A Fluid-like Model of Vibrated Granular Layers: Linear Stability, Kinks & Oscillons

A continuum model is proposed for the dynamics of shallow non-cohesive granular layers driven by vertical vibration of a horizontal rigid plate in vacuo. The granular mass is assumed to behave as a Newtonian liquid while in flight and to exhibit a solid-like rebound from the vibrating plate. The simple version explored here involves two constant parameters, a kinematic viscosity and a coefficient of restitution. According to the present model, the sensitive dependence of the contact dynamics on layer shape is the primary source of instability and pattern formation, while the lack of cohesive stress rules out the Rayleigh-Taylor instability driving Faraday patterns on liquid layers. This is borne out by a linear stability analysis which indicates that flat granular layers are linearly stable against free-surface perturbations and, hence, that the instability of vibrated granular layers is nonlinear in origin. An effort is made to capture certain aspects of the high Reynolds number nonlinear dynamics, namely spatially localized ”kinks” and ”oscillons”, based on purely rectilinear (”antiplane”) motion with constant layer thickness. Three different numerical methods were employed (1) computation of spatial amplitudes in a timeperiodic solutions, (2) discretization of the underlying PDEs, with the resulting ODEs in time treated by an event-detecting integrator, and (3) a variant of the latter based on discretization of a well-known Green’s function. Inaccuracies in contact detection appear to give disagreements between the different methods, such that Method (2) produced stable f/2 kinks and oscillons only if the variant (3) was employed, while Method (1) gave rise to stable oscillons but apparently unstable kinks. Also, while qualitatively similar, the numerically simulated oscillons exhibit differences from those observed experimentally. The present findings call for further work on the numerical methods, an investigation of more complex solutions involving lateral motion and varicose layers, together with improvements in the constitutive model for the granular layer, such as the incorporation of a velocity-dependent restitution. Preprint submitted to Elsevier Science 30 June 2008