Wavelength Selection in a Vibrated Granular Layer
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Wavelength Selection in a Vibrated Granular Layer Eric Clément, Laurent Labous Laboratoire des Milieux Désordonnés et Hétérogènes – UMR 7603 Université Pierre et Marie Curie- Boîte 86 4, Place Jussieu, F-75252 Paris, France ABSTRACT We present a numerical study of a surface instability occurring in a bidimensional vibrated granular layer . The driving mechanism for the formation of stationary waves is closely followed. Two regimes of wavelength selection are identified : a dispersive regime and a saturation regime. For the saturation regime, a connection is established between the pattern formation and an intrinsic instability occurring spontaneously in dissipative gases. We also address the importance of the detailed dissipation laws determining the wavelength values. INTRODUCTION In a series of experiments Melo et al.[1] reported a pattern forming instability taking place in a vibrated thin layer of grains (for a recent report see [2] and refs. inside). The apparent phenomenology of the patterns (squares, stripes, hexagons) is strongly reminiscent of the outcome of a parametric instability occurring in vibrated fluid layers called the Faraday instability[3]. Experiments showing parametric surface patterns were also reported in 2D granular layers [4] . Numerical simulations were performed using an event driven algorithm in a 2D[5] and in a 3D geometry[6]. Several theoretical viewpoints were proposed to address the pattern formation issue (see [7] and refs inside) but there is so far no clear vision of the basic mechanisms driving this instability. Here, we report some results on an extensive numerical study we performed using an optimized version of an event-driven algorithm. We address specifically the wavelength selection problem algorithm. In these proceedings we just summarize the principal results. We suggest for more details to look in reference [8] where an extensive report is provided. NUMERICAL SIMULATIONS The system we investigate consists of N beads in a container of size L constrained to move in 2D. The bottom plate moves vertically with a trajectory z(t)= asinωt (a is the amplitude and f=ω/2π, the frequency. The lateral boundary conditions are periodic. The collision interactions stem from a collision matrix described in [9] (see refs and details in [6]). The collision parameters are a frontal restitution ε coefficient, a tangential restitution coefficient β (with a maximal value β0) and a friction coefficient µ. The frontal restitution coefficient is taken to decrease with velocity: ε(U)=1- ε0(U/U*)1/5 with U the relative velocity in the normal direction and U* = 1m/s. We use a cut-off velocity (Ucut=10-5m/s) for which ε=1. We use ε0=0.4 (for beadplate collisions, this coefficient is set equal to 0. The other parameters are β0=0.0 and µ =0.2.
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