The temporal evolution of a dilute granular gas, both in a compressible flow (uniform longitudinal flow) and in an incompressible flow (uniform shear flow), is investigated by means of the direct simulation Monte Carlo method to solve the Boltzmann equation. Emphasis is laid on the identification of a first "kinetic" stage (where the physical properties are strongly dependent on the initial state) subsequently followed by an unsteady "hydrodynamic" stage (where the momentum fluxes are well-defined non-Newtonian functions of the rate of strain). The simulation data are seen to support this two-stage scenario. Furthermore, the rheological functions obtained from simulation are well described by an approximate analytical solution of a model kinetic equation.
We experimentally study the transition from steady flow to unsteady flow in a quasi-2D granular heap when small amounts of water are added to monodisperse glass spheres. Particles flow uniformly down both sides of the heap for low water content, but unsteady flow occurs as the water content increases. The unsteady flow mode consists of a non-depositing downslope avalanche and an upslope propagating granular jump. The transition from steady to unsteady flow occurs when the slope exceeds a critical angle as a result of water-induced cohesion. Under unsteady flow conditions, the deposited heap co…
The uniform longitudinal flow is characterized by a linear longitudinal velocity field $u_x(x,t)=a(t)x$, where $a(t)={a_0}/({1+a_0t})$ is the strain rate, a uniform density $n(t)\propto a(t)$, and a uniform granular temperature $T(t)$. Direct simulation Monte Carlo solutions of the Boltzmann equation for inelastic hard spheres are presented for three (one positive and two negative) representative values of the initial strain rate $a_0$. Starting from different initial conditions, the temporal evolution of the reduced strain rate $a^*\propto a_0/\sqrt{T}$, the non-Newtonian viscosity, the secon…
Hydrodynamic equations for granular gases driven by the Fokker-Planck operator are derived. Transport coefficients appeared in Navier-Stokes order change from the values of a free cooling state to those of a steady state.
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. T…
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.
We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases as introduced J. J. Brey, J. Dufty and A. Santos (1999), by showing that the Homogenous Cooling State attracts any solution at an algebraic rate.
The long-time behavior of the current auto-correlation functions for the velocity, the shear stress and the heat flux is investigated in freely cooling granular gases. It is found that the correlation functions for the velocity and the shear stress have the long-time tails obeying $τ^{-d/2}$, while the correlation function of heat flux decays as $τ^{-(d+2)/2} \exp(-ζ^* τ)$ with the dimensionless cooling rate $ζ^*$, the spatial dimension $d$ and the scaled time $τ$ in terms of the collision frequency. The result of our numerical simulation of the freely cooling granular gases is consistent with…
An impact of particles' roughness on the self-diffusion coefficient in granular gases is investigated. For a simplified collision model where the normal and tangential restitution coefficients are assumed to be constant we develop an analytical theory for the diffusion coefficient, which takes into account non-Maxwellain form of the velocity-angular velocity distribution function. We perform molecular dynamics simulations for a gas in a homogeneous cooling state and study the dependence of the self-diffusion coefficient on restitution coefficients. Our theoretical results are in a good agreeme…
Motivated by recent experiments reporting non-Gaussian velocity distributions in driven dilute granular materials, we study by numerical simulation the properties of 2D inelastic gases. We find theoretically that the form of the observed velocity distribution is governed primarily by the coefficient of restitution $η$ and $q=N_H/N_C$, the ratio between the average number of heatings and the average number of collisions in the gas. The differences in distributions we find between uniform and boundary heating can then be understood as different limits of $q$, for $q \gg 1$ and $q \lesssim 1$ res…
While memory effects have been reported for dense enough disordered systems such as glasses, we show here by a combination of analytical and simulation techniques that they are also intrinsic to the dynamics of dilute granular gases. By means of a certain driving protocol, we prepare the gas in a state where the granular temperature $T$ coincides with its long time limit. However, $T$ does not subsequently remain constant, but exhibits a non-monotonic evolution before reaching its non-equilibrium steady value. The corresponding so-called Kovacs hump displays a normal behavior for weak dissipat…
The expansion of the velocity distribution function for the homogeneous cooling state (HCS) in a Sonine polynomial series around a Maxwellian is shown to be divergent, though Borel resummable. A convergent expansion for the HCS has been devised and employed to obtain the HCS velocity distribution function and (using it) the linear transport coefficients for a three dimensional monodisperse granular gas of smooth inelastic spheres, for all physical values of the coefficient of normal restitution. The results are in very good agreement with findings of DSMC simulations.
