Stability of plane Couette flow of a granular material

Abstract

This paper presents a linear stability analysis of plane Couette flow of a granular material using a kinetic-theory-based model for the rheology of the medium. The stability analysis, restricted to two-dimensional disturbances, is carried out for three illustrative sets of grain and wall properties which correspond to the walls being perfectly adiabatic, and sources and sinks of fluctuational energy. When the walls are not adiabatic and the Couette gap H is sufficiently large, the base state of steady fully developed flow consists of a slowly deforming ‘plug’ layer where the bulk density is close to that of maximum packing and a rapidly shearing layer where the bulk density is considerably lower. The plug is adjacent to the wall when the latter acts as a sink of energy and is centred at the symmetry axis when it acts as a source of energy. For each set of properties, stability is determined for a range of H and the mean solids fraction [barvee ]. For a given value of [barvee ], the flow is stable if H is sufficiently small; as H increases it is susceptible to instabilities in the form of cross-stream layering waves with no variation in the flow direction, and stationary and travelling waves with variation in the flow and gradient directions. The layering instability prevails over a substantial range of H and [barvee ] for all sets of wall properties. However, it grows far slower than the strong stationary and travelling wave instabilities which become active at larger H. When the walls act as energy sinks, the strong travelling wave instability is absent altogether, and instead there are relatively slow growing long-wave instabilities. For the case of adiabatic walls there is another stationary instability for dilute flows when the grain collisions are quasi-elastic; these modes become stable when grain collisions are perfectly elastic or very inelastic. Instability of all modes is driven by the inelasticity of grain collisions.

Shear-induced crystallization of a dense rapid granular flow: Hydrodynamics beyond the melting point

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“…One can obtain this solution directly by putting λ = 0 in Eq. (20). Therefore, the assumption of a nonzero λ [the specific form of viscosity divergence, Eq.…”

“…(20), while T 0 = 1−n 0 = ǫ * 2 . Linearizing the hydrodynamic equations with respect to small perturbations, we arrive at…”

Stability analysis of rapid granular chute flows: formation of longitudinal vortices

“…Another well-known source of instability in granular flows is inelasticity. Studies on two-dimensional shear flows have shown that the dissipation due to inelastic collisions can lead to the formation of clusters (Savage 1992;Chi-Hwa Wang et al 1997;Alam & Nott 1998;Nott et al 1999). In order to better understand the role of dissipation in our problem, we have performed the stability analysis by setting to zero the collisional rate of energy dissipation γ in the linearized energy equation.…”

Simulation of pattern dynamics of cohesive granular particles under a plane shear

“…When a shear is applied to the granular system, high-density region, called "shear band", appears [7]. There exist many papers to estimate the transport coefficients by using kinetic theory [8,9,10,11,12,13] and to analyze the pattern dynamics by using continuum mechanics [14,15,16,17,18,19,20,21,22].…”