Instabilities of granular material undergoing vertical vibrations: a uniformly driven layer

Abstract

In this paper, the stability of a uniformly driven granular layer is examined by linear stability analysis. This includes two main steps: first the base state at various values of mass holdup ($M_{t}$) and energy input ($Q_{t}$) is calculated; and, secondly, small perturbations are introduced to verify the stability of the base state by solving the linearized governing equations and corresponding boundary conditions. Results from the base-state solution show that, for a given pair of $M_{t}$ and $Q_{t}$, solid fraction tends to increase at first up the layer height and then decrease after a certain vertical position. In contrast, granular temperature decreases rapidly from the bottom plate to the top surface. The stability diagram is constructed by checking the eigenvalues at different points in the ($M_{t}$, $Q_{t}$) plane, and their dependence on the operating conditions and materials properties is also investigated. For the unstable regime, pattern formation is illustrated with the variation of solid fraction with vertical position. For the layer mode, there are no variations at different horizontal positions. In contrast, a periodic feature is found for the stationary mode in which alternating particle clusters and voids are observed in the horizontal direction. By introducing perturbations in different directions, we have produced surface patterns such as stripes, squares and hexagons. Besides the solid fraction distribution, other variables such as the profiles of velocities and granular temperatures are also examined.