Hydrodynamics of an open vibrated granular system

Rescaling invariance and anomalous energy transport in a small vertical column of grains

It is well known that energy dissipation and finite size can deeply affect the dynamics of granular matter, often making usual hydrodynamic approaches problematic. Here we report on the experimental investigation of a small model system, made of ten beads constrained into a 1D geometry by a narrow vertical pipe and shaken at the base by a piston excited by a periodic wave. Recording the beads motion with a high frame rate camera allows to investigate in detail the microscopic dynamics and test hydrodynamic and kinetic models. Varying the energy, we explore different regimes from fully fluidized to the edge of condensation, observing good hydrodynamic behavior down to the edge of fluidization, despite the small system size. Density and temperature fields for different system energies can be collapsed by suitable space and time rescaling, and the expected constitutive equation holds very well when the particle diameter is considered. At the same time, the balance between dissipated and fed energy is not well described by commonly adopted dependence due to the up-down symmetry breaking. Our observations, supported by the measured particle velocity distributions, show a different phenomenological temperature dependence, which yields equation solutions in agreement with experimental results.

Hydrodynamics of granular particles on a line

We investigate a lattice model representing a granular gas in a thin channel. We deduce the hydrodynamic description for the model from the microscopic dynamics in the large-system limit, including the lowest finite-size corrections. The main prediction from hydrodynamics, when finite-size corrections are neglected, is the existence of a steady "uniform longitudinal flow" (ULF), with the granular temperature and the velocity gradient both uniform and directly related. Extensive numerical simulations of the system show that such a state can be observed in the bulk of a finite-size system by attaching two thermostats with the same temperature at its boundaries. The relation between the ULF state and the shocks appearing in the late stage of a cooling gas of inelastic hard rods is discussed.

Effect of particle stiffness on contact dynamics and rheology in a dense granular flow

Dense granular flows have been well described by the Bagnold rheology, even when the particles are in the multibody contact regime and the coordination number is greater than 1. This is surprising, because the Bagnold law should be applicable only in the instantaneous collision regime, where the time between collisions is much larger than the period of a collision. Here, the effect of particle stiffness on rheology is examined. It is found that there is a rheological threshold between a particle stiffness of 10^{4}-10^{5} for the linear contact model and 10^{5}-10^{6} for the Hertzian contact model above which Bagnold rheology (stress proportional to square of the strain rate) is valid and below which there is a power-law rheology, where all components of the stress and the granular temperature are proportional to a power of the strain rate that is less then 2. The system is in the multibody contact regime at the rheological threshold. However, the contact energy per particle is less than the kinetic energy per particle above the rheological threshold, and it becomes larger than the kinetic energy per particle below the rheological threshold. The distribution functions for the interparticle forces and contact energies are also analyzed. The distribution functions are invariant with height, but they do depend on the contact model. The contact energy distribution functions are well fitted by Gamma distributions. There is a transition in the shape of the distribution function as the particle stiffness is decreased from 10^{7} to 10^{6} for the linear model and 10^{8} to 10^{7} for the Hertzian model, when the contact number exceeds 1. Thus, the transition in the distribution function correlates to the contact regime threshold from the binary to multibody contact regime, and is clearly different from the rheological threshold. An order-disorder transition has recently been reported in dense granular flows. The Bagnold rheology applies for both the ordered and disordered states, even though the rheological constants differ by orders of magnitude. The effect of particle stiffness on the order-disorder transition is examined here. It is found that when the particle stiffness is above the rheological threshold, there is an order-disorder transition as the base roughness is increased. The order-disorder transition disappears after the crossover to the soft-particle regime when the particle stiffness is decreased below the rheological threshold, indicating that the transition is a hard-particle phenomenon.

Criticality in Vibrated Frictional Flows at a Finite Strain Rate

We evidence critical fluctuations in the strain rate of granular flows that are weakly vibrated. Strikingly, the critical point arises at finite values of the mean strain rate and vibration strength, far from the yielding critical point at a zero flow rate. We show that the global rheology, as well as the amplitude and correlation time of the fluctuations, are consistent with a mean-field, Landau-like description, where the strain rate and the stress act as conjugated variables. We introduce a general model which captures the observed phenomenology and argue that this type of critical behavior generically arises when self-fluidization competes with friction.

