Gas Of Inelastic Hard Spheres Research Articles

Rheology of granular particles immersed in a molecular gas under uniform shear flow.

Non-Newtonian transport properties of a dilute gas of inelastic hard spheres immersed in a molecular gas are determined. We assume that the granular gas is sufficiently rarefied, and hence the state of the molecular gas is not disturbed by the presence of the solid particles. In this situation, one can treat the molecular gas as a bath (or thermostat) of elastic hard spheres at a given temperature. Moreover, in spite of the fact that the number density of grains is quite small, we take into account their inelastic collisions among themselves in its kinetic equation. The system (granular gas plus a bath of elastic hard spheres) is subjected to a simple (or uniform) shear flow. In the low-density regime, the rheological properties of the granular gas are determined by solving the Boltzmann kinetic equationby means of Grad's moment method. These properties turn out to be highly nonlinear functions of the shear rate and the remaining parameters of the system. Our results show that the kinetic granular temperature and the non-Newtonian viscosity present a discontinuous shear thickening effect for sufficiently high values of the mass ratio m/m_{g} (m and m_{g} being the mass of grains and gas particles, respectively). This effect becomes more pronounced as the mass ratio m/m_{g} increases. In particular, in the Brownian limit (m/m_{g}→∞) the expressions of the non-Newtonian transport properties derived here are consistent with those previously obtained by considering a coarse-grained approach where the effect of gas phase on grains is through an effective force. Theoretical results are compared against computer simulations, showing an excellent agreement.

Mobility and diffusion of intruders in granular suspensions: Einstein relation

The Enskog kinetic equation is considered to determine the diffusion D and mobility λ transport coefficients of intruders immersed in a granular gas of inelastic hard spheres (grains). Intruders and grains are in contact with a thermal bath, which plays the role of a background gas. As usual, the influence of the latter on the dynamics of intruders and grains is accounted for via a viscous drag force plus a stochastic Langevin-like term proportional to the background temperature T b. In this case, the starting kinetic equations are the Enskog and Enskog–Lorentz equations for grains and intruders, respectively, with the addition of Fokker–Planck terms to each one of the above equations. The transport coefficients λ and D are determined by solving the Enskog–Lorentz kinetic equation by means of the Chapman–Enskog method adapted to inelastic collisions. As for elastic collisions, both transport coefficients are given in terms of the solutions of two integral equations, which are approximately solved up to the second order in a Sonine polynomial expansion. Theoretical results are compared against numerical solutions of the inelastic Enskog equation by means of the direct simulation Monte Carlo method. In general, good agreement between theory and simulations is found, especially in the case of the second Sonine approximation. Knowledge of the coefficients λ and D allows us to assess the departure of the (conventional) Einstein relation ϵ=D/(Tbλ) from 1. As expected from previous results for driven granular gases, it is shown that when the bath temperature T b is replaced by the intruder temperature T 0 in the Einstein relation, the origin of the deviation of ε from 1 is only due to the non-Maxwellian behavior of the reference state of intruders (measured by the cumulant c 0). Since the magnitude of c 0 is in general very small, deviations of the (modified) Einstein relation ϵ0=D/(T0λ) from 1 cannot be detected in computer simulations of dilute granular gases. This conclusion agrees well with previous computer simulation results.

Kinetic theory of granular particles immersed in a molecular gas

The transport coefficients of a dilute gas of inelastic hard spheres immersed in a gas of elastic hard spheres (molecular gas) are determined. We assume that the number density of the granular gas is much smaller than that of the surrounding molecular gas, so that the latter is not affected by the presence of the granular particles. In this situation, the molecular gas may be treated as a thermostat (or bath) of elastic hard spheres at a fixed temperature. The Boltzmann kinetic equation is the starting point of the present work. The first step is to characterise the reference state in the perturbation scheme, namely the homogeneous state. Theoretical results for the granular temperature and kurtosis obtained in the homogeneous steady state are compared against Monte Carlo simulations showing a good agreement. Then, the Chapman–Enskog method is employed to solve the Boltzmann equation to first order in spatial gradients. In dimensionless form, the Navier–Stokes–Fourier transport coefficients of the granular gas are given in terms of the mass ratio$m/m_g$($m$and$m_g$being the masses of a granular and a gas particle, respectively), the (reduced) bath temperature and the coefficient of restitution. Interestingly, previous results derived from a suspension model based on an effective fluid–solid interaction force are recovered in the Brownian limit ($m/m_g \to \infty$). Finally, as an application of the theory, a linear stability analysis of the homogeneous steady state is performed showing that this state is always linearly stable.

