Constitutive relations for compressible granular flow in the inertial regime

Abstract

Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the µ(I), Φ(I)-rheology, which postulates that the bulk friction coefficient µ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction φ are functions of the inertial number I only. Although the µ(I), Φ(I)-rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the µ(I), Φ(I)-rheology that does not suffer from such defects is proposed. In the framework of compressible I-dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established µ(I) and Φ(I) relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.

Self-channelisation and levee formation in monodisperse granular flows

“…Understanding how to model this quantitatively would probably require a compressible granular theory (see e.g. Barker et al 2017;Heyman et al 2017;Schaeffer et al 2019) that could span the quasi-static and inertial regimes (Chialvo, Sun & Sundaresan 2012). There is, however, a long way to go before either of these effects can be included in depth-averaged avalanche models, and the existing theory appears to be able to make accurate predictions without them.…”

Coupling rheology and segregation in granular flows

“…2017; Schaeffer et al. 2019) the and laws only hold at steady state, and so the general form of the diffusivity (3.6) applies.…”

“…2017; Schaeffer et al. 2019) or non-locality (Henann & Kamrin 2013). However, in this paper the partially regularized -rheology is chosen for the bulk flow, both for simplicity and because it is most readily compatible with existing numerical methods and particle segregation models.…”

The compressible granular collapse in a fluid as a continuum: validity of a Navier–Stokes model with , -rheology

“…It should be noted that there are various possibilities to combine critical state theory and the μ(I), φ(I)-rheology. An alternative approach including bulk viscosity is provided, for example, by Schaeffer et al (2019).…”