Thermal Convection in Granular Gases with Dissipative Lateral Walls

We consider a granular gas under the action of gravity, fluidized by a vibrating base. We show that a horizontal temperature gradient, here induced by limiting dissipative lateral walls (DLW), leads always to a granular thermal convection (DLW TC) that is essentially different from ordinary bulk-buoyancy-driven convection (BBD TC). In an experiment where BBD TC is inhibited, by reducing gravity with an inclined plane, we always observe a DLW TC cell next to each lateral wall. Such a cell squeezes towards the nearest wall as the gravity and/or the number of grains increase. Molecular dynamics simulations reproduce the experimental results and indicate that at large gravity or number of grains the DLW TC is barely detectable.

Pattern transition, microstructure, and dynamics in a two-dimensional vibrofluidized granular bed

Experiments are conducted in a two-dimensional monolayer vibrofluidized bed of glass beads, with a goal to understand the transition scenario and the underlying microstructure and dynamics in different patterned states. At small shaking accelerations (Γ=Aω^{2}/g<1, where A and ω=2πf are the amplitude and angular frequency of shaking and g is the gravitational acceleration), the particles remain attached to the base of the vibrating container; this is known as the solid bed (SB). With increasing Γ (at large enough shaking amplitude A/d) and/or with increasing A/d (at large enough Γ), the sequence of transitions/bifurcations unfolds as follows: SB ("solid bed") to BB ("bouncing bed") to LS ("Leidenfrost state") to "2-roll convection" to "1-roll convection" and finally to a gas-like state. For a given length of the container, the coarsening of multiple convection rolls leading to the genesis of a "single-roll" structure (dubbed the multiroll transition) and its subsequent transition to a granular gas are two findings of this work. We show that the critical shaking intensity (Γ_{BB}^{LS}) for the BB→LS transition has a power-law dependence on the particle loading (F=h_{0}/d, where h_{0} is the number of particle layers at rest and d is the particle diameter) and the shaking amplitude (A/d). The characteristics of BB and LS states are studied by calculating (i) the coarse-grained density and temperature profiles and (ii) the pair correlation function. It is shown that while the contact network of particles in the BB state represents a hexagonal-packed structure, the contact network within the "floating cluster" of the LS resembles a liquid-like state. An unsteadiness of the Leidenfrost state has been uncovered wherein the interface (between the floating cluster and the dilute collisional layer underneath) and the top of the bed are found to oscillate sinusoidally, with the oscillation frequency closely matching the frequency of external shaking. Therefore, the granular Leidenfrost state is a period-1 wave as is the case for the BB state.

Statistics of the dissipated energy in driven diffusive systems

Understanding the physics of non-equilibrium systems remains one of the major open questions in statistical physics. This problem can be partially handled by investigating macroscopic fluctuations of key magnitudes that characterise the non-equilibrium behaviour of the system of interest; their statistics, associated structures and microscopic origin. During the last years, some new general and powerful methods have appeared to delve into fluctuating behaviour that have drastically changed the way to address this problem in the realm of diffusive systems: macroscopic fluctuation theory (MFT) and a set of advanced computational techniques that make it possible to measure the probability of rare events. Notwithstanding, a satisfactory theory is still lacking in a particular case of intrinsically non-equilibrium systems, namely those in which energy is not conserved but dissipated continuously in the bulk of the system (e.g. granular media). In this work, we put forward the dissipated energy as a relevant quantity in this case and analyse in a pedagogical way its fluctuations, by making use of a suitable generalisation of macroscopic fluctuation theory to driven dissipative media.

