Shocks in vertically oscillated granular layers

State-dependent driving: a route to non-equilibrium stationary states

We study three different experiments that involve dry friction and periodic driving, and which employ both single- and many-particle systems. These experimental set-ups, besides providing a playground for investigation of frictional effects, are relevant in broad areas of science and engineering. Across all these experiments, we monitor the dynamics of objects placed on a substrate that is being moved in a horizontal manner. The driving couples to the degrees of freedom of the substrate and this coupling in turn influences the motion of the objects. Our experimental findings suggest emergence of stationary-states with non-trivial features. We invoke a minimalistic phenomenological model to explain our experimental findings. Within our model, we treat the injection of energy into the system to be dependent on its dynamical state, whereby energy injection is allowed only when the system is in its suitable-friction state. Our phenomenological model is built on the fact that such a state-dependent driving results in a force that repeatedly toggles the frictional states in time and serves to explain our experimental findings.

Discharge of a granular silo under mechanical vibrations

In this paper, we study the flow rate of model granular material in a silo under the influence of mechanical vibrations. Experimental measurements and discrete element simulations (DEM) are performed in a quasi-2D silo. The influence on the flow rate of the opening size and the vibration applied on the entire silo is studied. Two distinct regimes are evidenced, governed by the Froude number Fr and the relative frequency Ω. In the first regime, a decreased flow rate is observed when increasing the vibration intensity. This behavior is explained by the presence of reorganizations induced by the vibration, leading to a more homogeneous but also slower flow. In the second regime, an increased flow rate is evidenced when increasing the vibration intensity. We find this behavior comes from the intermittent nature of the flow, where the flow rate is directly controlled by the propagation of shock waves all along the silo.

Velocity Distribution of a Homogeneously Cooling Granular Gas

In contrast to molecular gases, granular gases are characterized by inelastic collisions and require therefore permanent driving to maintain a constant kinetic energy. The kinetic theory of granular gases describes how the average velocity of the particles decreases after the driving is shut off. Moreover, it predicts that the rescaled particle velocity distribution will approach a stationary state with overpopulated high-velocity tails as compared to the Maxwell-Boltzmann distribution. While this fundamental theoretical result was reproduced by numerical simulations, an experimental confirmation is still missing. Using a microgravity experiment that allows the spatially homogeneous excitation of spheres via magnetic fields, we confirm the theoretically predicted exponential decay of the tails of the velocity distribution.

The role of initial speed in projectile impacts into light granular media

Projectile impact into a light granular material composed of expanded polypropylene (EPP) particles is investigated systematically with various impact velocities. Experimentally, the trajectory of an intruder moving inside the granular material is monitored with a recently developed non-invasive microwave radar system. Numerically, discrete element simulations together with coarse-graining techniques are employed to address both dynamics of the intruder and response of the granular bed. Our experimental and numerical results of the intruder dynamics agree with each other quantitatively and are in congruent with existing phenomenological model on granular drag. Stepping further, we explore the ‘microscopic’ origin of granular drag through characterizing the response of granular bed, including density, velocity and kinetic stress fields at the mean-field level. In addition, we find that the dynamics of cavity collapse behind the intruder changes significantly when increasing the initial speed . Moreover, the kinetic pressure ahead of the intruder decays exponentially in the co-moving system of the intruder. Its scaling gives rise to a characteristic length scale, which is in the order of intruder size. This finding is in perfect agreement with the long-scale inertial dissipation type that we find in all cases.

Reticulate collisional structure in boundary-driven granular gases

We report a peculiar head-on collision network between two vibrating boundaries in experiments performed during a parabolic flight and in a laboratory using horizontal vibration. This structure is a new ordering, which is due to an orientation correlation between the relative position and velocity of any particle pair. It weakens the collision frequency and produces a long-range boundary effect. Moreover, we find the molecular chaos assumption is violated in a larger portion of the phase space. Using an anisotropic distribution model, we modify angular integration results and compare them to the results of the kinetic theory.

