Sung Joon Moon

Shocks in vertically oscillated granular layers

We study shock formation in vertically oscillated granular layers, using both molecular dynamics simulations and numerical solutions of continuum equations to Navier-Stokes order. A flat layer of grains is thrown up from an oscillating plate during each oscillation cycle and collides with the plate later in the cycle. The collisions produce layer compaction near the plate and a high temperature shock front that rapidly propagates upward through the layer. The shock is highly time dependent, propagating through the layer in only a quarter of the cycle. We compare numerical solutions of the continuum equations to molecular dynamics simulations that assume binary, instantaneous collisions between frictionless, inelastic hard spheres. The two simulations yield results for the shock position, shape, and speed that agree well. An investigation of the effect of inelasticity shows that the shock velocity increases continuously with decreasing inelasticity; the elastic limit is not singular.

Velocity distributions and correlations in homogeneously heated granular media.

We compare the steady state velocity distributions from our three-dimensional inelastic hard sphere molecular dynamics simulation for homogeneously heated granular media, with the predictions of a mean field-type Enskog-Boltzmann equation for inelastic hard spheres [T. P. C. van Noije and M. H. Ernst, Granular Matter 1, 57 (1998)]. Although we find qualitative agreement for all values of density and inelasticity, the quantitative disagreement approaches approximately 40% at high inelasticity or density. By contrast the predictions of the pseudo-Maxwell molecule model [J. A. Carrillo, C. Cercignani, and I. M. Gamba, Phys. Rev. E, 62, 7700 (2000)] are both qualitatively and quantitatively different from those of our simulation. We also measure short-range and long-range velocity correlations exhibiting nonzero correlations at contact before the collision, and being consistent with a slow algebraic decay over a decade in the unit of the diameter of the particle, proportional to r(-(1+alpha)), where 0.2<alpha<0.3. The existence of these correlations implies the failure of the molecular chaos assumption and the mean field approximation, which is responsible for the quantitative disagreement of the inelastic hard sphere kinetic theory.

Role of friction in pattern formation in oscillated granular layers.

Particles in granular flows are often modeled as frictionless (smooth) inelastic spheres; however, there exist no frictionless grains, just as there are no elastic grains. Our molecular dynamics simulations reveal that friction is essential for realistic modeling of vertically oscillated granular layers: simulations of frictionless particles yield patterns with an onset at a container acceleration about 30% smaller than that observed in experiments and simulations with friction. More importantly, even though square and hexagonal patterns form for a wide range of the oscillation parameters in experiments and in our simulations of frictional inelastic particles, only stripe patterns form in the simulations without friction, even if the inelasticity is increased to obtain as much dissipation as in frictional particles. We also consider the effect of particle friction on the shock wave that forms each time the granular layer strikes the container. While a shock wave still forms for frictionless particles, the spatial and temporal dependence of the hydrodynamic fields differ for the cases with and without friction.

Phase bubbles and spatiotemporal chaos in granular patterns

We use inelastic hard sphere molecular dynamics simulations and laboratory experiments to study patterns in vertically oscillated granular layers. The simulations and experiments reveal that phase bubbles spontaneously nucleate in the patterns when the container acceleration amplitude exceeds a critical value, about 7 g, where the pattern is approximately hexagonal, oscillating at one-fourth the driving frequency (f/4). A phase bubble is a localized region that oscillates with a phase opposite (differing by pi) to that of the surrounding pattern; a localized phase shift is often called an arching in studies of two-dimensional systems. The simulations show that the formation of phase bubbles is triggered by undulation at the bottom of the layer on a large length scale compared to the wavelength of the pattern. Once formed, a phase bubble shrinks as if it had a surface tension, and disappears in tens to hundreds of cycles. We find that there is an oscillatory momentum transfer across a kink, and the shrinking is caused by a net collisional momentum inward across the boundary enclosing the bubble. At increasing acceleration amplitudes, the patterns evolve into randomly moving labyrinthian kinks (spatiotemporal chaos). We observe in the simulations that f/3 and f/6 subharmonic patterns emerge as primary instabilities, but that they are unstable to the undulation of the layer. Our experiments confirm the existence of transient f/3 and f/6 patterns.

Lattice dynamics and melting of a nonequilibrium pattern.

We present a new description of nonequilibrium square patterns as a harmonically coupled crystal lattice. In a vertically oscillating granular layer, different transverse normal modes of the granular square-lattice pattern are observed for different driving frequencies (f(d)) and accelerations. The amplitude of a mode can be further excited by either frequency modulation of f(d) or reduction of friction between the grains and the plate. When the mode amplitude becomes large, the lattice melts (disorders), in accord with the Lindemann criterion for melting in two dimensions.

Steady-state velocity distributions of an oscillated granular gas.

We use a three-dimensional molecular dynamics simulation to study the single particle distribution function of a dilute granular gas driven by a vertically oscillating plate at high accelerations (15g-90g). We find that the density and the temperature fields are essentially time-invariant above a height of about 40 particle diameters, where typically 20% of the grains are contained. These grains form the nonequilibrium steady-state granular gas with a Knudsen number unity or greater. In the steady-state region, the probability distribution function of the horizontal velocity c(x) (scaled by the local horizontal temperature) is found to be nearly independent of height, even though the hydrodynamic fields vary with height. We find that the high energy tails of the distribution function are described by a stretched exponential approximately exp(-Bcalphax), where alpha depends on the restitution coefficient e and falls in the range 1.2<alpha<1.6. However, alpha does not vary significantly for a wide range of friction coefficient values. We find that the distribution function of a frictionless inelastic hard sphere model can be made similar to that of a frictional model by adjusting e. However, there is no single value of e that mimics the frictional model over a range of heights.

Kink-induced transport and segregation in oscillated granular layers.

We use experiments and molecular dynamics simulations of vertically oscillated granular layers to study horizontal particle segregation induced by a kink (a boundary between domains oscillating out of phase). Counterrotating convection rolls carry the larger particles in a bidisperse layer along the granular surface to a kink, where they become trapped. The convection originates from avalanches that occur inside the layer, along the interface between solidified and fluidized grains. The position of a kink can be controlled by modulation of the container frequency, making possible systematic harvesting of the larger particles.