Memory effects in the relaxation of a confined granular gas

The accuracy of a model to describe the horizontal dynamics of a confined quasi-two-dimensional system of inelastic hard spheres is discussed by comparing its predictions for the relaxation of the temperature in a homogenous system with molecular dynamics simulation results for the original system. A reasonably good agreement is found. Next the model is used to investigate the peculiarities of the nonlinear evolution of the temperature when the parameter controlling the energy injection is instantaneously changed while the system was relaxing. This can be considered as a nonequilibrium generalization of the Kovacs effect. It is shown that, in the low-density limit, the effect can be accurately described by using a simple kinetic theory based on the first Sonine approximation for the one-particle distribution function. Some possible experimental implications are indicated.

Nuclear magnetic resonance measurements of velocity distributions in an ultrasonically vibrated granular bed

We report the results of nuclear magnetic resonance imaging experiments on granular beds of mustard grains fluidized by vertical vibration at ultrasonic frequencies. The variation of both granular temperature and packing fraction with height was measured within the three-dimensional cell for a range of vibration frequencies, amplitudes and numbers of grains. Small increases in vibration frequency were found--contrary to the predictions of classical 'hard-sphere' expressions for the energy flux through a vibrating boundary--to result in dramatic reductions in granular temperature. Numerical simulations of the grain-wall interactions, using experimentally determined Hertzian contact stiffness coefficients, showed that energy flux drops significantly as the vibration period approaches the grain-wall contact time. The experiments thus demonstrate the need for new models for 'soft-sphere' boundary conditions at ultrasonic frequencies.

Homogeneous steady state of a confined granular gas

The nonequilibrium statistical mechanics and kinetic theory for a model of a confined quasi-two-dimensional gas of inelastic hard spheres is presented. The dynamics of the particles includes an effective mechanism to transfer the energy injected in the vertical direction to the horizontal degrees of freedom. The Enskog approximation is formulated and used as the basis to investigate the temperature and the distribution function of the steady state eventually reached by the system. An exact scaling of the distribution function of the system having implications on the form of its moments is pointed out. The theoretical predictions are compared with numerical results obtained by a particle simulation method, and a good agreement is found.

Thermal convection and temperature inhomogeneity in a vibrofluidized granular bed: the influence of sidewall dissipation

Using a vertically vibrated, fully three-dimensional granular system, we investigate the impact of dissipative interactions between the particles in the system and the vertical sidewalls bounding it. We find that sidewall dissipation influences various properties of the bed including, but not limited to, the spatial distribution of granular temperatures, the functional form of velocity distributions, and the strength of convection. Simple, monotonic relationships are observed for all the aforementioned properties, including a striking linear relationship between convection strength and wall dissipation. We conclude that sidewall effects are not limited to the vicinity of the walls themselves, but extend into the bulk of the system and hence must be considered even in relatively wide, three-dimensional systems. We also propose the possibility of using the alteration of sidewall material as a method of "tuning" certain system parameters in situations where changing the bulk properties or driving parameters of a granular system may be undesirable.

Dynamics of the wet granular Leidenfrost phenomenon

By event-driven molecular dynamics simulations, we study the Leidenfrost effect for wet granular matter driven from below. In marked contrast to all earlier studies on other fluids, the dense plug hovering on the hot gas cushion undergoes an undamped oscillation. The location of the Hopf bifurcation leading to this oscillation is strongly dependent on the inelasticity of the grain impacts. The vertical separation into a gas phase with a condensed plug hovering above it is particularly pronounced due to the cohesiveness of the granulate. For sufficiently large system sizes, the Rayleigh-Taylor instability terminates the oscillatory state at late times.

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.

Fluctuation-dissipation relations for motions of center of mass in driven granular fluids under gravity

We investigated the validity of fluctuation-dissipation relations in the nonequilibrium stationary state of fluidized granular media under gravity by two independent approaches, based on theory and numerical simulations. A phenomenological Langevin-type theory describing the fluctuation of center of mass height, which was originally constructed for a one-dimensional granular gas on a vibrating bottom plate, was generalized to any dimensionality, even for the case in which the vibrating bottom plate is replaced by a thermal wall. The theory predicts a fluctuation-dissipation relation known to be satisfied at equilibrium, with a modification that replaces the equilibrium temperature by an effective temperature defined by the center of mass kinetic energy. To test the validity of the fluctuation-dissipation relation, we performed extensive and accurate event-driven molecular dynamics simulations for the model system with a thermal wall at the bottom. The power spectrum and response function of the center of mass height were measured and closely compared with theoretical predictions. It is shown that the fluctuation-dissipation relation for the granular system is satisfied, especially in the high-frequency (short time) region, for a wide range of system parameters. Finally, we describe the relationship between systematic deviations in the low-frequency (long time) region and the time scales of the driven granular system.

