Onset of patterns in an oscillated granular layer: Continuum and molecular dynamics simulations

Abstract

We study the onset of patterns in vertically oscillated layers of frictionless dissipative particles. Using both numerical solutions of continuum equations to Navier-Stokes order and molecular dynamics (MD) simulations, we find that standing waves form stripe patterns above a critical acceleration of the cell. Changing the frequency of oscillation of the cell changes the wavelength of the resulting pattern; MD and continuum simulations both yield wavelengths in accord with previous experimental results. The value of the critical acceleration for ordered standing waves is approximately 10% higher in molecular dynamics simulations than in the continuum simulations, and the amplitude of the waves differs significantly between the models. The delay in the onset of order in molecular dynamics simulations and the amplitude of noise below this onset are consistent with the presence of fluctuations which are absent in the continuum theory. The strength of the noise obtained by fit to Swift-Hohenberg theory is orders of magnitude larger than the thermal noise in fluid convection experiments, and is comparable to the noise found in experiments with oscillated granular layers and in recent fluid experiments on fluids near the critical point. Good agreement is found between the mean field value of onset from the Swift-Hohenberg fit and the onset in continuum simulations. Patterns are compared in cells oscillated at two different frequencies in MD; the layer with larger wavelength patterns has less noise than the layer with smaller wavelength patterns.

Patterns and collective behavior in granular media: Theoretical concepts

“…Full three-dimensional continuum simulations based on the granular hydrodynamics equations (5), (6), (7) were performed by Bougie et al (2005). Quantitative agreement was found between this description and eventdriven molecular dynamics simulations and experiments FIG.…”

“…In the context of phase-separation instability Meerson et al (2004) raised the non-trivial question of the origin of giant fluctuations and break-down of hydrodynamic description in granular systems near the threshold of instability (see also (Bougie et al, 2005;Goldman et al, 2004) on the effect of fluctuations in multilayers). Remarkably, for the granular gas confined between two oscillating walls predicted on the basis of event-driven simulations a novel oscillatory instability for the position of the dense cluster.…”

“…Quantitative agreement was found between this description and eventdriven molecular dynamics simulations and experiments FIG. 21 Dispersion relation for stipes near the onset according to continuum granular hydrodynamics equations and molecular dynamics simulations compared with experimental data, from Bougie et al (2005).…”

“…Granular hydrodynamics equations in the form (5),(6),(7) and their modifications are widely used in the engineering community to describe a variety of large-scale granular flows, especially for design of gas-fluidized bed reactors (Gidaspow, 1994). In the physics community granular hydrodynamics is used to understand various instabilities in relatively small-scale flows, such as flow past obstacle (Rericha et al, 2002), convection (Livne et al, 2002a,b), floating clusters , longitudinal rolls Pouliquen, 2002, 2003), patterns in vibrated layers (Bougie et al, 2005) and others. However, Eqs.…”

“…The apparatus of fluctuating hydrodynamics which was developed in particular for description of transition to rolls in Rayleigh-Bénard convection (Swift and Hohenberg, 1977), has been recently applied to the granular patterns (Bougie et al, 2005;Goldman et al, 2004). The Swift-Hohenberg theory is based on the equation for the order parameter ψ,…”