A robot made of robots: Emergent transport and control of a smarticle ensemble

Robot locomotion is typically generated by coordinated integration of single-purpose components, like actuators, sensors, body segments, and limbs. We posit that certain future robots could self-propel using systems in which a delineation of components and their interactions is not so clear, becoming robust and flexible entities composed of functional components that are redundant and generic and can interact stochastically. Control of such a collective becomes a challenge because synthesis techniques typically assume known input-output relationships. To discover principles by which such future robots can be built and controlled, we study a model robophysical system: planar ensembles of periodically deforming smart, active particles—smarticles. When enclosed, these individually immotile robots could collectively diffuse via stochastic mechanical interactions. We show experimentally and theoretically that directed drift of such a supersmarticle could be achieved via inactivation of individual smarticles and used this phenomenon to generate endogenous phototaxis. By numerically modeling the relationship between smarticle activity and transport, we elucidated the role of smarticle deactivation on supersmarticle dynamics from little data—a single experimental trial. From this mapping, we demonstrate that the supersmarticle could be exogenously steered anywhere in the plane, expanding supersmarticle capabilities while simultaneously enabling decentralized closed-loop control. We suggest that the smarticle model system may aid discovery of principles by which a class of future “stochastic” robots can rely on collective internal mechanical interactions to perform tasks.

Internal and surface waves in vibrofluidized granular materials: Role of cohesion

Wave phenomena in vibrofluidized dry and partially wet granular materials confined in a quasi-two-dimensional geometry are investigated with numerical simulations considering individual particles as hard spheres. Short-ranged cohesive interactions arising from the formation of liquid bridges between adjacent particles are modeled by changing the velocity-dependent coefficient of restitution. Such a change effectively suppresses the formation of surface waves, in agreement with previous experimental observations. The difference in pattern creation arises from the suppressed momentum transfer due to wetting and it can be quantitatively understood from an analysis of binary impacts.

Density-wave fronts on the brink of wet granular condensation

Density-wave fronts in a vibrofluidized wet granular layer undergoing a gas-liquid-like transition are investigated experimentally. The threshold of the instability is governed by the amplitude of the vertical vibrations. Fronts, which are curved into a spiral shape, propagate coherently along the circular rim of the container with leading edges. They are stable beyond a critical distance from the container center. Based on an analysis of the emerging time and length scales, we propose a model for the pattern formation by considering the competition between the time scale for the condensation of wet granular particles from a gas-like state and that of the energy injection resisting this process.

A review on locomotion robophysics: the study of movement at the intersection of robotics, soft matter and dynamical systems

Discovery of fundamental principles which govern and limit effective locomotion (self-propulsion) is of intellectual interest and practical importance. Human technology has created robotic moving systems that excel in movement on and within environments of societal interest: paved roads, open air and water. However, such devices cannot yet robustly and efficiently navigate (as animals do) the enormous diversity of natural environments which might be of future interest for autonomous robots; examples include vertical surfaces like trees and cliffs, heterogeneous ground like desert rubble and brush, turbulent flows found near seashores, and deformable/flowable substrates like sand, mud and soil. In this review we argue for the creation of a physics of moving systems-a 'locomotion robophysics'-which we define as the pursuit of principles of self-generated motion. Robophysics can provide an important intellectual complement to the discipline of robotics, largely the domain of researchers from engineering and computer science. The essential idea is that we must complement the study of complex robots in complex situations with systematic study of simplified robotic devices in controlled laboratory settings and in simplified theoretical models. We must thus use the methods of physics to examine both locomotor successes and failures using parameter space exploration, systematic control, and techniques from dynamical systems. Using examples from our and others' research, we will discuss how such robophysical studies have begun to aid engineers in the creation of devices that have begun to achieve life-like locomotor abilities on and within complex environments, have inspired interesting physics questions in low dimensional dynamical systems, geometric mechanics and soft matter physics, and have been useful to develop models for biological locomotion in complex terrain. The rapidly decreasing cost of constructing robot models with easy access to significant computational power bodes well for scientists and engineers to engage in a discipline which can readily integrate experiment, theory and computation.

Survey of shock-wave structures of smooth-particle granular flows

We show the effects of simulated supersonic granular flow made up of smooth particles passing over two prototypical bodies: a wedge and a disk. We describe a way of computationally identifying shock wave locations in granular flows and tabulate the shock wave locations for flow over wedges and disks. We quantify the shock structure in terms of oblique shock angle for wedge impediments and shock standoff distance for disk impediments. We vary granular flow parameters including upstream volume fraction, average upstream velocity, granular temperature, and the collision coefficient of restitution. Both wedges and disks have been used in the aerospace community as prototypical impediments to flowing air in order to investigate the fundamentally different shock structures emanating from sharp and blunt bodies, and we present these results in order to increase the understanding of the fundamental behavior of supersonic granular flow.