Segregation of an intruder in a heated granular dense gas

A recent segregation criterion [Phys. Rev. E 78, 020301(R) (2008)] based on the thermal diffusion factor Λ of an intruder in a heated granular gas described by the inelastic Enskog equation is revisited. The sign of Λ provides a criterion for the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE). The present theory incorporates two extra ingredients not accounted for by the previous theoretical attempt. First, the theory is based upon the second Sonine approximation to the transport coefficients of the mass flux of the intruder. Second, the dependence of the temperature ratio (intruder temperature over that of the host granular gas) on the solid volume fraction is taken into account in the first and second Sonine approximations. In order to check the accuracy of the Sonine approximation considered, the Enskog equation is also numerically solved by means of the direct simulation Monte Carlo method to get the kinetic diffusion coefficient D(0). The comparison between theory and simulation shows that the second Sonine approximation to D(0) yields an improvement over the first Sonine approximation when the intruder is lighter than the gas particles in the range of large inelasticity. With respect to the form of the phase diagrams for the BNE-RBNE transition, the kinetic theory results for the factor Λ indicate that while the form of these diagrams depends sensitively on the order of the Sonine approximation considered when gravity is absent, no significant differences between both Sonine solutions appear in the opposite limit (gravity dominates the thermal gradient). In the former case (no gravity), the first Sonine approximation overestimates both the RBNE region and the influence of dissipation on thermal diffusion segregation.

Fluctuating hydrodynamics in a vertically vibrated granular fluid with gravity

We investigate hydrodynamic fluctuations in a two-dimensional granular fluid excited by a vibrating base and in the presence of gravity, focusing on the transverse velocity modes. Since the system is inhomogeneous, we measure fluctuations in horizontal layers whose width is smaller than the characteristic hydrodynamic lengths: they can be considered as almost-homogeneous subsystems. The large time decay of the autocorrelations of modes is exponential and compatible with vorticity diffusion due to shear viscosity, as in equilibrium fluids. The velocity structure factor, which strongly deviates from the equilibrium constant behavior, is well reproduced by an effective fluctuating hydrodynamics described by two noise terms: the first associated with vorticity diffusion and the second with the local energy exchange, which have internal and external character, respectively.

Cooling rates and energy partition in inhomogeneous fluidized granular mixtures

The local cooling rates of the components of a vibrated binary granular mixture in a steady state are investigated. The accuracy of the expression obtained by assuming a local homogeneous cooling state distribution of the gas is analyzed by comparing it with molecular dynamics simulation results. A good agreement is observed. Also, the profiles of the partial temperatures are compared with the theoretical prediction following from the application of the Chapman-Enskog method to solve the kinetic Enskog equations of the mixture. In this case, the agreement is satisfactory if the boundary layers near the walls are excluded. The implications of the results are discussed.

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.

Onset of convection in strongly shaken granular matter

Strongly vertically shaken granular matter can display a density inversion: A high-density cluster of beads is elevated by a dilute gaslike layer of fast beads underneath ("granular Leidenfrost effect"). For even stronger shaking the granular Leidenfrost state becomes unstable and granular convection rolls emerge. This transition resembles the classical onset of convection in fluid heated from below at some critical Rayleigh number. The same transition is seen in molecular dynamics (MD) simulations of the shaken granular material. The critical shaking strength for the onset of granular convection can be calculated from a linear stability analysis of a hydrodynamiclike model of the granular flow. Experiment, MD simulations, and theory quantitatively agree.

