Dilute Granular Gas Research Articles

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.

Orientational Correlation and Velocity Distributions in Uniform Shear Flow of a Dilute Granular Gas

Using particle simulations of the uniform shear flow of a rough dilute granular gas, we show that the translational and rotational velocities are strongly correlated in direction, but there is no orientational correlation-induced singularity at perfectly smooth (beta=-1) and rough (beta=1) limits for elastic collisions (e=1); both the translational and rotational velocity distribution functions remain close to a Gaussian for these two limiting cases. Away from these two limits, the orientational as well as spatial velocity correlations are responsible for the emergence of non-Gaussian high-velocity tails. The tails of both distribution functions follow stretched exponentials, with the exponents depending on normal (e) and tangential (beta) restitution coefficients.

Gas-liquid Like Phase Transition in Granular Gases Under Zero Gravity

颗粒体系是远离平衡态的复杂散耗体系，在外力驱动下，能表现出类似于固体、液体或气体的特性。在外力驱动下，能表现出类似于固体、液体、或气体的特性，在稀疏的颗粒气体中，由于耗散，体系中常常会形成颗粒在局部的凝聚。这种凝聚行为具有自发对称性破缺的特征，使得颗粒气体中自动形成高密度区域和低密度区域共存的非均匀状态，类似于气体中液滴的形成。本文利用零重力条件下的三维颗粒气体理论模型进行计算和分析，揭示了颗粒气体这一相变行为的不稳定性根源及类气液相变本质，给出了颗粒气体中这种相分离发生的临界条件，并通过了分子动力学模拟进行检验，结果从定性上和定量上都能够很好地吻合，为进一步的空间实验提供了理论依据和相关实验参数。

Formation of density singularities in ideal hydrodynamics of freely cooling inelastic gases: A family of exact solutions

We employ granular hydrodynamics to investigate a paradigmatic problem of clustering of particles in a freely cooling dilute granular gas. We consider large-scale hydrodynamic motions where the viscosity and heat conduction can be neglected, and one arrives at the equations of ideal gas dynamics with an additional term describing bulk energy losses due to inelastic collisions. We employ Lagrangian coordinates and derive a broad family of exact nonstationary analytical solutions that depend only on one spatial coordinate. These solutions exhibit a new type of singularity, where the gas density blows up in a finite time when starting from smooth initial conditions. The density blowups signal formation of close-packed clusters of particles. As the density blow-up time tc is approached, the maximum density exhibits a power law ∼(tc−t)−2. The velocity gradient blows up as ∼−(tc−t)−1 while the velocity itself remains continuous and develops a cusp (rather than a shock discontinuity) at the singularity. The gas temperature vanishes at the singularity, and the singularity follows the isobaric scenario: the gas pressure remains finite and approximately uniform in space and constant in time close to the singularity. An additional exact solution shows that the density blowup, of the same type, may coexist with an “ordinary” shock, at which the hydrodynamic fields are discontinuous but finite. We confirm stability of the exact solutions with respect to small one-dimensional perturbations by solving the ideal hydrodynamic equations numerically. Furthermore, numerical solutions show that the local features of the density blowup hold universally, independently of details of the initial and boundary conditions.

Average energy and fluctuations of a granular gas at the threshold of the clustering instability

The behavior of an isolated dilute granular gas near the threshold of its clustering instability is investigated by means of fluctuating hydrodynamics and the direct simulation Monte Carlo method. The theoretical predictions from the former are shown to be in good agreement with the simulation results. The total energy of the the system is renormalized by the fluctuations of the vorticity field. Moreover, the scaled second moment of the energy fluctuations exhibits a power-law divergent behavior.