We study velocity statistics of electrostatically driven granular gases. For two different experiments: (i) non-magnetic particles in a viscous fluid and (ii) magnetic particles in air, the velocity distribution is non-Maxwellian, and its high-energy tail is exponential, P(v) ~ exp(-|v|). This behavior is consistent with kinetic theory of driven dissipative particles. For particles immersed in a fluid, viscous damping is responsible for the exponential tail, while for magnetic particles, long-range interactions cause the exponential tail. We conclude that velocity statistics of dissipative gas…
Dry, freely evolving granular materials in a dilute gaseous state coalesce into dense clusters only due to dissipative interactions. This clustering transition is important for a number of problems ranging from geophysics to cosmology. Here we show that the evolution of a dilute, freely cooling granular gas is determined in a universal way by the ratio of inertial flow and thermal velocities, that is, the Mach number. Theoretical calculations and direct numerical simulations of the granular Navier--Stokes equations show that irrespective of the coefficient of restitution, density or initial ve…
Two granular gases separated by an adiabatic piston and initially in the same macroscopic state are considered. It is found that a phase transition with an spontaneous symmetry breaking occurs. When the mass of the piston is increased beyond a critical value, the piston moves to a stationary position different from the middle of the system. The transition is accurately described by a simple kinetic model that takes into account the velocity fluctuations of the piston. Interestingly, the final state is not characterized by the equality of the temperatures of the subsystems but by the cooling ra…
We study dynamics of freely cooling granular gases in two-dimensions using large-scale molecular dynamics simulations. We find that for dilute systems the typical kinetic energy decays algebraically with time, E(t) ~ t^{-1}, in the long time limit. Asymptotically, velocity statistics are characterized by a universal Gaussian distribution, in contrast with the exponential high-energy tails characterizing the early homogeneous regime. We show that in the late clustering regime particles move coherently as typical local velocity fluctuations, Delta v, are small compared with the typical velocity,…
We study the equilibrium behavior of one-dimensional granular clusters and one-particle granular gases for a variety of velocity dependent coefficients of restitution $r$. We obtain equations describing of the long time behavior for the cluster's pressure, r.m.s. velocity and granular inter-spacing. We show that for extremely long times, clusters with velocity dependent coefficients of restitution are unstable and dissolve into homogeneous, quasi-elastic gases, but clusters with velocity independent $r$ are permanent. This is in accordance with hydrodynamic studies pointing to the transient na…
Rheology of a dilute cohesive granular gas is theoretically and numerically studied. The flow curve between the shear viscosity and the shear rate is derived from the inelastic Boltzmann equation for particles having square-well potentials in a simple shear flow. It is found that (i) the stable uniformly sheared state only exists above a critical shear rate and (ii) the viscosity in the uniformly sheared flow is almost identical to that for uniformly sheared flow of hard core granular particles. Below the critical shear rate, clusters grow with time, in which the viscosity can be approximated …
We evidence a Kovacs-like memory effect in a uniformly driven granular gas. A system of inelastic hard particles, in the low density limit, can reach a non-equilibrium steady state when properly forced. By following a certain protocol for the drive time dependence, we prepare the gas in a state where the granular temperature coincides with its long time value. The temperature subsequently does not remain constant, but exhibits a non-monotonic evolution with either a maximum or a minimum, depending on the dissipation, and on the protocol. We present a theoretical analysis of this memory effect,…
We report the statistical properties of spherical steel particles rolling on an inclined surface being driven by an oscillating wall. Strong dissipation occurs due to collisions between the particles and rolling and can be tuned by changing the number density. The velocities of the particles are observed to be correlated over large distances comparable to the system size. The distribution of velocities deviates strongly from a Gaussian. The degree of the deviation, as measured by the kurtosis of the distribution, is observed to be as much as four times the value corresponding to a Gaussian, si…
We report the results of an experimental investigation of segregation in a binary mixture of dry particles subjected to horizontal oscillatory excitation. The thin layer of particles was driven by the stick--slip frictional interaction with the surface of a horizontal tray. As the packing fraction of the mixture was increased the evolution of distinct phases was observed. We identified them as a binary gas, segregation liquid and segregation crystal and provide both microscopic and macroscopic measures to identify their properties. Finally, we draw some analogies between segregation in our gra…
We study the behavior of an assembly of $N$ granular particles contained in two compartments within a simple kinetic approach. The particles belonging to each compartment collide inelastically with each other and are driven by a stochastic heat bath. In addition, the fastest particles can change compartment at a rate which depends on their kinetic energy. Via a Boltzmann velocity distribution approach we first study the dynamics of the model in terms of a coupled set of equations for the populations in the containers and their granular temperatures and find a crossover from a symmetric high-te…