Hydrodynamic character of the nonequipartition of kinetic energy in binary granular gases

The influence of the heating mechanism on the kinetic energy densities of the components of a vibrated granular mixture is investigated. Collisions of the particles with the vibrating wall are inelastic and characterized by two coefficients of normal restitution, one for each of the two species. By means of molecular-dynamics simulations, it is shown that the nonequipartition of kinetic energy is not affected by the differential mechanism of energy injection aside the usual boundary layer around the wall. The macroscopic state of the mixture in the bulk is defined by intensive variables that do not include the partial granular temperatures of the components.

Vibrated granular gas confined by a piston

The steady state of a vibrated granular gas confined by a movable piston on top is discussed. Particular attention is given to the hydrodynamic boundary conditions to be used when solving the inelastic Navier-Stokes equations. The relevance of an exact general condition relating the grain fluxes approaching and moving away from each of the walls is emphasized. It is shown how it can be used to get a consistent hydrodynamic description of the boundaries. The obtained expressions for the fields do not contain any undetermined parameter. Comparison of the theoretical predictions with molecular-dynamics simulation results is carried out, and a good agreement is observed for low density and not too large inelasticity. A practical way of introducing small finite density corrections to the dilute limit theory is proposed to improve the accuracy of the theory.

Driven inelastic-particle systems with drag

We study steady states of the motion of a large number of particles in a closed box that are excited by a vibrating boundary and experience a linear drag force from the interstitial fluid. The dissipation in such systems arises from two main sources: Inelasticity in particle collisions and the effects of interstitial fluid on the particles. In many applications, order of magnitude estimates suggest that the dissipation due to interstitial fluid effects may greatly exceed that due to inelasticity and one is naturally led to neglect inelastic effects. In this study, we show that, if one adopts a linear drag force and inelastic effects are neglected, a steady state only exists when the vibration speed of the boundary is below a critical value. For vibration speeds above this critical value, no steady state exists since the kinetic energy of the particles grows without bound. We show that, for vibration speeds above the critical value, inelastic effects must be included to obtain a steady state even if order of magnitude estimates suggest they are negligible. Numerical simulations confirm these theoretical predictions. We also show that inclusion of apparently small nonlinear drag terms can also play a similar role in preventing the kinetic energy of the particles growing without bound.

Linear response and hydrodynamics for granular fluids

A formal derivation of linear hydrodynamics for a granular fluid is given. The linear response to small spatial perturbations of a homogeneous reference state is studied in detail, using methods of nonequilibrium statistical mechanics. A transport matrix for macroscopic excitations in the fluid is defined in terms of the response functions. An expansion in the wave vector to second order allows identification of all phenomenological susceptibilities and transport coefficients through Navier-Stokes order in terms of appropriate time correlation functions. The transport coefficients in this representation are the generalization to granular fluids of the familiar Helfand and Green-Kubo relations for normal fluids. The analysis applies to a variety of collision rules. Important differences in both the analysis and results from those for normal fluids are identified and discussed. A scaling limit is described corresponding to the conditions under which idealized inelastic hard sphere models can apply. Further details and interpretation are provided in the paper following this one, by specialization to the case of smooth, inelastic hard spheres with constant coefficient of restitution.

Enskog theory for polydisperse granular mixtures. II. Sonine polynomial approximation

The linear integral equations defining the Navier-Stokes (NS) transport coefficients for polydisperse granular mixtures of smooth inelastic hard disks or spheres are solved by using the leading terms in a Sonine polynomial expansion. Explicit expressions for all the NS transport coefficients are given in terms of the sizes, masses, compositions, density, and restitution coefficients. In addition, the cooling rate is also evaluated to first order in the gradients. The results hold for arbitrary degree of inelasticity and are not limited to specific values of the parameters of the mixture. Finally, a detailed comparison between the derivation of the current theory and previous theories for mixtures is made, with attention paid to the implication of the various treatments employed to date.

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.

Hydrodynamic profiles for an impurity in an open vibrated granular gas

The hydrodynamic state of an impurity immersed in a low density granular gas is analyzed. Explicit expressions for the temperature and density fields of the impurity in terms of the hydrodynamic fields of the gas are derived. It is shown that the ratio between the temperatures of the two components, measuring the departure from the energy equipartition, only depends on the mechanical properties of the particles, being therefore constant in the bulk of the system. This ratio plays an important role in determining the density profile of the intruder and its position with respect to the gas, since it determines the sign of the pressure diffusion coefficient. The theoretical predictions are compared with molecular dynamics simulation results for the particular case of the steady state of an open vibrated granular system in the absence of macroscopic fluxes, and a satisfactory agreement is found.