First-order Chapman–Enskog velocity distribution function in a granular gas

A method is devised to measure the first-order Chapman–Enskog (CE) velocity distribution function associated with the heat flux in a dilute granular gas. The method is based on the application of a homogeneous, anisotropic velocity-dependent external force which produces heat flux in the absence of gradients. The form of the force is found under the condition that, in the linear response regime, the deviation of the velocity distribution function from that of the homogeneous cooling state obeys the same linear integral equation as the one derived from the conventional CE expansion. The direct simulation Monte Carlo method is used to solve the corresponding Boltzmann equation and measure the dependence of the (modified) thermal conductivity on the coefficient of normal restitution α . Comparison with previous simulation data obtained from the Green–Kubo relations [J.J. Brey, M.J. Ruiz-Montero, P. Mayner, M.I. García de Soria, J. Phys.: Condens. Matter 17 (2005) S2489] shows an excellent agreement, both methods consistently showing that the first Sonine approximation dramatically overestimates the thermal conductivity for high inelasticity ( α ≲ 0.7 ). Since our method is tied to the Boltzmann equation, the results indicate that the failure of the first Sonine approximation is not due to velocity correlation effects absent in the Boltzmann framework. This is further confirmed by an analysis of the first-order CE velocity distribution function and its three first Sonine coefficients obtained from the simulations.

Collision frequencies and energy flux in a dilute granular gas

Recent experimental study of a granular gas fluidized by vibrations in a low gravity environment has reported that the collision frequency nu(rho) of the particles with the container boundary scales roughly like N(alpha) with alpha=0.6 +/- 0.1, where N is the number of particles. Using numerical simulations, we show that this scaling is observed on a wide range of N, both for nu(rho) and for the particle-particle collision frequency nu(c). Simple scaling arguments show that this behavior is related to the energy flux in the granular gas, from injection by the moving boundary to dissipation by inelastic collisions. We predict in the dilute limit that the collision frequencies scale such as (sqrt)N are in fair agreement with experimental measurements.

Hydrodynamic modes, Green–Kubo relations, and velocity correlations in dilute granulargases

It is shown that the hydrodynamic modes of a dilute granular gas of inelastic hard spherescan be identified, and calculated in the long wavelength limit. Assuming they dominate atlong times, formal expressions for the Navier–Stokes transport coefficients are derived.They can be expressed in a form that generalizes the Green–Kubo relations formolecular systems, and it is shown that they can also be evaluated by means ofN-particle simulation methods. The form of the hydrodynamic modes to zeroth order in thegradients is used to detect the presence of inherent velocity correlations in thehomogeneous cooling state, even in the low density limit. They manifest themselves in thefluctuations of the total energy of the system. The theoretical predictions are shown to bein agreement with molecular dynamics simulations. Relevant related questions deservingfurther attention are pointed out.

On cyclic oscillation in granular gas

In a Maxwell’s demon experiment on a dilute granular gas made up of two species of particles placed in two closely connected containers, a cyclic oscillation similar to the chemical oscillations is recognized that the two species of particles cluster alternately in the two containers. Then the nonlinear behavior is qualitatively explained based on a flux model.

Simulation study of the Green-Kubo relations for dilute granular gases

The Green-Kubo relations for dilute granular gases are employed to compute their transport coefficients by means of the direct simulation Monte Carlo method. This requires not only to follow the dynamics of the system, but also to identify some modified fluxes appearing in the time-correlation functions. The results are compared with those obtained from the Boltzmann equation by means of the Chapman-Enskog procedure in the first Sonine approximation. A good agreement is found for the shear viscosity over a wide range of inelasticities. Nevertheless, for the two transport coefficients associated with the heat flux, significant discrepancies appear for strong inelasticity. Their origin is discussed, showing that they are partially due to the presence of velocity correlations in the homogeneous cooling state of a dilute granular fluid.

Energy fluctuations in the homogeneous cooling state of granular gases.

