Stability of unbounded and bounded granular shear flows

Abstract

Recent computer simulations of granular flows (Hopkins & Louge 1991; Goldhirsch, Kui & Zanetti 1993) have identified an interesting picture of cluster formation. In this thesis, we have attempted to explain particle clustering as an instability of the mean flow. We argue that density fluctuations can be thought of as a precursor of clustering instability in that clusters might appear in the flow due to non-linear saturation of infinitesimal perturbations. We focus on two aspects that have not been studied earlier, namely the effect of friction and the effect of wall properties on flow instabilities. In the first part of this work, the linear stability of unbounded granular shear flow is considered with a model that integrates in a simple manner the stress generation due to grain friction with the kinetic stress that arises from streaming and collisions of grains. As in the purely kinetic case, instability only occurs when disturbances are in the form of layering modes, i.e., when the wavenumber for the streamwise direction is zero (k? = 0). The flow is always stable to disturbances in the form of non-layering modes (k? ? 0). The influence of friction on stability was found to depend on the coefficient of restitution for grain collisions, e?. When e? is small, friction has the effect of stabilizing the flow. When e? is large, friction enhances instability growth rate. An important point is that unbounded shear flow in the limit of perfectly elastic collisions (e? = 1) is stable regardless of the value of the friction coefficient. We show that transient growth of disturbance is possible because the associated linear operator is non-normal; friction has the effect of enhancing the initial growth rate as well as the maximum transient growth, especially at high solids fraction. In the second part, we consider bounded shear flow (plane Couette flow) in the absence of gravity, sustained by the relative motion of two plane parallel walls. A kinetic-theory-based model for the rheology of the medium is assumed. We find plug formation to be a generic feature of the fully developed shear flow for large Couette gap H when the mean solids fraction (F) exceeds a minimum threshold. Within the plug, particles are densely packed with solids fraction close to that at the maximum packing limit, and the shear rate almost zero. The stability of bounded shear flow is considered for two classes of walls, namely, the adiabatic and non-adiabatic walls. The walls are thermally passive for the former, while they act as the sources or sinks of pseudo-thermal energy for the latter. For the case of adiabatic walls, the base state is that of uniform shear and constant density across the Couette gap. This solution is stable if F is sufficiently small, and is susceptible to instabilities in the form of (non-oscillatory) layering, stationary and traveling waves as H increases. For non-adiabatic walls, the presence of plugs in the base states strongly influences stability, and some features are distinctly different from those of the uniform shear case. An important departure from the uniform shear case is that the flow is unstable to layering disturbances for a finite range of H only. Instabilities in the form of stationary and traveling waves are also present. The time evolution of layering instabilities in the adiabatic case results in highly segregated solutions that bifurcate from the uniform shear branch. These segregated solutions resemble those in the non-adiabatic case, and on comparing the stability of analogous base states of the adiabatic and non-adiabatic cases, we find no qualitative differences. As in unbounded shear flow, grain inelasticity is essential for the instability in that the flow becomes stable in the perfect elastic limit (e? = 1).