Kinetic model for a confined quasi-two-dimensional gas of inelastic hard spheres

The local balance equations for the density, momentum, and energy of a dilute gas of elastic or inelastic hard spheres, strongly confined between two parallel hard plates are obtained. The starting point is a Boltzmann-like kinetic equation, recently derived for this system. As a consequence of the confinement, the pressure tensor and the heat flux contain, in addition to the terms associated to the motion of the particles, collisional transfer contributions, similar to those that appear beyond the dilute limit. The complexity of these terms, and of the kinetic equation itself, compromise the potential of the equation to describe the rich phenomenology observed in this kind of systems. For this reason, a simpler model equation based on the Boltzmann equation is proposed. The model is formulated to keep the main properties of the underlying equation, and it is expected to provide relevant information in more general states than the original equation. As an illustration, the solution describing a macroscopic state with uniform temperature, but a density gradient perpendicular to the plates is considered. This is the equilibrium state for an elastic system, and the inhomogeneous cooling state for the case of inelastic hard spheres. The results are in good agreement with previous results obtained directly from the Boltzmann equation.

Coarsening in granular systems

We review a few representative examples of granular experiments or models where phase separation, accompanied by domain coarsening, is a relevant phenomenon. We first elucidate the intrinsic non-equilibrium, or athermal, nature of granular media. Thereafter, dilute systems, the so-called “granular gases”, are discussed: idealized kinetic models, such as the gas of inelastic hard spheres in the cooling regime, are the optimal playground to study the slow growth of correlated structures, e.g., shear patterns, vortices, and clusters. In fluidized experiments, liquid–gas or solid–gas separations have been observed. In the case of monolayers of particles, phase coexistence and coarsening appear in several different setups, with mechanical or electrostatic energy input. Phenomenological models describe, even quantitatively, several experimental measures, both for the coarsening dynamics and for the dynamic transition between different granular phases. The origin of the underlying bistability is in general related to negative compressibility from granular hydrodynamics computations, even if the understanding of the mechanism is far from complete. A relevant problem, with important industrial applications, is related to the demixing or segregation of mixtures, for instance in rotating tumblers or on horizontally vibrated plates. Finally, the problem of compaction of highly dense granular materials, which is relevant in many practical situations, is usually described in terms of coarsening dynamics: there, bubbles of misaligned grains evaporate, allowing the coalescence of optimally arranged islands and a progressive reduction of the total occupied volume. On décrit quelques exemples représentatifs d'expériences et de modèles concernant des matériaux granulaires, dans lesquels apparaît une séparation de phases avec formation de domaines de plus en plus grands ( coarsening ). On précise d'abord la nature intrinsèque des matériaux granulaires hors de l'équilibre, dans lesquels la température n'intervient pas. On discute ensuite les systèmes dilués appelés «gaz granulaires» : des modèles cinétiques idéalisés, tels que des gaz de sphères dures inélastiques en cours de refroidissement, sont l'outil idéal pour étudier la croissance lente de structures corrélées, par exemple des figures de cisaillement, des tourbillons et des amas. Dans des expériences en milieu liquide, on a observé une séparation liquide–gaz ou solide–gaz. Dans le cas de monocouches de particules, la coexistence de phase et la croissance des domaines apparaît dans diverses situations, quand on fournit de l'énergie mécanique ou électrostatique. Des modèles phénoménologiques décrivent, au moins qualitativement, divers résultats expérimentaux, concernant la dynamique de coarsening aussi bien que la transition dynamique entre les différentes phases granulaires. L'origine de la bistabilité semble liée à une compressibilité négative, révélée par des calculs d'hydrodynamique granulaire, mais le phénomène est loin d'être bien compris. Un problème important, notamment pour ses applications industrielles, est celui de la décomposition des mélanges ou de leur ségrégation, par exemple dans des tambours tournants ou sur des plaques vibrantes horizontales. Enfin, le problème du compactage des matériaux granulaires très denses, qui se pose dans bien des situations pratiques, est ordinairement décrit comme un coarsening : des bulles de grains mal alignés s'évaporent, ce qui permet la coalescence des îlots bien disposés et une réduction progressive du volume total occupé.

