Stability of Granular Materials under Vertical Vibrations

The influence of periodic vibrations on the granular flow of materials is of great interests to scientists and engineers due to both theoretical and practical reasons. In this paper, the stability of a vertically vibrated granular layer is examined by linear stability analysis. This includes two major steps, firstly, the base state at various values of mass holdup (Mt) and energy input (Qt) is calculated and secondly, small perturbations are introduced to verify the stability of the base state by solving the resultant eigenvalue problem derived from the linearized governing equations and corresponding boundary conditions. Results from the base state solution show that, for a given pair of Mt and Qt, solid fraction tends to increase at first along the layer height and then decrease after a certain vertical position while granular temperature decreases rapidly from the bottom plate to the top surface. This may be due to the existence of inelastic collisions between particles that dissipate the energy input from the bottom. It is also found that more energy input results in a lower solid fraction and a higher granular temperature. The stability diagram is constructed by checking the stability property at different points in the Mt-Qt plane. For a fixed Mt, the base state is stable at low energy inputs, and becomes unstable if Qt is larger than a critical value Qtc1. A higher value of Mt corresponds to a larger Qtc1. There also exists a critical mass holdup (Mtc), for Mt larger than Mtc, the patterns corresponding to the instabilities are standing waves (stationary mode); otherwise the flat layer appears (layer mode). Moreover, the stationary mode turns into the layer mode when Qt is increased beyond a critical value Qtc2. These findings agree with the experimental observations of other researchers (Hsiau and Pan, 1998). The effects of restitution coefficients (ep, ew) and material properties (dp, ρp) on the stability diagram are also investigated. Together with Mt and Qt these variables can be classified into two groups, i.e. the stabilizing factors (Mt, dp, ρp) and the destabilizing factors (Qt, ep, ew). The stability of the system is enhanced with increasing stabilizing factors and decreasing destabilizing factors.