Abstract
In light of the successes of the Navier–Stokes equations in the study of fluid flows, similar continuum treatment of granular materials is a long-standing ambition. This is due to their wide-ranging applications in the pharmaceutical and engineering industries as well as to geophysical phenomena such as avalanches and landslides. Historically this has been attempted through modification of the dissipation terms in the momentum balance equations, effectively introducing pressure and strain-rate dependence into the viscosity. Originally, a popular model for this granular viscosity, the Coulomb rheology, proposed rate-independent plastic behaviour scaled by a constant friction coefficient ${\it\mu}$. Unfortunately, the resultant equations are always ill-posed. Mathematically ill-posed problems suffer from unbounded growth of short-wavelength perturbations, which necessarily leads to grid-dependent numerical results that do not converge as the spatial resolution is enhanced. This is unrealistic as all physical systems are subject to noise and do not blow up catastrophically. It is therefore vital to seek well-posed equations to make realistic predictions. The recent ${\it\mu}(I)$-rheology is a major step forward, which allows granular flows in chutes and shear cells to be predicted. This is achieved by introducing a dependence on the non-dimensional inertial number $I$ in the friction coefficient ${\it\mu}$. In this paper it is shown that the ${\it\mu}(I)$-rheology is well-posed for intermediate values of $I$, but that it is ill-posed for both high and low inertial numbers. This result is not obvious from casual inspection of the equations, and suggests that additional physics, such as enduring force chains and binary collisions, becomes important in these limits. The theoretical results are validated numerically using two implicit schemes for non-Newtonian flows. In particular, it is shown explicitly that at a given resolution a standard numerical scheme used to compute steady-uniform Bagnold flow is stable in the well-posed region of parameter space, but is unstable to small perturbations, which grow exponentially quickly, in the ill-posed domain.
Highlights
The Groupement de Recherche Milieux Divisés collated experimental data and discrete element simulations obtained in six different steady flow configurations (GDR-MiDi 2004) and interpreted them with a view to determining the rheology of granular materials
In the dense inertial regime GDR-MiDi (2004) used dimensional analysis to postulate a local rheology in which μeff = τ /P was a function of I, i.e. the effective friction μ = μ(I)
This paper shows that in two dimensions the μ(I)-rheology (GDR-MiDi 2004; Jop et al 2005, 2006) is well-posed for a large intermediate range of inertial numbers, provided μ is sufficiently large, but that for both high and low inertial numbers the equations are ill-posed
Summary
The Groupement de Recherche Milieux Divisés collated experimental data and discrete element simulations obtained in six different steady flow configurations (GDR-MiDi 2004) and interpreted them with a view to determining the rheology of granular materials. Well-posed and ill-posed behaviour of the μ(I)-rheology for granular flow 801 Substitution of the operator L from (2.33) gives the growth rate λ(ξ ) = η0 q |ξ |2 (ξ · Aξ ) − |ξ |4 + r(ξ ⊥ · Aξ )2 |ξ |2 − q(ξ · Aξ )
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