Dynamics of a freely evolving, two-dimensional granular medium

Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. II. Stability analysis

Conditions for the stability under linear perturbations around the homogeneous cooling state are studied for dilute granular gases of inelastic and rough hard disks or spheres with constant coefficients of normal (α) and tangential (β) restitution. After a formally exact linear stability analysis of the Navier-Stokes-Fourier hydrodynamic equations in terms of the translational (d_{t}) and rotational (d_{r}) degrees of freedom, the transport coefficients derived in the companion paper [A. Megías and A. Santos, "Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. I. Transport coefficients" Phys. Rev. E 104, 034901 (2021)10.1103/PhysRevE.104.034901] are employed. Known results for hard spheres [Garzó, Santos, and Kremer, Phys. Rev. E 97, 052901 (2018)10.1103/PhysRevE.97.052901] are recovered by setting d_{t}=d_{r}=3, while novel results for hard disks (d_{t}=2, d_{r}=1) are obtained. In the latter case, a high-inelasticity peculiar region in the (α,β) parameter space is found, inside which the critical wave number associated with the longitudinal modes diverges. Comparison with event-driven molecular dynamics simulations for dilute systems of hard disks at α=0.2 shows that this theoretical region of absolute instability may be an artifact of the extrapolation to high inelasticity of the approximations made in the derivation of the transport coefficients, although it signals a shrinking of the conditions for stability. In the case of moderate inelasticity (α=0.7), however, a good agreement between the theoretical predictions and the simulation results is found.

On the apparent yield stress in non-Brownian magnetorheological fluids

We use simulations to probe the flow properties of dense two-dimensional magnetorheological fluids. Prior results from both experiments and simulations report that the shear stress σ scales with strain rate [small gamma, Greek, dot above] as σ ∼ [small gamma, Greek, dot above]^1-Δ, with values of the exponent ranging between 2/3 < Δ ≤ 1. However it remains unclear what properties of the system select the value of Δ, and in particular under what conditions the system displays a yield stress (Δ = 1). To address these questions, we perform simulations of a minimalistic model system in which particles interact via long ranged magnetic dipole forces, finite ranged elastic repulsion, and viscous damping. We find a surprising dependence of the apparent exponent Δ on the form of the viscous force law. For experimentally relevant values of the volume fraction ϕ and the dimensionless Mason number Mn (which quantifies the competition between viscous and magnetic stresses), models using a Stokes-like drag force show Δ ≈ 0.75 and no apparent yield stress. When dissipation occurs at the contact, however, a clear yield stress plateau is evident in the steady state flow curves. In either case, increasing ϕ towards the jamming transition suffices to induce a yield stress. We relate these qualitatively distinct flow curves to clustering mechanisms at the particle scale. For Stokes-like drag, the system builds up anisotropic, chain-like clusters as Mn tends to zero (vanishing strain rate and/or high field strength). For contact damping, by contrast, there is a second clustering mechanism due to inelastic collisions.

Shock propagation in locally driven granular systems

We study shock propagation in a system of initially stationary hard spheres that is driven by a continuous injection of particles at the origin. The disturbance created by the injection of energy spreads radially outward through collisions between particles. Using scaling arguments, we determine the exponent characterizing the power-law growth of this disturbance in all dimensions. The scaling functions describing the various physical quantities are determined using large-scale event-driven simulations in two and three dimensions for both elastic and inelastic systems. The results are shown to describe well the data from two different experiments on granular systems that are similarly driven.

Effective potentials in a bidimensional vibrated granular gas

We present a numerical study of the spatial correlations of a quasi-two-dimensional granular fluid kept in a nonstatic steady state via vertical shaking. The simulations explore a wide range of vertical accelerations, restitution coefficients, and packing fractions, always staying below the crystallization limit. From the simulations we obtain the relevant pair distribution functions (PDFs), and effective potentials for the interparticle interaction are extracted from these PDFs via the Ornstein-Zernike equation with the Percus-Yevick closure. The correlations in the granular structures originating from these effective potentials are checked against the originating PDF using standard Monte Carlo simulations, and we find in general an excellent agreement. The resulting effective potentials show an increase of the spatial correlation at contact with the decreasing values of the restitution coefficient, and a tendency of the potentials to display deeper wells for more dissipative dynamics. A general exception to this trend appears for a range of values of the forcing, which depends on the restitution coefficient, but not on the density, where resonant bouncing increases correlations, resulting in deeper potential wells. The nature of these resonances is explored and shown to be the result of synchronization in the parabolic flights of the particles.

Kinetic theory for dilute cohesive granular gases with a square well potential

We develop the kinetic theory of dilute cohesive granular gases in which the attractive part is described by a square well potential. We derive the hydrodynamic equations from the kinetic theory with the microscopic expressions for the dissipation rate and the transport coefficients. We check the validity of our theory by performing the direct simulation Monte Carlo.