Self-organized shocks in the sedimentation of a granular gas

A granular gas in gravity heated from below develops a certain stationary density profile. When the heating is switched off, the granular gas collapses. We investigate the process of sedimentation using computational hydrodynamics, based on the Jenkins-Richman theory, and find that the process is significantly more complex than generally acknowledged. In particular, during its evolution, the system passes several stages which reveal distinct spatial regions of inertial (supersonic) and diffusive (subsonic) dynamics. During the supersonic stages, characterized by Mach>1, the system develops supersonic shocks which are followed by a steep front of the hydrodynamic fields of temperature and density, traveling upward.

Compaction of highly porous granular matter by impacts on a hard wall

Using a granular-mechanics code, we study the impact of a highly porous granular body on a hard wall. The projectile consists of monodisperse adhesive micrometer-sized silica grains. For the impact velocities studied, v<0.5m/s, the sample does not fragment, but is compacted. We find that the compaction is proportional to the impact speed. The proportionality constant increases with decreasing porosity. However, the compaction is inhomogeneous and decreases with distance from the target. A compaction wave runs through the aggregate; it slows down while the compaction becomes less efficient.

Unstable blast shocks in dilute granular flows

Shocks and blasts can be readily obtained in granular flows be they dense or dilute. Here, by examining the propagation of a blast shock in a dilute granular flow, we show that such a front is unstable with respect to transverse variations of the density of grains. This instability has a well-defined wavelength which depends on the density of the medium and has an amplitude which grows as an exponential of the distance traveled. These features can be understood using a simple model for the shock front, including dissipation which is inherent to granular flows. While this instability bears much resemblance to that anticipated in gases, it is distinct and has special features we discuss here.

The origin of granular convection in vertically vibrated particle beds: The differential shear flow field

This paper investigates the particle scale dynamics of granular convection in vertically vibrated granular beds. The onset of the convection is found to coincide with the noticeable particle transverse migrations from the side walls towards the centre of the bed, which only take place in the wake of the gravity wave front dividing the upward moving particles and the falling ones. The mechanism driving the particle inward flows and thus sustaining the complete convection rolls can be understood in light of a convection model based on void penetration. This stochastic convection model reveals that the underlying driving force is a distinctive differential shear flow field arising from the combined effect of frictional holdback by the walls and the downward pull of gravity. The changes of the convection pattern with inceasing acceleration amplitude, in terms of the convection strength and the thickness of the bottom of the convection rolls, can be accounted for by this model.

Time dependence and density inversion in simulations of vertically oscillated granular layers

We study a layer of grains atop a plate which oscillates sinusoidally in the direction of gravity, using three-dimensional, time-dependent numerical solutions of continuum equations to Navier-Stokes order as well as hard-sphere molecular dynamics simulations. For high accelerational amplitudes of the plate, the layer exhibits a steady-state "density inversion" in which a high-density portion of the layer is supported by a lower-density portion. At low accelerational amplitudes, the layer exhibits oscillatory time dependence that is strongly correlated to the motion of the plate. We show that continuum simulations yield results consistent with molecular dynamics results in both regimes.

Liquid-gas phase separation in confined vibrated dry granular matter

A new phase transition is observed experimentally in a dry granular gas subject to vertical vibration between two horizontal plates. Molecular dynamics simulations of this system allow us to investigate the observed phase separation in detail. We find a high-density, low temperature liquid, coexisting with a low-density, high temperature gas moving coherently. The importance of the coherent motion for phase separation is investigated using frequency modulation.

Continuum simulations of shocks and patterns in vertically oscillated granular layers

We study interactions between shocks and standing-wave patterns in vertically oscillated layers of granular media using three-dimensional, time-dependent numerical solutions of continuum equations to Navier-Stokes order. We simulate a layer of grains atop a plate that oscillates sinusoidally in the direction of gravity. Standing waves form stripe patterns when the accelerational amplitude of the plate's oscillation exceeds a critical value. Shocks also form with each collision between the layer and the plate; we show that pressure gradients formed by these shocks cause the flow to reverse direction within the layer. This reversal leads to an oscillatory state of the pattern that is subharmonic with respect to the plate's oscillation. Finally, we study the relationship between shocks and patterns in layers oscillated at various frequencies and show that the pattern wavelength increases monotonically as the shock strength increases.

