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 $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-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 $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-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 $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ 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.
Highlights
The original incompressible μ(I)-rheology was proposed (GDR MiDi 2004; Jop, Forterre & Pouliquen 2006) to describe granular flow in which the particles are rigid and the solids volume fraction φ is constant and uniform
The purpose of this paper is to develop a viable alternative to μ(I), Φ(I)-rheology that preserves its successes
The Inertial compressible I-dependent rheology (iCIDR) equations, introduced in this paper, provide a continuum model for fluid-like inertial flows of rigid spherical particles that lie below a critical volume fraction, above which the compressibility of the grains becomes important (Otsuki & Hayakawa 2009; Chialvo et al 2012)
Summary
The original incompressible μ(I)-rheology was proposed (GDR MiDi 2004; Jop, Forterre & Pouliquen 2006) to describe granular flow in which the particles are rigid and the solids volume fraction φ is constant and uniform. The key physical insight behind the theory was that, under these circumstances, the only non-dimensional groups are the bulk friction coefficient μ (the ratio τ /p of shear stress to pressure) and the inertial number I = dγ √ p/ρ∗ (1.1). Where d is the grain diameter, γis the shear rate, p is the pressure and ρ∗ is the intrinsic grain density. This approach has led to significant progress in modelling granular flows (e.g. Lagrée, Staron & Popinet 2011; Staron, Lagrée & Popinet 2012; Gray & Edwards 2014; Barker & Gray 2017). Barker et al (2015) identified conditions on μ(I) and I under which incompressible μ(I)-rheology is ill-posed
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