Transport coefficients of solid particles immersed in a viscous gas

Transport properties of a suspension of solid particles in a viscous gas are studied. The dissipation in such systems arises from two sources: inelasticity in particle collisions and viscous dissipation due to the effect of the gas phase on the particles. Here we consider a simplified case in which the mean relative velocity between the gas and solid phases is taken to be zero, such that "thermal drag" is the only remaining gas-solid interaction. Unlike the previous, more general, treatment of the drag force [Garzó et al., J. Fluid Mech. 712, 129 (2012)]JFLSA70022-112010.1017/jfm.2012.404, here we take into account contributions to the (scaled) transport coefficients η^{*} (shear viscosity), κ^{*} (thermal conductivity), and μ^{*} (Dufour-like coefficient) coming from the temperature dependence of the (dimensionless) friction coefficient γ^{*} characterizing the amplitude of the drag force. At moderate densities, the thermal drag model (which is based on the Enskog kinetic equation) is solved by means of the Chapman-Enskog method and the Navier-Stokes transport coefficients are determined in terms of the coefficient of restitution, the solid volume fraction, and the friction coefficient. The results indicate that the effect of the gas phase on η^{*} and μ^{*} is non-negligible (especially in the case of relatively dilute systems) while the form of κ^{*} is the same as the one obtained in the dry granular limit. Finally, as an application of these results, a linear stability analysis of the hydrodynamic equations is carried out to analyze the conditions for stability of the homogeneous cooling state. A comparison with direct numerical simulations shows a good agreement for conditions of practical interest.

Simulation of cohesive fine powders under a plane shear

Three-dimensional molecular-dynamics simulations of cohesive dissipative powders under a plane shear are performed. We find the various phases depending on the dimensionless shear rate and the dissipation rate as well as the density. We also find that the shape of clusters depends on the initial condition of velocities of particles when the dissipation is large. Our simple stochastic model reproduces the non-Gaussian velocity distribution function appearing in the coexistence phase of a gas and a plate.

Effect of cohesion on shear banding in quasistatic granular materials

Cohesive powders have widely different bulk behavior due to their peculiar interactions. We use discrete element simulations to investigate the effect of contact cohesion on the steady state flow of dense powders in a slowly sheared split-bottom Couette cell, which imposes a wide stable shear band. The intensity of cohesive forces can be quantified by the granular Bond number (Bo), namely the ratio between maximum attractive force and average force due to external compression. We find that the shear banding phenomenon is almost independent of cohesion for Bond numbers Bo<1, however for Bo≥1 cohesive forces start to play an important role, as both width and center position of the band increase. Inside the shear band, the mean normal contact force is independent of cohesion and depends only on the confining stress. In contrast, when the behavior is analyzed focusing on the eigendirections of the local strain rate tensor, a dependence on cohesion shows up. Forces carried by contacts along the compressive and tensile directions are symmetric about the mean force (larger and smaller respectively), while the force along the third, neutral direction follows the mean force. This anisotropy of the force network increases with cohesion, just like the heterogeneity in all (compressive, tensile and neutral) directions.

Higher-order effects on orientational correlation and relaxation dynamics in homogeneous cooling of a rough granular gas

The orientational or angular correlation between the directions of the translational and rotational motions is analyzed theoretically for the homogeneous cooling state of a rough granular gas. The dynamical equations are derived using an approximate form of the single-particle distribution function that incorporates angular correlations. The goal is to assess the effects of higher-order angular corrections for which both quadratic- and quartic-order terms (in translational and rotational velocities of particles) are retained in the perturbation expansion of the distribution function. We show that higher-order corrections can markedly affect the steady-state orientational correlation when the normal restitution coefficient is moderate or small, and this effect is more prominent for nearly smooth particles. The transient evolution of orientational correlation is found to be significantly affected by higher-order terms. In particular the higher-order orientational correlations can dominate over the leading-order contribution during short times even in the quasi-elastic limit, although the steady correlation remains unaffected by such corrections in the same limit. The buildup of correlations during the transient stage seems to be closely tied to the evolution of the ratio between the rotational and translational temperatures. It is demonstrated that the transient dynamics of the temperature ratio and its steady state remain insensitive to higher-order angular correlation.

