Convection and Diffusion in Patterns in Oscillated Granular Media

Abstract

Motions of individual particles within the stripe and square patterns formed in oscillated granular media are studied using numerical simulations. Our eventdriven molecular dynamics simulations yield standing wave patterns in good accord with those observed in experiments at the same frequency and acceleration amplitude. The patterns are subharmonic and so return to their initial macroscopic state after two external cycles. However, simulations reveal that individual particles do not return to their initial position. In addition to diffusive motion, an organized flow of particles within the patterns is found; associated with each peak and each valley of the pattern is a pair of counterrotating convection rolls. The diffusion is anisotropic: transport perpendicular to stripes is enhanced over that parallel to stripes. This enhancement is computed as a function of the layer depth, acceleration amplitude, frequency, and coefficient of restitution of the particles, and is attributed to the effect of the advective motion. Velocity distributions, granular temperature, and the dependence of the diffusion coefficient parallel to the stripes on the average granular temperature are studied.

Transport coefficients for granular media from molecular dynamics simulations

“…We have found that the simulations quantitatively reproduce experiments on standing waves in oscillated granular media, producing the correct wave patterns and wavelengths [9] and secondary instabilities [4]. With the numerical simulation, we not only have access to experimentally unmeasurable quantities [10], but we also can study systems that are not experimentally realizable, allowing us to test both the assumptions and results of the granular kinetic theory. Once the constitutive relations have been tested and possibly enhanced through the use of particle simulations, direct comparison between continuum theory and laboratory experiment becomes possible.…”

“…In simulations of oscillated granular media, the results are not sensitively dependent on v a or β, and ǫ = 0.7 produces good agreement with experiment [4,9,10]. In addition to forestalling collapse, variation of e allows us to further probe granular kinetic theory, which assumes that e ≈ 1.…”

“…In our model the collisions are instantaneous and binary; they conserve momentum and dissipate energy. Particles are assumed to be frictionless; particle friction can be incorporated into kinetic theories [25,26], and was included in the simulations of oscillated granular media, [9,10,4], but introduces complications that we wish to avoid for this study.…”

Wavelength scaling and square/stripe and grain mobility transitions in vertically oscillated granular layers

“…Pattern formation in granular media can perhaps be described by continuum equations analogous to those used in fluid system [9][10][11], but the continuum equations remain largely untested by experiment. Aspects of granular patterns have been described using phenomenological models [12][13][14], but making the connection to real systems requires an understanding of granular media at a more microscopic level.…”

Granular circulation in a cylindrical pan: Simulations of reversing radial and tangential flows

“…The advantages of this model for the coefficient of restitution are discussed and demonstrated in Refs. [15,18,19,20,21,22]. We have set ǫ = 0.6, because this value gives the best agreement in terms of particle compaction observed in similarly agitated states in experiments with nylon balls and in our simulations.…”