The free evolution of inelastic particles in one dimension is studied by means of Molecular Dynamics (MD), of an inelastic pseudo-Maxwell model and of a lattice model, with emphasis on the role of spatial correlations. We present an exact solution of the 1d granular pseudo-Maxwell model for the scaling distribution of velocities and discuss how this model fails to describe correctly the homogeneous cooling stage of the 1d granular gas. Embedding the pseudo-Maxwell gas on a lattice (hence allowing for the onset of spatial correlations), we find a much better agreement with the MD simulations ev…
We consider the velocity distribution for a granular gas of inelastic hard spheres described by the Boltzmann equation. We investigate both the free of forcing case and a system heated by a stochastic force. We propose a new method to compute the first correction to Gaussian behavior in a Sonine polynomial expansion quantified by the fourth cumulant $a_2$. Our expressions are compared to previous results and to those obtained through the numerical solution of the Boltzmann equation. It is numerically shown that our method yields very accurate results for small velocities of the rescaled distri…
Fluctuating hydrodynamics is used to describe the total energy fluctuations of a freely evolving gas of inelastic hard spheres near the threshold of the clustering instability. They are shown to be governed by vorticity fluctuations only, that also lead to a renormalization of the average total energy. The theory predicts a power-law divergent behavior of the scaled second moment of the fluctuations, and a scaling property of their probability distribution, both in agreement with simulations results. A more quantitative comparison between theory and simulation for the critical amplitudes and t…
We study fluidized granular gases in a stationary state determined by the balance between an external driving and the bulk dissipation. The two considered situations are inspired by recent experiments, where the gravity plays a major role as a driving mechanism: in the first case gravity acts only in one direction and the bottom wall is vibrated, in the second case gravity acts in both directions and no vibrating walls are present. Simulations performed under the molecular chaos assumption show averaged profiles of density, velocity and granular temperature which are in good agreement with the…
The empirical observation of aggregation of dielectric particles under the influence of electrostatic forces lies at the origin of the theory of electricity. The growth of clusters formed of small grains underpins a range of phenomena from the early stages of planetesimal formation to aerosols. However, the collective effects of Coulomb forces on the nonequilibrium dynamics and aggregation process in a granular gas -- a model representative of the above physical processes -- have so far evaded theoretical scrutiny. Here, we establish a hydrodynamic description of aggregating granular gases tha…
We study inelastic gases in two dimensions using event-driven molecular dynamics simulations. Our focus is the nature of the stationary state attained by rare injection of large amounts of energy to balance the dissipation due to collisions. We find that under such extreme driving, with the injection rate much smaller than the collision rate, the velocity distribution has a power-law high energy tail. The numerically measured exponent characterizing this tail is in excellent agreement with predictions of kinetic theory over a wide range of system parameters. We conclude that driving by rare bu…
In contrast to molecular gases, granular gases are characterized by inelastic collisions and require therefore permanent driving to maintain a constant kinetic energy. The kinetic theory of granular gases describes how the average velocity of the particles decreases after the driving is shut off. Moreover it predicts that the rescaled particle velocity distribution will approach a stationary state with overpopulated high-velocity tails as compared to the Maxwell-Boltzmann distribution. While this fundamental theoretical result was reproduced by numerical simulations, an experimental confirmati…
In general, the total kinetic energy in a multicomponent granular gas of inelastic and rough hard spheres is unequally partitioned among the different degrees of freedom. On the other hand, partial energy equipartition can be reached, in principle, under appropriate combinations of the mechanical parameters of the system. Assuming common values of the coefficients of restitution, we use kinetic-theory tools to determine the conditions under which the components of a granular mixture in the homogeneous cooling state have the same translational and rotational temperatures as those of a one-compo…
The fundamental question addressed in this paper is whether the partial Chapman--Enskog expansion $P_{xy}=-\sum_{k=0}^\infty η_k ({\partial u_x}/{\partial y})^{2k+1}$ of the shear stress converges or not for a gas of inelastic hard spheres. By using a simple kinetic model it is shown that, in contrast to the elastic case, the above series does converge, the radius of convergence increasing with inelasticity. It is argued that this paradoxical conclusion is not an artifact of the kinetic model and can be understood in terms of the time evolution of the scaled shear rate in the uniform shear flo…
The evolution to the steady state of a granular gas subject to simple shear flow is analyzed by means of computer simulations. It is found that, regardless of its initial preparation, the system reaches (after a transient period lasting a few collisions per particle) a non-Newtonian (unsteady) hydrodynamic regime, even at strong dissipation and for states where the time scale associated with inelastic cooling is shorter than the one associated with the irreversible fluxes. Comparison with a simplified rheological model shows a good agreement.