Inhomogeneous quasistationary state of dense fluids of inelastic hard spheres

We study closed dense collections of freely cooling hard spheres that collide inelastically with constant coefficient of normal restitution. We find inhomogeneous states (ISs) where the density profile is spatially nonuniform but constant in time. The states are exact solutions of nonlinear partial differential equations that describe the coupled distributions of density and temperature valid when inelastic losses of energy per collision are small. The derivation is performed without modeling the equations' coefficients that are unknown in the dense limit (such as the equation of state) using only their scaling form specific for hard spheres. Thus the IS is the exact state of this dense many-body system. It captures a fundamental property of inelastic collections of particles: the possibility of preserving nonuniform temperature via the interplay of inelastic cooling and heat conduction that generalizes previous results. We perform numerical simulations to demonstrate that arbitrary initial state evolves to the IS in the limit of long times where the container has the geometry of the channel. The evolution is like a gas-liquid transition. The liquid condenses in a vanishing part of the total volume but takes most of the mass of the system. However, the gaseous phase, which mass grows only logarithmically with the system size, is relevant because its fast particles carry most of the energy of the system. Remarkably, the system self-organizes to dissipate no energy: The inelastic decay of energy is a power law [1+t/t(c)](-2), where t(c) diverges in the thermodynamic limit. This is reinforced by observing that for supercritical systems the IS coincide in most of the space with the steady states of granular systems heated at one of the walls. We discuss the relation of our results to the recently proposed finite-time singularity in other container's geometries.

Impact dynamics of granular jets with noncircular cross sections

Using high-speed photography, we investigate two distinct regimes of the impact dynamics of granular jets with noncircular cross sections. In the steady-state regime, we observe the formation of thin granular sheets with anisotropic shapes and show that the degree of anisotropy increases with the aspect ratio of the jet's cross section. Our results illustrate the liquidlike behavior of granular materials during impact and demonstrate that a collective hydrodynamic flow emerges from strongly interacting discrete particles. We discuss the analogy between our experiments and those from the Relativistic Heavy Ion Collider, where similar anisotropic ejecta from a quark-gluon plasma have been observed in heavy-ion impact.

Energy decay in three-dimensional freely cooling granular gas

The kinetic energy of a freely cooling granular gas decreases as a power law t(-θ) at large times t. Two theoretical conjectures exist for the exponent θ. One based on ballistic aggregation of compact spherical aggregates predicts θ=2d/(d+2) in d dimensions. The other based on Burgers equation describing anisotropic, extended clusters predicts θ=d/2 when 2≤d≤4. We do extensive simulations in three dimensions to find that while θ is as predicted by ballistic aggregation, the cluster statistics and velocity distribution differ from it. Thus, the freely cooling granular gas fits to neither the ballistic aggregation or a Burgers equation description.

Collision dynamics of particle clusters in a two-dimensional granular gas

In a granular gas, inelastic collisions produce an instability in which the constituent particles cluster heterogeneously. These clusters then interact with each other, further decreasing their kinetic energy. We report experiments of the free collisions of dense clusters of particles in a two-dimensional geometry. The particles are composed of solid CO(2), which float nearly frictionlessly on a hot surface due to sublimated vapor. After two dense clusters of ≈100 particles collide, there are two distinct stages of evolution. First, the translational kinetic energy rapidly decreases by over 90% as a "jamming front" sweeps across each cluster. Subsequently, the kinetic energy decreases more slowly as the particles approach the container boundaries. In this regime, the measured velocity distributions are non-Gaussian with long tails. Finally, we compare our experiments to computer simulations of colliding, two-dimensional, granular clusters composed of circular, viscoelastic particles with friction.

Energy loss at propagating jamming fronts in granular gas clusters

We explore the initial moments of impact between two dense granular clusters in a two-dimensional geometry. The particles are composed of solid CO(2) and are levitated on a hot surface. Upon collision, the propagation of a dynamic "jamming front" produces a distinct regime for energy dissipation in a granular gas in which the translational kinetic energy decreases by over 90%. Experiments and associated simulations show that the initial loss of kinetic energy obeys a power law in time ΔE = -Kt(3/2), a form that can be predicted from kinetic arguments.

Inelastic collapse in one-dimensional driven systems under gravity

We study inelastic collapse in a one-dimensional N-particle system when the system is driven from below under gravity. We investigate the hard-sphere limit of inelastic soft-sphere systems by numerical simulations to find how the collision rate per particle n(coll) increases as a function of the elastic constant of the sphere k when the restitution coefficient e is kept constant. For systems with large enough N>/~20, we find three regimes in e depending on the behavior of n(coll) in the hard-sphere limit: (i) an uncollapsing regime for 1≥e>e(c1), where n(coll) converges to a finite value, (ii) a logarithmically collapsing regime for e(c1)>e>e(c2), where n(coll) diverges as n(coll)~logk, and (iii) a power-law collapsing regime for e(c2)>e>0, where n(coll) diverges as n(coll)~k(α) with an exponent α that depends on N. The power-law collapsing regime shrinks as N decreases and seems not to exist for the system with N=3, while, for large N, the size of the uncollapsing and the logarithmically collapsing regime decreases as e(c1)=/~1-2.6/N and e(c2)=/~1-3.0/N. We demonstrate that this difference between large and small systems exists already in the inelastic collapse without external drive and gravity.

