Transient development of instabilities in a uniformly driven layer

Abstract

We present a nonlinear stability analysis on a uniformly driven granular layer to investigate the transient development of the instabilities induced by small perturbations. A continuum model based on the grain kinetic theory was adopted to trace the pattern evolution from the original unstable base state to a new base state and was numerically solved using a finite-element method. In the stability diagram characterized by two operating parameters, dimensionless mass holdup (Mt), and energy input (Qt), one point (Mt=4.75, Qt=58.31) in the stationary mode is selected to examine the fate of the two-dimensional density wave found from the linear stability analysis. Upon perturbing the base state, the bed height, solid fraction, particle velocities, and granular temperature profiles move away from their original to new, steady values after undergoing some oscillations. The stripes of solid fraction and granular temperature profiles finally change into a layer-like pattern, while the top surface is not uniform anymore but appears as periodic peaks and valleys due to the existence of uneven vertical velocities during the evolution. It is found that the finite amplitude of the initial perturbations is important for the pattern transition, and the evolution time increases with the increasing energy input and the increasing collision restitution coefficient. With the top surface of the granular layer being fixed at its original position, the oscillatory feature in the evolution curves of mass and temperature profiles disappears, and a lower solid fraction is obtained as compared to that observed in the presence of a moving upper boundary. © 2006 American Institute of Physics.