Effects of thermal noise on pattern onset in continuum simulations of shaken granular layers

The author investigates the onset of patterns in vertically oscillated layers of dissipative particles using numerical solutions of continuum equations to Navier-Stokes order. Above a critical accelerational amplitude of the cell, standing waves form stripe patterns which oscillate subharmonically with respect to the cell. Continuum simulations neglecting interparticle friction yield pattern wavelengths consistent with experiments using frictional particles. However, the critical acceleration for standing-wave formation is approximately 10% lower in continuum simulations without added noise than in molecular-dynamics simulations. This Brief Report incorporates fluctuating hydrodynamics theory into continuum simulations by adding noise terms with no fit parameters; this modification yields a critical acceleration in agreement with molecular-dynamics simulations.

Blast shocks in quasi-two-dimensional supersonic granular flows

In a thin, dilute, and fast flowing granular layer, the impact of a small sphere generates a fast growing hole devoid of matter. The growth of this hole is studied in detail, and its dynamics is found to mimic that of blast shocks in gases. This dynamics can be decomposed into two stages: a fast initial stage (the blast) and a slower growth regime whose growth velocity is given by the speed of sound in the medium used. A simple model using ingredients already invoked for the case of blast shocks in gases but including the inelastic nature of collisions between grains accounts accurately for our results. The system studied here allows for a detailed study of the full dynamics of a blast as it relaxes from a strong to a weak shock and later to an acoustic disturbance.

Three-dimensional simulations of a vertically vibrated granular bed including interstitial air

We present a numerical study of the effect of interstitial air on a vertically vibrated granular bed within one period of oscillation. We use a three-dimensional molecular-dynamics simulation including air phenomenologically. The simulations are validated with experiments made with spherical glass beads in a rectangular container. After validation, results are reported for a granular column of 9000 grains and approximately 50 layers deep (at rest), agitated with a sinusoidal excitation with maximal acceleration 4.7g at 11.7 Hz. We report the evolution of density, granular temperature, and coordination number within a vibration cycle, and the effect of interstitial air on those parameters. In three-dimensional computer simulations we found that the presence of interstitial air can promote the collective motion of the granular material as a whole.

Experimental measurements of the collapse of a two-dimensional granular gas under gravity

We experimentally measure the decay of a quasi-two-dimensional granular gas under gravity. A granular gas is created by vibrofluidization, after which the energy input is halted, and the time-dependent statistical properties of the decaying gas are measured with video particle tracking. There are two distinct cooling stages separated by a high temperature settling shock. In the final stage, the temperature of a fluid packet decreases as a power law T proportional, variant(t{c}-t);{alpha} just before the system collapses to a static state. The measured value of alpha ranges from 3.3 to 6.1 depending on the height, significantly higher than the exponent of 2 found in theoretical work on this problem [D. Volfson, B. Meerson, and L. S. Tsimring, Phys Rev. E 73, 61305 (2006)]. We also address the question of whether the collapse occurs simultaneously at different heights in the system.

Visualization of collisional substructure in granular shock waves

We study shock wave formation and propagation in an experimental vertically driven quasi-two-dimensional granular gas. We measure the moments of the single particle velocity distribution as a function of space and time. The space-time fields of the velocity moments show acoustic waves with a serrated substructure on the scale of a particle diameter. We show that this substructure is the result of collisional transport in which sequential collisions each transport momentum and energy by one particle diameter.

Self-assembling of dry and cohesive non-Brownian spheres

Numerous experimental and computational studies have been carried out in recent years to understand the mechanisms governing the compaction of granular systems. Here the problem is further investigated from a different perspective. We compact spheres by a vibrational annealing method and show how the interactions between them and the walls determine the final structure. Dry spheres self-assemble only in body-centered-tetragonal structures, while cohesive ones surpass such density and reach the most compact face-centered-cubic phase. We argue that such polymorphism is due to a molecularlike behavior induced by a compensation mechanism between free and vibrational energies.