Hydrodynamic modes in a confined granular fluid

Confined granular fluids, placed in a shallow box that is vibrated vertically, can achieve homogeneous stationary states due to energy injection mechanisms that take place throughout the system. These states can be stable even at high densities and inelasticities allowing for a detailed analysis of the hydrodynamic modes that govern the dynamics of granular fluids. By analyzing the decay of the time correlation functions it is shown that there is a crossover from a quasielastic regime in which energy evolves as a slow mode to an inelastic regime with energy slaved to the other conserved fields. The two regimes have well differentiated transport properties and in the inelastic regime the dynamics can be described by a reduced hydrodynamics with modified longitudinal viscosity and sound speed. The crossover between the two regimes takes place at a wave vector that is proportional to the inelasticity. A two-dimensional granular model, with collisions that mimic the energy transfers that take place in a confined system, is studied by means of microscopic simulations. The results show excellent agreement with the theoretical framework and allow validation of hydrodynamiclike models.

Shock propagation in granular flow subjected to an external impact

We analyze a recent experiment [Boudet, Cassagne, and Kellay, Phys. Rev. Lett. 103, 224501 (2009)] in which the shock created by the impact of a steel ball on a flowing monolayer of glass beads is studied quantitatively. We argue that radial momentum is conserved in the process and hence show that in two dimensions the shock radius increases in time t as a power law t{1/3}. This is confirmed in event driven simulations of an inelastic hard sphere system. The experimental data are compared with the theoretical prediction and are shown to compare well at intermediate times. At long times the experimental data exhibit a crossover to a different scaling behavior. We attribute this to the problem becoming effectively three dimensional due to the accumulation of particles at the shock front and propose a simple hard sphere model that incorporates this effect. Simulations of this model capture the crossover seen in the experimental data.

Assessing a hydrodynamic description for instabilities in highly dissipative, freely cooling granular gases

An intriguing phenomenon displayed by granular flows and predicted by kinetic-theory-based models is the instability known as particle "clustering," which refers to the tendency of dissipative grains to form transient, loose regions of relatively high concentration. In this work, we assess a modified-Sonine approximation recently proposed [Garzó, Santos, and Montanero, Physica A 376, 94 (2007)] for a granular gas via an examination of system stability. In particular, we determine the critical length scale associated with the onset of two types of instabilities--vortices and clusters--via stability analyses of the Navier-Stokes-order hydrodynamic equations by using the expressions of the transport coefficients obtained from both the standard and the modified-Sonine approximations. We examine the impact of both Sonine approximations over a range of solids fraction φ<0.2 for small restitution coefficients e = 0.25-0.4, where the standard and modified theories exhibit discrepancies. The theoretical predictions for the critical length scales are compared to molecular dynamics (MD) simulations, of which a small percentage were not considered due to inelastic collapse. Results show excellent quantitative agreement between MD and the modified-Sonine theory, while the standard theory loses accuracy for this highly dissipative parameter space. The modified theory also remedies a high-dissipation qualitative mismatch between the standard theory and MD for the instability that forms more readily. Furthermore, the evolution of cluster size is briefly examined via MD, indicating that domain-size clusters may remain stable or halve in size, depending on system parameters.

Bouncing, rolling, energy flows, and cluster formation in a two-dimensional vibrated granular gas

We study the formation of crystalline clusters for a two-dimensional (2D) sinusoidally vibrated granular gas, with maximum vertical acceleration smaller than gravity, using fully 3D simulations. It is found that this phenomenon arises from the spontaneous segregation of the granulate into two dynamical modes: one of grains that bounce in synchrony with the motion of the sustaining plate ("bouncers") and another of grains that cease to bounce and simply rolls on the plate, without ever loosing contact with it ("rollers"). These two dynamical categories are quite robust with respect to perturbations. The populations for bouncers and rollers depend on the preparation of the granulate and can be made to take arbitrary values in all the range of accelerations where both dynamical modes are present. It is found that the dynamical mode with the largest population coalesces in clusters under the influence of the other mode, whose grains act as a higher pressure gas that compresses the clusters. In this way it is possible to produce clusters of rollers or clusters of bouncers. A gas made of grains from only one dynamical class shows only weak density fluctuations. When the occupation fractions for both modes are similar, one observes segregation and clusters of both types. The clustering of the gas is monitored using both the average coordination number and the local hexatic order parameter ψ(6). Energy flows in the plane are monitored, and it is shown that roller-bouncer collisions increase horizontal kinetic energy, while all other types of collisions reduce this energy. We find that friction with the substrate is the main sink of horizontal energy for these granular gases.

Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas

We show that, in dimension higher than one, heat diffusion and viscosity cannot arrest thermal collapse in a freely evolving dilute granular gas, even in the absence of gravity. Thermal collapse involves a finite-time blowup of the gas density. It was predicted earlier in ideal, Euler hydrodynamics of dilute granular gases in the absence of gravity, and in nonideal, Navier-Stokes granular hydrodynamics in the presence of gravity. We determine, analytically and numerically, the dynamic scaling laws that characterize the gas flow close to collapse. We also investigate bifurcations of a freely evolving dilute granular gas in circular and wedge-shaped containers. Our results imply that, in general, thermal collapse can only be arrested when the gas density becomes comparable with the close-packing density of grains. This provides a natural explanation to the formation of densely packed clusters of particles in a variety of initially dilute granular flows.