Granular Leidenfrost effect: experiment and theory of floating particle clusters

Granular material is vertically vibrated in a 2D container: above a critical shaking strength, and for a sufficient number of beads, a crystalline cluster is elevated and supported by a dilute gaseous layer of fast beads underneath. We call this phenomenon the granular Leidenfrost effect. The experimental observations are explained by a hydrodynamic model featuring three dimensionless control parameters: the energy input S, the number of particle layers F, and the inelasticity of the particle collisions epsilon. The (S,F) phase diagram, in which the Leidenfrost state lies between the purely solid and gas phases, shows accurate agreement between experiment and theory.

Energy partition and segregation for an intruder in a vibrated granular system under gravity

The difference of temperatures between an impurity and the surrounding gas in an open vibrated granular system is studied. It is shown that, in spite of the high inhomogeneity of the state, the temperature ratio remains constant in the bulk of the system. The lack of energy equipartition is associated to the change of sign of the pressure diffusion coefficient for the impurity at certain values of the parameters of the system, leading to a segregation criterium. The theoretical predictions are consistent with previous experimental results, and also in agreement with molecular dynamics simulation results reported in this Letter.

Convection in vibrated annular granular beds

The response to vibration of a granular bed, consisting of a standard cylindrical geometry but with the addition of a dissipative cylindrical inner wall, has been investigated both experimentally (using positron emission particle tracking) and numerically (using hard sphere molecular dynamics simulation). The packing fraction profiles and granular temperature distributions (in both vertical and horizontal directions) were determined as a function of height and distance from the axis. The two sets of results were in reasonable agreement. The molecular dynamics simulations were used to explore the behavior of the granular bed in the inner wall-outer wall coefficient of restitution phase space. It was observed that one could control the direction of the toroidal convection rolls by manipulating the relative dissipation at the inner and outer walls via the coefficients of restitution, and with several layers of grains it was seen that double convection rolls could also be formed, a result that was subsequently confirmed experimentally.

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.

Competitive clustering in a bidisperse granular gas: experiment, molecular dynamics, and flux model

A compartmentalized bidisperse granular gas clusters competitively [Phys. Rev. Lett. 89, 214301 (2002)]]: By tuning the shaking strength, the clustering can be directed either towards the compartment initially containing mainly small particles or to the compartment containing mainly large particles. Here, the conditions under which this competitive clustering occurs are studied experimentally, numerically (by means of molecular dynamics simulations), and analytically. A minimal model is derived that quantitatively accounts for the observed phenomena.

Anisotropic energy distribution in three-dimensional vibrofluidized granular systems

We examine the energy flows in a three-dimensional model of a granular system consisting of N inelastic hard spheres contained in an open cylinder of radius R under the influence of gravity. Energy is supplied to the system in the vertical direction by a vibrating base and is transferred to the perpendicular directions through particle-particle collisions. We examine how the local and global dissipation of energy by particle-particle and particle-wall collisions depends on the number of particles, the velocity of the vibrating base, and the restitution coefficients.

Gaussian kinetic model for granular gases

A kinetic model for the Boltzmann equation is proposed and explored as a practical means to investigate the properties of a dilute granular gas. It is shown that all spatially homogeneous initial distributions approach a universal "homogeneous cooling solution" after a few collisions. The homogeneous cooling solution (HCS) is studied in some detail and the exact solution is compared with known results for the hard sphere Boltzmann equation. It is shown that all qualitative features of the HCS, including the nature of overpopulation at large velocities, are reproduced by the kinetic model. It is also shown that all the transport coefficients are in excellent agreement with those from the Boltzmann equation. Also, the model is specialized to one having a velocity independent collision frequency and the resulting HCS and transport coefficients are compared to known results for the Maxwell model. The potential of the model for the study of more complex spatially inhomogeneous states is discussed.

