particle size distribution (PSD). Continuous PSDs are ubiquitous, though existing kinetic-theory-based, hydrodynamic models for rapid granular flows are limited to a discrete number of species. The objective of this work is twofold: (i) to determine the number of discrete species required to accurately approximate a continuous PSD and (ii) to validate these results via a comparison with molecular dynamics (MD) simulations of continuous PSDs. With regard to the former, several analytic (Gaussian and lognormal) and experimental (coal and lunar soil simulants) distributions are investigated. Transport coefficients (pressure, shear viscosity, etc.) of the granular mixture given by the polydisperse theory of Garzó et al. [“Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport,” Phys. Rev. E 76, 031303 (2007)10.1103/PhysRevE.76.031303;Garzó et al. “Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport,” Phys. Rev. E 76, 031304 (2007)10.1103/PhysRevE.76.031304] are compared using an increasing number of species s to approximate the given PSD. These discrete approximations are determined by matching the first 2s moments of the approximation and the given continuous distribution. Relatively few species are required to approximate moderately wide distributions (Gaussian, lognormal), whereas even wider distributions (coal and lunar soil simulants) require a larger number of species. Regarding the second objective, a comparison between MD simulations and kinetic-theory predictions for a simple shear flow of both Gaussian and lognormal PSDs reveal essentially no loss of accuracy stemming from the polydisperse theory itself (as compared to theories for monodisperse systems) or from the discrete approximations of continuous PSDs used in the polydisperse theory.