Starting from the hierarchy of equations for microscopic densities in phase space, a general theory for fluctuations and correlations in a dilute granular gas of hard particles is developed. Then, the particular case of the homogeneous cooling state is addressed. Explicit expressions for some distributions describing the presence of velocity correlations and their dynamics are obtained. These correlations are inherent to the dissipative dynamics of the collisions. The implications for the behavior of the total energy of the system are analyzed and the results are expressed in terms of a fluctuation-dissipation theorem. The theoretical predictions are shown to be in agreement with results obtained by molecular dynamics simulations, which also indicate that energy fluctuations are well described by a Gaussian distribution.

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.

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.

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.

Hydrodynamics and transport coefficients for dilute granular gases

The hydrodynamics of granular gases of viscoelastic particles, whose collision is described by an impact-velocity dependent coefficient of restitution, which is developed using the Chapman-Enskog approach. We derive the hydrodynamic equations and the according transport coefficients with the assumption that the shape of the velocity distribution function follows adiabatically the decaying temperature. We show numerically that this approximation is justified up to intermediate dissipation. The transport coefficients and the coefficient of cooling are expressed in terms of the elastic and dissipative parameters of the particle material and by the gas parameters. The dependence of these coefficients on temperature differs qualitatively from that obtained with the simplifying assumption of a constant coefficient of restitution which was used in previous studies. The approach formulated for gases of viscoelastic particles may be applied also for other impact-velocity dependencies of the restitution coefficient.

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.

Velocity fluctuations in electrostatically driven granular media.

We study experimentally the particle velocity fluctuations in an electrostatically driven dilute granular gas. The velocity distributions have strong deviations from a Maxwellian form over a wide range of parameters. We have found that the tails of the distribution functions are consistent with a stretched exponential law with typical exponents of the order 3/2. Molecular dynamic simulations shows qualitative agreement with experimental data. Our results suggest that this non-Gaussian behavior is typical of most inelastic gases with both short- and long-range interactions.

Hydrodynamics of rapid granular flow of inelastic particles into vacuum

The problem of expansion of a dilute granular gas consisting of smooth, inelastic hard spheres into vacuum was investigated by three different methods: both (1) analytical and (2) computational (CFD) treatments of a hydrodynamic model, and (3) by Discrete Element Method (DEM) simulation. Furthermore, the systems were followed for long times, over which the granular kinetic energy decreases by several hundredfold. The hydrodynamic model assumes that the particles are uniformly distributed in space, yet, in the DEM simulations, the particles are free to cluster. Thus the comparison allows us to evaluate the effects of cluster formation on the system. All three methods give quantitatively similar results for the escape momentum, energy evolution, and for the hydrodynamic velocity distribution, even for restitution coefficients as small as e =0 .8. The maximum deviation between the escape momentum computed from the hydrodynamic model and from DEM is shown to be no more than 4%. This means that cluster formation exerts only a relatively minor effect on the hydrodynamic quantities, suggesting to us that the hydrodynamic model in combination with the CFD algorithm may provide an efficient tool in other related problems. In addition, it is one of few cases where hydrodynamic theory and DEM agree quantitatively.

Instability and spatial correlations in a dilute granular gas

Direct Monte Carlo simulation is used to investigate the stability of a dilute freely evolving granular gas of hard disks. The boundary between stability and instability in the plane (α,L), where α is the restitution coefficient and L the size of the system, has been delineated. Instability is associated with the buildup of spatial correlations, which describes the formation of velocity vortices in the system. The simulation results are compared with theoretical predictions presented recently, and a good agreement is found.

Spatial correlations in dilute granular flows: A kinetic model study

The development of spatial correlations in the hydrodynamic fields of a freely evolving granular fluid is studied. The analysis is based on a model kinetic theory for dilute granular gases, which includes an equation for the pair correlation function. For small inelasticity, the results agree with those derived previously from fluctuating hydrodynamics, and also with preliminary calculations using ring kinetic theory for inelastic hard spheres. A critical discussion of the domain of validity of a perturbative calculation of the hydrodynamic modes is given.