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.

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.

Mass and size effects in three-dimensional vibrofluidized granular mixtures

We examine the steady state properties of binary systems of driven inelastic hard spheres. The spheres, which move under the influence of gravity, are contained in a vertical cylinder with a vibrating base. We computed the trajectories of the spheres using an event-driven molecular dynamics algorithm. In the first part of the study, we chose simulation parameters that match those of experiments published by Wildman and Parker. Various properties computed from the simulation including the density profile, granular temperature, and circulation pattern are in good qualitative agreement with the experiments. We then studied the effect of varying the mass ratio and the size ratio independently while holding the other parameters constant. The mass and size ratio are shown to affect the distribution of the energy. The changes in the energy distributions affect the packing fraction and temperature of each component. The temperature of the heavier component has a nonlinear dependence on the mass of the lighter component, while the temperature of the lighter component is approximately proportional to its mass. The temperature of both components is inversely dependent on the size of the smaller component.

Close-packed floating clusters: granular hydrodynamics beyond the freezing point?

Monodisperse granular flows often develop regions with hexagonal close packing of particles. We investigate this effect in a system of inelastic hard spheres driven from below by a "thermal" plate. Molecular dynamics simulations show, in a wide range of parameters, a close-packed cluster supported by a low-density region. Surprisingly, the steady-state density profile, including the close-packed cluster part, is well described by a variant of Navier-Stokes granular hydrodynamics (NSGH). We suggest a simple explanation for the success of NSGH beyond the freezing point.

Velocity distribution of fluidized granular gases in the presence of gravity

The velocity distribution of a fluidized dilute granular gas in the direction perpendicular to the gravitational field is investigated by means of molecular dynamics simulations. The results indicate that the velocity distribution can be exactly described neither by a Gaussian nor by a stretched exponential law. Moreover, it does not exhibit any kind of scaling. In fact, the actual shape of the distribution depends on the number of monolayers at rest, on the restitution coefficient and on the height at what it is measured. The role played by the number of particle-particle collisions as compared with the number of particle-wall collisions is discussed.

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.

Molecular dynamics simulations of vibrated granular gases

We present molecular dynamics simulations of monodisperse or bidisperse inelastic granular gases driven by vibrating walls, in two dimensions (without gravity). Because of the energy injection at the boundaries, a situation often met experimentally, density and temperature fields display heterogeneous profiles in the direction perpendicular to the walls. A general equation of state for an arbitrary mixture of fluidized inelastic hard spheres is derived and successfully tested against numerical data. Single-particle velocity distribution functions with non-Gaussian features are also obtained, and the influence of various parameters (inelasticity coefficients, density, etc.) are analyzed. The validity of a recently proposed random restitution coefficient model is assessed through the study of projected collisions onto the direction perpendicular to that of energy injection. For the binary mixture, the nonequipartition of translational kinetic energy is studied and compared both to experimental data and to the case of homogeneous energy injection ("stochastic thermostat"). The rescaled velocity distribution functions are found to be very similar for both species.

Wall-enhanced convection in vibrofluidized granular systems

An event-driven molecular dynamics simulation of inelastic hard spheres contained in a cylinder and subject to strong vibration reproduces accurately experimental results [R. D. Wildman et al., Phys. Rev. Lett. 86, 3304 (2001)] for a system of vibrofluidized glass beads. In particular, we are able to obtain the velocity field and the density and temperature profiles observed experimentally. In addition, we show that the appearance of convection rolls is strongly influenced by the value of the sidewall-particle restitution coefficient. Suggestions for observing more complex convection patterns are proposed.

Measurements of grain motion in a dense, three-dimensional granular fluid

We have used an NMR technique to measure the short-time, three-dimensional displacement of grains in a system of mustard seeds vibrated vertically at 15 g. The technique averages over a time interval in which the grains move ballistically, giving a direct measurement of the granular temperature profile. The dense, lower portion of the sample is well described by a recent hydrodynamic theory for inelastic hard spheres. Near the free upper surface the mean free path is longer than the particle diameter and the hydrodynamic description fails.