Finite-time collapse and localized states in the dynamics of dissipative gases

A study of the gas dynamics of a dilute collection of the inelastically colliding hard spheres is presented. When diffusive processes are neglected the gas density blows up in a finite time. The blowup is the mathematical expression for one of the possible mechanisms for cluster formation in dissipative gases. The way diffusive processes smooth the singularity has been studied. Exact localized states of the gas when heat diffusion balances nonlinear cooling are obtained. The presented results generalize previous findings for planar flows.

Dispersion of particles by spontaneous interparticle percolation through unconsolidated porous media

We have performed extensive experimental and numerical studies of spontaneous percolation of small beads through an unconsolidated porous media made with large glass beads. In this paper, an experimental setup and a fast "discrete element method" algorithm are presented to deal with large numbers of particles during our interparticle percolation phenomenon studies. In all the experimental and numerical analyses, the size ratio between the moving beads and the stable packing was chosen larger than the geometrical trapping threshold: xi_{c}=(2/sqrt[3]-1);{-1}=6.464... . We measure the longitudinal and transverse dispersion coefficients versus the height of the porous medium or the number of falling small beads. The influence of bead properties such as density, diameter, or restitution coefficients was investigated by using either steel or glass beads. The individual description of these effects and their explanations were made possible by confrontation and coupling between experimental and numerical results. Indeed, with our numerical model, individual analysis of the effects of these mechanical or geometrical parameters were made possible and carried out.

Effective potentials of dissipative hard spheres in granular matter

We present an experimental study of the spatial correlations of a quasi-two-dimensional dissipative gas kept in a non-static steady state via vertical shaking. From high temporal resolution images we obtain the Pair Distribution Function (PDF) for granular species with different restitution coefficients. Effective potentials for the interparticle interaction are extracted using the Ornstein-Zernike equation with the Percus-Yevick closure. From both the PDFs and the corresponding effective potentials, we find a clear increase of the spatial correlation at contact with the decreasing values of the restitution coefficient.

Collision statistics in sheared inelastic hard spheres

The dynamics of sheared inelastic-hard-sphere systems is studied using nonequilibrium molecular-dynamics simulations and direct simulation Monte Carlo. In the molecular-dynamics simulations Lees-Edwards boundary conditions are used to impose the shear. The dimensions of the simulation box are chosen to ensure that the systems are homogeneous and that the shear is applied uniformly. Various system properties are monitored, including the one-particle velocity distribution, granular temperature, stress tensor, collision rates, and time between collisions. The one-particle velocity distribution is found to agree reasonably well with an anisotropic Gaussian distribution, with only a slight overpopulation of the high-velocity tails. The velocity distribution is strongly anisotropic, especially at lower densities and lower values of the coefficient of restitution, with the largest variance in the direction of shear. The density dependence of the compressibility factor of the sheared inelastic-hard-sphere system is quite similar to that of elastic-hard-sphere fluids. As the systems become more inelastic, the glancing collisions begin to dominate over more direct, head-on collisions. Examination of the distribution of the times between collisions indicates that the collisions experienced by the particles are strongly correlated in the highly inelastic systems. A comparison of the simulation data is made with direct Monte Carlo simulation of the Enskog equation. Results of the kinetic model of Montanero [J. Fluid Mech. 389, 391 (1999)] based on the Enskog equation are also included. In general, good agreement is found for high-density, weakly inelastic systems.

Equivalence of the freely cooling granular gas to the sticky gas

A freely cooling granular gas with a velocity-dependent restitution coefficient is studied in one dimension. The restitution coefficient becomes near elastic when the relative velocity of the colliding particles is less than a velocity scale delta . Different statistical quantities, namely, density distribution, occupied and empty cluster length distributions, and spatial density and velocity correlation functions, are obtained using event driven molecular dynamic simulations. We compare these with the corresponding quantities of the sticky gas (inelastic gas with zero coefficient of restitution). We find that in the inhomogeneous cooling regime, for times smaller than a crossover time t{1} , where t{1} approximately delta;{-1} , the behavior of the granular gas is equivalent to that of the sticky gas. When delta-->0 , then t{1}-->infinity and, hence, the results support an earlier claim that the freely cooling inelastic gas is described by the inviscid Burgers equation. For a real granular gas with finite delta , the existence of the time scale t{1} shows that, for large times, the granular gas is not described by the inviscid Burgers equation.