Scaling and dynamics of sphere and disk impact into granular media

Direct measurements of the acceleration of spheres and disks impacting granular media reveal simple power law scalings along with complex dynamics which bear the signatures of both fluid and solid behavior. The penetration depth scales linearly with impact velocity while the collision duration is constant for sufficiently large impact velocity. Both quantities exhibit power law dependence on sphere diameter and density, and gravitational acceleration. The acceleration during impact is characterized by two jumps: a rapid, velocity-dependent increase upon initial contact and a similarly sharp depth-dependent decrease as the impacting object comes to rest. Examination of the measured forces on the sphere in the vicinity of these features leads to an experimentally based granular force model for collision. We discuss our findings in the context of recently proposed phenomenological models that capture qualitative dynamical features of impact but fail both quantitatively and in their inability to capture significant acceleration fluctuations that occur during penetration and which depend on the impacted material.

Interaction between intruders in vibrated granular beds

Two neutrally buoyant intruder particles in a granular bed fluidized by vertical, sinusoidal vibration are known to interact with each other over a range of about five intruder diameters. Using molecular dynamics simulations, we investigate in detail the spatial and temporal nature of this interaction. We show that the force of attraction between intruders can be calculated from the local density and kinetic energy using a simple equation of state. Moreover, the interaction can be changed from attractive to repulsive by reducing the coefficient of restitution between the intruders and host particles, one of the key results of this work.

NMR measurements and hydrodynamic simulations of phase-resolved velocity distributions within a three-dimensional vibrofluidized granular bed

We report the results of nuclear magnetic resonance imaging experiments on vertically vibrated granular beds of mustard grains. A novel spin-echo velocity profiling technique was developed that allows granular temperature, mean velocity and packing fraction distributions within the three-dimensional cell to be measured as a function of both vertical position and vibration phase. Bimodal velocity distributions were observed at certain portions of the vibration cycle, and in general the ability to acquire time-resolved data demonstrated the significant distortions to the velocity distributions and the systematic errors in calculated temperature distributions that may arise with time-averaged measurements. The experimental behaviour was compared with predictions from a time-varying one-dimensional hydrodynamic model using the experimental parameters as input to the code. In both cases, damping of longitudinal sound waves was linked to significant volume heating effects, which contrasts with the usual heat transport mechanism (i.e. diffusion from the boundaries) currently assumed in most steady-state models. This leads to a new explanation for the counterintuitive upturn in granular temperature in vibrofluidized granular beds, based on amplification and damping of sound waves in the high-altitude region.

Stationary temperature profiles in a liquid nanochannel: comparisons between molecular-dynamics simulation and classical hydrostatics

We compare the results of three-dimensional molecular-dynamics (MD) simulations of a Lennard-Jones (LJ) liquid with a hydrostatic (HS) solution of a high temperature liquid channel which is surrounded by a fluid at lower temperature. The maximum temperature gradient, dT/dx , between the two temperature regions ranged from infinity (step function) to dT/dx=0.1 (in the usual LJ units). Because the systems were in stationary-nonequilibrium states with no fluid flow, both MD simulation and the HS solution gave flat profiles for the normal pressure in all temperature-gradient cases. However, the other quantities showed differences between the two methods. The MD-derived density was found to oscillate over the length of ca. 8 LJ particle diameters from the boundary plane in the system with the infinite temperature gradient, while the HS-derived density showed simply a stepwise profile. The MD simulation also showed another anomaly near the boundary in potential energy. We have found systems in which the HS treatment works well and those where the HS approach breaks down, and therefore established the minimum length scale for the HS treatment to be valid. We also compare the kinetic temperature and the configurational temperature in these systems, and show that these can differ in the transition zone between the two temperatures.

Mechanisms in the size segregation of a binary granular mixture

A granular mixture of particles of two sizes that is shaken vertically will in most cases segregate. If the larger particles accumulate at the top of the sample, this is called the Brazil-nut effect (BNE); if they accumulate at the bottom, it is called the reverse Brazil-nut effect (RBNE). While this process is of great industrial importance in the handling of bulk solids, it is not well understood. In recent years ten different mechanisms have been suggested to explain when each type of segregation is observed. However, the dependence of the mechanisms on driving conditions and material parameters and hence their relative importance is largely unknown. In this paper we present experiments and simulations where both types of particles are made from the same material and shaken under low air pressure, which reduces the number of mechanisms to be considered to seven. We observe both BNE and RBNE by varying systematically the driving frequency and amplitude, diameter ratio, ratio of total volume of small to large particles, and overall sample volume. All our results can be explained by a combination of three mechanisms: a geometrical mechanism called void filling, transport of particles in sidewall-driven convection rolls, and thermal diffusion, a mechanism predicted by kinetic theory.

