A new kinetic theory model of granular flows that incorporates particle stiffness

Abstract

Granular materials of practical interest in general have finite stiffness; therefore, the particle collision is a process that takes finite time to complete. Soft-sphere Discrete Element Method (DEM) simulations suggest that there are three regimes for granular shear flows: inertial regime (or rapid flow regime), elastic regime (or quasistatic regime), and the transition regime (or elastic-inertial regime). If we use tf to represent the mean free flight time for a particle between two consecutive collisions and tc to represent the binary collision duration, these regimes are implicitly related to the ratio tc/tf. Granular flows can be successfully predicted by the classical Kinetic Theory (KT) when they are in the inertial regime of low particle-particle collision frequencies and short time contacts (tc/tf ≈ 0). However, we find that KT becomes less accurate in the transition regime where the collision duration tc is no longer small compared with the collision interval tf (tc/tf > 0.05). To address this issue, we develop a soft-sphere KT (SSKT) model that takes particle stiffness k as an input parameter since tc/tf is mainly determined by k. This is achieved by proposing a modified expression for the collision frequency and introducing an elastic granular temperature Te. Compared with the classical KT that only considers the kinetic granular temperature Tk, a redefined total granular temperature (Tg = Tk + Te/3) that takes both kinetic fluctuation energy and elastic potential energy into consideration is used in the SSKT model. The model is developed for identical frictionless particles with the linear-spring-dashpot collision scheme; however, it can be extended to frictional systems as well after the modification of constitutive equations. We show that the proposed SSKT extends the applicability of the KT framework to the transition regime without losing significant accuracy. The rheological crossover has been explained physically, and the regime boundaries that separate the inertial regime and the elastic regime are quantitatively determined, showing good agreement with the previous regime map that was based on the DEM simulations. Our SSKT predictions also show that for unsteady flows such as homogeneous cooling, the particle stiffness could have a large impact on the granular flow behavior due to the energy transfer between Te and Tk.