Shear state of freely evolving granular gases

Hydrodynamic equations are used to identify the final state reached by a freely evolving granular gas above but close to its shear instability. The theory predicts the formation of a two bands shear state with a steady density profile. There is a modulation between temperature and density profiles as a consequence of the energy balance, the density fluctuations remaining small, without producing clustering. Moreover, the time dependence of the velocity field can be scaled out with the squared root of the average temperature of the system. The latter follows the Haff law, but with an effective cooling rate that is smaller than that of the free homogeneous state. The theoretical predictions are compared with numerical results for inelastic hard disks obtained by using the direct Monte Carlo simulation method, and a good agreement is obtained for low inelasticity.

Sediment transport in the saltation regime

We present a study of sediment transport in the creeping and saltation regimes. In our model, a bed of particles is simulated with the conventional event-driven method. The particles are considered as hard disks in a two-dimensional domain with periodic boundary conditions in the horizontal direction. The flow of the fluid over this bed of particles is modeled by imposing a force on each particle that depends on the velocity of the fluid at its height above the bed. We consider two velocity profiles for the fluid, parabolic and logarithmic. The first one models laminar flow and the second corresponds to turbulent flow. For each case we investigate the behavior of the saturated flux. We find that for the logarithmic profile, the saturated flux shows a quadratic increase with the strength of the flow, and for parabolic profile, a cubic increase. The velocity distribution functions are used to interpret the results.

Frequency-dependent fluctuation-dissipation relations in granular gases

The Green-Kubo relation for two models of granular gases is discussed. In the Maxwell model in any dimension, the effective temperature obtained from the Green-Kubo relation is shown to be frequency independent and equal to the average kinetic energy, known as the granular temperature. In the second model analyzed, a mean-field granular gas, the collision rate of a particle is taken to be proportional to its velocity. The Green-Kubo relation in the high-frequency limit is calculated for this model, and the effective temperature in this limit is shown to be equal to the granular temperature. This result, taken together with previous results showing a difference between the effective temperature at zero frequency (the Einstein relation) and the granular temperature, shows that the Green-Kubo relation for granular gases is violated.

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.

Attempted density blowup in a freely cooling dilute granular gas: hydrodynamics versus molecular dynamics

It has been recently shown [I. Fouxon, Phys. Rev. E 75, 050301(R) (2007); I. Fouxon, Phys. Fluids 19, 093303 (2007)] that, in the framework of ideal granular hydrodynamics (IGHD), an initially smooth hydrodynamic flow of a granular gas can produce an infinite gas density in a finite time. Exact solutions that exhibit this property have been derived. Close to the singularity, the granular gas pressure is finite and almost constant. We report molecular dynamics (MD) simulations of a freely cooling gas of nearly elastically colliding hard disks, aimed at identifying the "attempted" density blowup regime. The initial conditions of the simulated flow mimic those of one particular solution of the IGHD equations that exhibits the density blowup. We measure the hydrodynamic fields in the MD simulations and compare them with predictions from the ideal theory. We find a remarkable quantitative agreement between the two over an extended time interval, proving the existence of the attempted blowup regime. As the attempted singularity is approached, the hydrodynamic fields, as observed in the MD simulations, deviate from the predictions of the ideal solution. To investigate the mechanism of breakdown of the ideal theory near the singularity, we extend the hydrodynamic theory by accounting separately for the gradient-dependent transport and for finite density corrections.

Nonlinear theory of nonstationary low Mach number channel flows of freely cooling nearly elastic granular gases

We employ hydrodynamic equations to investigate nonstationary channel flows of freely cooling dilute gases of hard and smooth spheres with nearly elastic particle collisions. This work focuses on the regime where the sound travel time through the channel is much shorter than the characteristic cooling time of the gas. As a result, the gas pressure rapidly becomes almost homogeneous, while the typical Mach number of the flow drops well below unity. Eliminating the acoustic modes and employing Lagrangian coordinates, we reduce the hydrodynamic equations to a single nonlinear and nonlocal equation of a reaction-diffusion type. This equation describes a broad class of channel flows and, in particular, can follow the development of the clustering instability from a weakly perturbed homogeneous cooling state to strongly nonlinear states. If the heat diffusion is neglected, the reduced equation becomes exactly soluble, and the solution develops a finite-time density blowup. The blowup has the same local features at singularity as those exhibited by the recently found family of exact solutions of the full set of ideal hydrodynamic equations [I. Fouxon, Phys. Rev. E 75, 050301(R) (2007); I. Fouxon,Phys. Fluids 19, 093303 (2007)]. The heat diffusion, however, always becomes important near the attempted singularity. It arrests the density blowup and brings about previously unknown inhomogeneous cooling states (ICSs) of the gas, where the pressure continues to decay with time, while the density profile becomes time-independent. The ICSs represent exact solutions of the full set of granular hydrodynamic equations. Both the density profile of an ICS and the characteristic relaxation time toward it are determined by a single dimensionless parameter L that describes the relative role of the inelastic energy loss and heat diffusion. At L>>1 the intermediate cooling dynamics proceeds as a competition between "holes": low-density regions of the gas. This competition resembles Ostwald ripening (only one hole survives at the end), and we report a particular regime where the "hole ripening" statistics exhibits a simple dynamic scaling behavior.