The transport coefficients for dilute granular gases of inelastic and rough hard disks or spheres with constant coefficients of normal ($α$) and tangential ($β$) restitution are obtained in a unified framework as functions of the number of translational ($d_t$) and rotational ($d_r$) degrees of freedom. The derivation is carried out by means of the Chapman--Enskog method with a Sonine-like approximation in which, in contrast to previous approaches, the reference distribution function for angular velocities does not need to be specified. The well-known case of purely smooth $d$-dimensional part…
We perform large-scale event-driven Molecular dynamics (MD) simulations for granular gases of particles interacting with the impact-velocity dependent restitution coefficient. We use the simplest first-principle collision model of viscoelastic spheres. Both cases of force-free and uniformly heated gases are studied. We formulate a simplified model of an effective constant restitution coefficient, which depends on a current granular temperature and compute the effective constant restitution coefficient, using the kinetic theory. We develop a theory of the velocity distribution function for driv…
We describe a special class of steady Couette flows in dilute granular gases admitting a non-Newtonian hydrodynamic description for strong dissipation. The class occurs when viscous heating exactly balances inelastic cooling, resulting in a uniform heat flux. It includes the Fourier flow of ordinary gases and the simple or uniform shear flow (USF) of granular gases as special cases. The rheological functions have the same values as in the USF and generalized thermal conductivity coefficients can be identified. These points are confirmed by molecular dynamics simulations, Monte Carlo simulation…
Simulation results of dense granular gases with particles of different size are compared with theoretical predictions concerning the pair-correlation functions, the collison rate, the energy dissipation, and the mixture pressure. The effective particle-particle correlation function, which enters the equation of state in the same way as the correlation function of monodisperse granular gases, depends only on the total volume fraction and on the dimensionless width A of the size-distribution function. The ''global equation of state'' is proposed, which unifies both the dilute and the dense regim…
We report a general class of steady and transient states of granular gases. We find that the kinetic theory of inelastic gases admits stationary solutions with a power-law velocity distribution, f(v) ~ v^(-sigma). The exponent sigma is found analytically and depends on the spatial dimension, the degree of inelasticity, and the homogeneity degree of the collision rate. Driven steady-states, with the same power-law tail and a cut-off can be maintained by injecting energy at a large velocity scale, which then cascades to smaller velocities where it is dissipated. Associated with these steady-stat…
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 over population at large velocities, are reproduced semi-quantitatively by th…
We study a dilute granular gas immersed in a thermal bath made of smaller particles with masses not much smaller than the granular ones in this work. Granular particles are assumed to have inelastic and hard interactions, losing energy in collisions as accounted by a constant coefficient of normal restitution. The interaction with the thermal bath is modeled by a nonlinear drag force plus a white-noise stochastic force. The kinetic theory for this system is described by an Enskog--Fokker--Planck equation for the one-particle velocity distribution function. To get explicit results of the temper…
A variant of the Direct Simulation Monte Carlo method is used to study the behavior of a granular gas, in two and three dimensions, under varying density, restitution coefficient, and inelasticity regimes, for realistic vibrating wall conditions. Our results agree quite well with recent experimental data. The role of the energy injection mechanism is discussed, as well as the behavior of state-functions, such as pressure, under realistic boundary conditions. Upon a density increase, we find signals of a clustering transition.