Shock wave propagation in vibrofluidized granular materials

Shock wave formation and propagation in vertically vibrated quasi-two-dimensional granular materials are studied by digital high speed photography. Steep density and temperature wave fronts form at the bottom of the granular layer when the layer collides with vibrating plate. Then the fronts propagate upwards through the layer. The temperature front is always in the transition region between the upward and downward granular flows. The effects of driving parameters and particle number on the shock are also explored.

Velocity distributions in dilute granular systems

We investigate the idea that velocity distributions in granular gases are determined mainly by eta, the coefficient of restitution and q, which measures the relative importance of heating (or energy input) to collisions. To this end, we study by numerical simulation the properties of inelastic gases as functions of eta, concentration phi, and particle number N with various heating mechanisms. For a wide range of parameters, we find Gaussian velocity distributions for uniform heating and non-Gaussian velocity distributions for boundary heating. Comparison between these results and velocity distributions obtained by other heating mechanisms and for a simple model of a granular gas without spatial degrees of freedom, shows that uniform and boundary heating can be understood as different limits of q, with q>>1 and q < or approximately 1 respectively. We review the literature for evidence of the role of q in the recent experiments.

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

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.

Comparisons between molecular dynamics and hydrodynamics treatment of nonstationary thermal processes in a liquid

Molecular dynamics (MD) and Navier-Stokes hydrodynamics have been performed to model thermal relaxation processes arising from an initially established nonequilibrium stationary state. A nanoscale two-layer Lennard-Jones (LJ) liquid system was constructed in which the two parts were initially at a different temperature, with a narrow transitional zone between the two layers that was spatially linear in temperature. The highest-temperature layer had widths of five or 20 LJ particle diameters. The hydrodynamics model used parametrized MD-derived transport coefficients and the LJ equation of state as input functions. The temporal and spatial temperature and density profiles produced by the two methods show good agreement, indicating that a hydrodynamics description is reliable even for nonstationary phenomena down to the scale of a few molecular diameters. We found that at certain locations the Navier-Stokes solution predicted that the pressure and temperature profiles relaxed in a damped oscillatory manner, which we could discern despite the fluctuations in the MD data.

Mach cone in a shallow granular fluid

We study the V -shaped wake (Mach cone) formed by a cylindrical rod moving through a thin, vertically vibrated granular layer. The wake, analogous to a shock (hydraulic jump) in shallow water, appears for rod velocities vR greater than a critical velocity c . We measure the half angle theta; of the wake as a function of vR and layer depth h . The angle satisfies the Mach relation, sin theta=c/vR , where c= square root of gh , even for h as small as one-particle diameter.

Noise, coherent fluctuations, and the onset of order in an oscillated granular fluid

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 an order of magnitude larger than the thermal noise in the most sensitive convecting fluid experiments to date; the effect of the granular noise is observable 20% below the onset of order.

NMR experiments on a three-dimensional vibrofluidized granular medium

A three-dimensional granular system fluidized by vertical container vibrations was studied using pulsed field gradient NMR coupled with one-dimensional magnetic resonance imaging. The system consisted of mustard seeds vibrated vertically at 50 Hz, and the number of layers N(l)<or=4 was sufficiently low to achieve a nearly time-independent granular fluid. Using NMR, the vertical profiles of density and granular temperature were directly measured, along with the distributions of vertical and horizontal grain velocities. The velocity distributions showed modest deviations from Maxwell-Boltzmann statistics, except for the vertical velocity distribution near the sample bottom, which was highly skewed and non-Gaussian. Data taken for three values of N(l) and two dimensionless accelerations Gamma=15,18 were fitted to a hydrodynamic theory, which successfully models the density and temperature profiles away from the vibrating container bottom. A temperature inversion near the free upper surface is observed, in agreement with predictions based on the hydrodynamic parameter micro which is nonzero only in inelastic systems.

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.

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.

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.