Spatial force correlations in granular shear flow. I. Numerical evidence

We investigate the emergence of correlations in granular shear flow. By increasing the density of a simulated granular flow, we observe a transition from a dilute regime, where interactions are dominated by binary collisions, to a dense regime characterized by large force networks and collective motions. With increasing density, interacting grains tend to form networks of simultaneous contacts due to the dissipative nature of collisions. We quantify the size of these networks by measuring two-point force correlations and find dramatic changes in the statistics of contact forces as the size of the networks increases.

Formation and evolution of density singularities in hydrodynamics of inelastic gases

We use hydrodynamics to investigate clustering in a gas of inelastically colliding spheres. The hydrodynamic equations exhibit a finite-time density blowup, where the gas pressure remains finite. The density blowup signals the formation of close-packed clusters. The blowup dynamics is universal and describable by exact analytic solutions continuable beyond the blowup time. These solutions show that dilute hydrodynamic equations yield a powerful effective description of a granular gas flow with close-packed clusters, described as finite-mass pointlike singularities of the density. This description is similar in spirit to the description of shocks in ordinary ideal gas dynamics.

Velocity distributions and aging in a cooling granular gas

We use large-scale molecular dynamics simulations to study freely evolving granular gases with dimensionality d=2,3 . The system dissipates kinetic energy (or cools) due to inelastic collisions between granular particles. The density and velocity fields are approximately homogeneous at early times, and the system is said to be in a homogeneous cooling state (HCS). However, fluctuations in the density and velocity fields grow, and the system evolves into an inhomogeneous cooling state (ICS). We study the nature of velocity distributions in both the HCS and ICS. We also investigate the aging property of the velocity autocorrelation function.

Velocity correlations in dense granular shear flows: effects on energy dissipation and normal stress

We study the effect of precollisional velocity correlations on granular shear flow by molecular dynamics simulations of an inelastic hard sphere system. Comparison of the simulations with kinetic theory reveals that the theory overestimates both the energy dissipation rate and the normal stress in the dense flow region. We find that the relative normal velocity of colliding particles is smaller than that expected from random collisions, and the discrepancies in the dissipation and the normal stress can be adjusted by introducing the idea of the collisional temperature, from which we conclude that the velocity correlation neglected in the kinetic theory is responsible for the discrepancies. Our analysis of the distributions of the precollisional velocity suggests that the correlation grows through multiple inelastic collisions during the time scale of the inverse of the shear rate. As for the shear stress, the discrepancy is also found in the dense region, but it depends strongly on the particle inelasticity.

Simplest piston problem. II. Inelastic collisions

We study the dynamics of three particles in a finite interval, in which two light particles are separated by a heavy "piston," with elastic collisions between particles but inelastic collisions between the light particles and the interval ends. A symmetry breaking occurs in which the piston migrates near one end of the interval and performs small-amplitude periodic oscillations on a logarithmic time scale. The properties of this dissipative limit cycle can be understood simply in terms of a effective restitution coefficient picture. Many dynamical features of the three-particle system closely resemble those of the many-body inelastic piston problem.

Instabilities in a free granular fluid described by the Enskog equation

A linear stability analysis of the hydrodynamic equations with respect to the homogeneous cooling state is carried out to identify the conditions for stability as functions of the wave vector, the dissipation, and the density. In contrast to previous studies, this description is based on the results derived from the Enskog equation for inelastic hard spheres [V. Garzó and J. W. Dufty, Phys. Rev. E 59, 5895 (1999)], which takes into account the dependence of the transport coefficients on dissipation. As expected, linear stability shows two transversal (shear) modes and a longitudinal "heat") mode to be unstable with respect to long enough wavelength excitations. Comparison with previous results (which neglect the influence of dissipation on transport) shows quantitative discrepancies for strong dissipation.

Inelastic gravitational billiards

We present an experimental investigation of gravitational billiards where the particle undergoes inelastic collisions with its boundary. The motion is mapped for an inelastic particle contained within parabolic, wedge, and hyperbolic boundaries. For the parabola, stable orbits are found and the wedge demonstrates a characteristic instability for its vertex angle. In the instance of the hyperbola, there are several features of the dynamics similar to the parabola at low driving and the wedge for higher driving. However, the low driving case for a hyperbola can only be completely understood by considering inelasticity effects predicted by a numerical simulation and the observation that the velocity dependent inelasticity allows the particle to sample several nearby trajectories for fixed driving.

