Emergence of order in an oscillated granular layer

2000

The formation of textured patterns has been predicted to occur in two stages. The first is an early time, domain-forming stage with dynamics characterized by a disorder function $\bar\delta (\beta) \sim t^{-\sigma_{E}}$, with $\sigma_{E} = {1/2}\beta$; this decay is universal. Coarsening of domains occurs in the second stage, in which $\bar\delta (\beta) \sim t^{-\sigma_{L}}$, where $\sigma_{L}$ is a nonlinear function of $\beta$ whose form is system and model dependent. Our experiments on a vertically oscillated granular layer are in accord with theory, yielding $\sigma_{E}\approx 0.5\beta$, and $\sigma_{L}$ a nonlinear function of $\beta$.

Physical Review Letters, 1995

Experiments on vertically oscillated granular layers in an evacuated container reveal a sequence of well-defined pattern bifurcations as the container acceleration is increased. Period doublings of the layer center of mass motion and a parametric wave instability interact to produce hexagons and more complicated patterns composed of distinct spatial domains of different relative phase separated by kinks (phase discontinuities). Above a critical acceleration, the layer becomes disordered in both space and time.

Physical Review E, 2001

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 7g, where the pattern is approximately hexagonal, oscillating at onefourth the driving frequency (f /4). A phase bubble is a localized region that oscillates with a phase opposite (differing by π) 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 this 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.

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 ͑MD͒ 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.

Lecture Notes in Physics, 1997

Sedimentary rocks have complicated permeability patterns arising from the geological processes that formed them. We concentrate on pattern formation in one particular geological process, avalanches (grainflow) in windblown or fluvial sands. We present a simple experiment and numerical model of how these avalanches cause segregation in particle size that lead to characteristic laminated patterns. We also address the longstanding question of how such patterns are generated. We analyze data on two sandstone samples from different, but similar, geological environments, and find that the permeability fluctuations display longrange power-law correlations characterized by an exponent H. For both samples, we find H ≈ 0.82 − 0.90. These permeability fluctuations significantly affect the flow of fluids through the rocks. We demonstrate this by investigating the influence of long-range correlation on percolation properties, like cluster morphology, and relate these properties to characteristics important for hydrocarbon recovery such as breakthrough time for injected fluids and recovery efficiency.

Physical Review Letters, 1998

Numerical simulations and laboratory experiments are conducted for thin layers of particles in a vertically oscillated container as a function of the frequency f, amplitude A, and depth H. The same standing wave patterns (stripes, squares, or hexagons oscillating at f͞2 or f͞4) and wavelengths are obtained in the simulations and experiments for a wide range of ͑ f, A͒ and two layer depths. Two model parameters are determined by fits at just two points ͑ f, A, H͒. Simulation results lead to heuristic arguments for the onset of patterns and the crossover from squares to stripes.

Physical Review E, 2001

Cell-filling spiral patterns are observed in a vertically oscillated layer of granular material when the oscillation amplitude is suddenly increased from below the onset of pattern formation into the region where stripe patterns appear for quasistatic increases in amplitude. These spirals are transients and decay to stripe patterns with defects. A transient spiral defect chaos state is also observed. We describe the behavior of the spirals, and the way in which they form and decay. Our results are compared with those for similar spiral patterns in Rayleigh-Bénard convection in fluids.

Physical Review Letters, 2004

We study fluctuations in a vertically oscillated layer of grains below the critical acceleration for the onset of ordered standing waves. As onset is approached, transient disordered waves with a characteristic length scale emerge and increase in power and coherence. The scaling behavior and the shift in the onset of order agrees with the Swift-Hohenberg theory for convection in fluids. However, the noise in the granular system is four orders of magnitude larger than the thermal noise in a convecting fluid, making the effect of granular noise observable even 20% below the onset of long range order.

Proceedings of the National Academy of Sciences, 1999

An important industrial problem that provides fascinating puzzles in pattern formation is the tendency for granular mixtures to de-mix or segregate. Small differences in either size or density lead to flow-induced segregation. Similar to fluids, noncohesive granular materials can display chaotic advection; when this happens chaos and segregation compete with each other, giving rise to a wealth of experimental outcomes. Segregated structures, obtained experimentally, display organization in the presence of disorder and are captured by a continuum flow model incorporating collisional diffusion and density-driven segregation. Under certain conditions, structures never settle into a steady shape. This may be the simplest experimental example of a system displaying competition between chaos and order.

HAL (Le Centre pour la Communication Scientifique Directe), 2006

We measure experimentally the rearrangements due to a small localized cyclic displacement applied to a packing of rigid grains under gravity in a 2D geometry. We analyze the evolution of the response to this perturbation by considering the individual particle displacement and the coarse grained displacement field, as well as the mean packing fraction and coordination number. We find that the displacement response is rather long ranged, and evolves considerably with the number of cycles. We show that a small difference in the preparation method (induced by tapping the container) leads to a significant modification in the response though the packing fraction changes are minute. Not only the initial response but also its further evolution change with preparation, demonstrating that the system still retains a memory of the initial preparation after many cycles. Nevertheless, after a sufficient number of cycles, the displacement response for both preparation methods converges to a nearly radial field with a 1/r decay from the perturbation source. The observed differences between the preparation methods seem to be related to the changes in the coordination number (which is more sensitive to the evolution of the packing than the packing fraction). Specifically, it may be understood as an effect of the breaking of local arches, which affects the lateral transmission of forces.