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. …
In a granular gas of rough particles the spin of a grain is correlated with its linear velocity. We develop an analytical theory to account for these correlations and compare its predictions to numerical simulations, using Direct Simulation Monte Carlo as well as Molecular Dynamics. The system is shown to relax from an arbitrary initial state to a quasi-stationary state, which is characterized by time-independent, finite correlations of spin and linear velocity. The latter are analysed systematically for a wide range of system parameters, including the coefficients of tangential and normal res…
A driven granular material, e.g. a vibrated box full of sand, is a stationary system which may be very far from equilibrium. The standard equilibrium statistical mechanics is therefore inadequate to describe fluctuations in such a system. Here we present numerical and analytical results concerning energy and injected power fluctuations. In the first part we explain how the study of the probability density function (pdf) of the fluctuations of total energy is related to the characterization of velocity correlations. Two different regimes are addressed: the gas driven at the boundaries and the h…
We investigate hydrodynamic noise in a dilute granular gas during the homogeneous cooling state, by means of a proper application of the Direct Simulation Monte Carlo (DSMC) algorithm. The DSMC includes a source of randomization which is not present in Molecular Dynamics (MD) for inelastic hard disks. Notwithstanding this difference, a fair quantitative agreement is found, including a violation of the fluctuation-dissipation relation for the noise amplitude of the same order observed in MD. This study suggests that deterministic collision dynamics is not an essential ingredient to reproduce, u…
We numerically and theoretically investigate how the softness of particles affects the rheology of sheared dilute granular gases. We find that the kinetic theory predicts the deviation of the flow curve from the Bagnold scaling, and it works well below a certain shear rate when we compare with the simulation results. It is also found that there is no theoretical solution above this shear rate, which is because the energy loss due to inelastic collisions cannot be balanced with the energy injection by the shear.
In contrast to normal fluids, a granular fluid under shear supports a steady state with uniform temperature and density since the collisional cooling can compensate locally for viscous heating. It is shown that the hydrodynamic description of this steady state is inherently non-Newtonian. As a consequence, the Newtonian shear viscosity cannot be determined from experiments or simulation of uniform shear flow. For a given degree of inelasticity, the complete nonlinear dependence of the shear viscosity on the shear rate requires the analysis of the unsteady hydrodynamic behavior. The relationshi…
The steady states of two vibrated granular gases separated by an adiabatic piston are investigated. The system exhibits a non-equilibrium phase transition with an spontaneous symmetry breaking. Even if the gases at both sides of the piston have the same number of particles and are mechanically identical, their steady volumes and temperatures can be rather different. The transition can be explained by a simple kinetic theory model expressing mechanical equilibrium and the energy balance occurring in the system. The model predictions are in good agreement with molecular dynamics simulation resul…
It is demonstrated that recent results on 1d granular gas in a container with a vibrating piston, which was modelled by a shock wave propagation, can be understood with a modelling using ideas coming from the "thermodynamics of a single particle". Defining e as the square root of the energetic-restitution coefficient of a single collision, and N as the total number of grains, the mean loss during a round trip of the momentum is calculated in the limit N(1-e)<<1. It is also demonstrated that the system cannot propagate sound waves nor shock waves in the limit of N(1-e)<<1 and that hydrodynamics…
The flow characterized by a linear longitudinal velocity field $u_x(x,t)=a(t)x$, where $a(t)={a_0}/({1+a_0t})$, a uniform density $n(t)\propto a(t)$, and a uniform temperature $T(t)$ is analyzed for dilute granular gases by means of a BGK-like model kinetic equation in $d$ dimensions. For a given value of the coefficient of normal restitution $α$, the relevant control parameter of the problem is the reduced deformation rate $a^*(t)=a(t)/ν(t)$ (which plays the role of the Knudsen number), where $ν(t)\propto n(t)\sqrt{T(t)}$ is an effective collision frequency. The relevant response parameter is…
Simulation of a system consisted of free particle bouncing on a vertically vibrated based is performed. Two different states, which are steady and unsteady energy transfer state are found. The vibrating based is hold at constant vibration frequency $f = 0.1$ as the vibration amplitude $A$ varied. Sinusoidal form is used. Granular temperature $T_g$ as function of based velocity and coefficient of restitution is used but shown no role in determining energy transfer state of the system. Peak of free particle trajectory $x_m$ around value 20 seperate region of the two states.