Hydrodynamic singularities and clustering in a freely cooling inelastic gas

We employ hydrodynamic equations to follow the clustering instability of a freely cooling dilute gas of inelastically colliding spheres into a well-developed nonlinear regime. We simplify the problem by dealing with a one-dimensional coarse-grained flow. We observe that at a late stage of the instability the shear stress becomes negligibly small, and the gas flows solely by inertia. As a result the flow formally develops a finite-time singularity, as the velocity gradient and the gas density diverge at some location. We argue that flow by inertia represents a generic intermediate asymptotic of unstable free cooling of dilute inelastic gases.

Hydrodynamics of probabilistic ballistic annihilation

We consider a dilute gas of hard spheres in dimension d> or =2 that upon collision either annihilate with probability p or undergo an elastic scattering with probability 1-p . For such a system neither mass, momentum, nor kinetic energy is a conserved quantity. We establish the hydrodynamic equations from the Boltzmann equation description. Within the Chapman-Enskog scheme, we determine the transport coefficients up to Navier-Stokes order, and give the closed set of equations for the hydrodynamic fields chosen for the above coarse-grained description (density, momentum, and kinetic temperature). Linear stability analysis is performed, and the conditions of stability for the local fields are discussed.

Steady-state representation of the homogeneous cooling state of a granular gas

The properties of a dilute granular gas in the homogeneous cooling state are mapped to those of a stationary state by means of a change in the time scale that does not involve any internal property of the system. The new representation is closely related with a general property of the granular temperature in the long time limit. The physical and practical implications of the mapping are discussed. In particular, simulation results obtained by the direct simulation Monte Carlo method applied to the scaled dynamics are reported. This includes ensemble averages and also the velocity autocorrelation function, as well as the self-diffusion coefficient obtained from the latter by means of the Green-Kubo representation. In all cases, the obtained results are compared with theoretical predictions.

Clustering in a one-dimensional inelastic lattice gas

We analyze a lattice model closely related to the one-dimensional inelastic gas with periodic boundary condition. The one-dimensional inelastic gas tends to form high density clusters of particles with almost the same velocity, separated by regions of low density; plotted as a function of particle indices, the velocities of the gas particles exhibit sharp gradients, which we call shocks. Shocks and clusters are seen to form in the lattice model too, although no true positions of the particles are taken into account. The locations of the shocks in terms of the particle index show remarkable independence on the coefficient of restitution and the sequence of collisions used to update the system, but they do depend on the initial configuration of the particle velocities. We explain the microscopic origin of the shocks. We show that dynamics of the velocity profile inside a cluster satisfies a simple continuum equation, thereby allowing us to study cluster-cluster interactions at late times.

Cluster growth in two- and three-dimensional granular gases

Dissipation in granular media leads to interesting phenomena such as cluster formation and crystallization in nonequilibrium dynamical states. The freely cooling system is examined concerning the energy decay and the cluster evolution in time, both in two and three dimensions. We also suggest an interpretation of the three-dimensional cluster growth in terms of percolation theory, but this point deserves further study.

Validity of the Boltzmann equation to describe low-density granular systems

The departure of a granular gas in the instable region of parameters from the initial homogeneous cooling state is studied. Results from molecular dynamics and from direct Monte Carlo simulation of the Boltzmann equation are compared. The results indicate that the Boltzmann equation accurately predicts the low-density limit of the system. The relevant role played by the parallelization of the velocities as time proceeds and the dependence of this effect on the density is analyzed in detail.

Velocity statistics in two-dimensional granular turbulence

We studied the macroscopic statistical properties on the freely evolving quasielastic hard disk (granular) system by performing a large-scale (up to a few million particles) event-driven molecular dynamics systematically and found it to be remarkably analogous to an enstrophy cascade process in the decaying two-dimensional fluid turbulence. There are four typical stages in the freely evolving inelastic hard disk system, which are homogeneous, shearing (vortex), clustering, and final state. In the shearing stage, the self-organized macroscopic coherent vortices become dominant. In the clustering stage, the energy spectra are close to the expectation of Kraichnan-Batchelor theory and the squared two-particle separation strictly obeys Richardson law.

Kinetics of inhomogeneous cooling in granular fluids

We study the dynamical behavior of a freely evolving granular gas, where the particles undergo inelastic collisions. The velocity and density fields exhibit complex pattern dynamics, which is reminiscent of phase ordering systems. For example, in the initial time regime, the density field stays (approximately) uniform, and the system is said to be in a homogeneous cooling state (HCS). At later times, the density field undergoes nonlinear clustering, and the system continues to lose energy in an inhomogeneous cooling state (ICS). We quantitatively characterize the HCS-->ICS crossover as a function of system parameters. Furthermore, we study nonlinear growth processes in the ICS by invoking analogies from studies of phase ordering dynamics.

Granular clustering in a hydrodynamic simulation

We examine the hydrodynamics of a granular gas using numerical simulation. We demonstrate the appearance of shearing and clustering instabilities predicted by linear stability analysis, and show that their appearance is directly related to the inelasticity of collisions in the material. We discuss the rate at which these instabilities arise and the manner in which clusters grow and merge.