Homogeneous steady state of a confined granular gas

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

Fluctuating hydrodynamics for dilute granular gases

Starting from the kinetic equations for the fluctuations and correlations of a dilute gas of inelastic hard spheres or disks, a Boltzmann-Langevin equation for the one-particle distribution function of the homogeneous cooling state is constructed. This equation is the linear Boltzmann equation with a fluctuating white-noise term. Balance equations for the fluctuating hydrodynamic fields are derived. New fluctuating forces appear as compared with the elastic limit. The particular case of the transverse velocity field is investigated in detail. Its fluctuations can be described by means of a Langevin equation, but exhibiting two main differences with the Landau-Lifshitz theory: the noise is not white, and its second moment is not determined by the shear viscosity. This shows that the fluctuation-dissipation relations for molecular fluids do not straightforwardly carry over to inelastic gases. The theoretical predictions are shown to be in good agreement with molecular-dynamics simulation results.

Does the Chapman–Enskog Expansion for Sheared Granular Gases Converge?

The fundamental question addressed in this Letter is whether or not the partial Chapman-Enskog expansion P(xy)= [see text for formula] of the shear stress converges 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 flow.

Mesoscopic Theory of Critical Fluctuations in Isolated Granular Gases

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 only by vorticity fluctuations 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 the form of the scaling function is also carried out.

System of elastic hard spheres which mimics the transport properties of a granular gas

The prototype model of a fluidized granular system is a gas of inelastic hard spheres (IHS) with a constant coefficient of normal restitution alpha. Using a kinetic theory description we investigate the two basic ingredients that a model of elastic hard spheres (EHS) must have in order to mimic the most relevant transport properties of the underlying IHS gas. First, the EHS gas is assumed to be subject to the action of an effective drag force with a friction constant equal to half the cooling rate of the IHS gas, the latter being evaluated in the local equilibrium approximation for simplicity. Second, the collision rate of the EHS gas is reduced by a factor (1/2)(1+alpha), relative to that of the IHS gas. Comparison between the respective Navier-Stokes transport coefficients shows that the EHS model reproduces almost perfectly the self-diffusion coefficient and reasonably well the two transport coefficients defining the heat flux, the shear viscosity being reproduced within a deviation less than 14% (for alpha > or = 0.5). Moreover, the EHS model is seen to agree with the fundamental collision integrals of inelastic mixtures and dense gases. The approximate equivalence between IHS and EHS is used to propose kinetic models for inelastic collisions as simple extensions of known kinetic models for elastic collisions.

Uniform shear flow in dissipative gases: Computer simulations of inelastic hard spheres and frictional elastic hard spheres

In the preceding paper, we have conjectured that the main transport properties of a dilute gas of inelastic hard spheres (IHSs) can be satisfactorily captured by an equivalent gas of elastic hard spheres (EHSs), provided that the latter are under the action of an effective drag force and their collision rate is reduced by a factor (1+alpha)/2 (where alpha is the constant coefficient of normal restitution). In this paper we test the above expectation in a paradigmatic nonequilibrium state, namely, the simple or uniform shear flow, by performing Monte Carlo computer simulations of the Boltzmann equation for both classes of dissipative gases with a dissipation range 0.5 < or = alpha < or = 0.95 and two values of the imposed shear rate a. It is observed that the evolution toward the steady state proceeds in two stages: a short kinetic stage (strongly dependent on the initial preparation of the system) followed by a slower hydrodynamic regime that becomes increasingly less dependent on the initial state. Once conveniently scaled, the intrinsic quantities in the hydrodynamic regime depend on time, at a given value of alpha, only through the reduced shear rate a*(t) is proportional to a/square root(T(t)), until a steady state, independent of the imposed shear rate and of the initial preparation, is reached. The distortion of the steady-state velocity distribution from the local equilibrium state is measured by the shear stress, the normal stress differences, the cooling rate, the fourth and sixth cumulants, and the shape of the distribution itself. In particular, the simulation results seem to be consistent with an exponential overpopulation of the high-velocity tail. These properties are common to both the IHS and EHS systems. In addition, the EHS results are in general hardly distinguishable from the IHS ones if alpha approximately > 0.7, so that the distinct signature of the IHS gas (higher anisotropy and overpopulation) only manifests itself at relatively high dissipations.

Poiseuille Flow in a Heated Granular Gas

The planar Poiseuille flow induced by a constant external field (e.g., gravity) has been the subject of recent interest in the case of molecular gases. One of the predictions from kinetic theory (confirmed by computer simulations) has been that the temperature profile exhibits a bimodal shape with a local minimum in the middle of the slab surrounded by two symmetric maxima, in contrast to the unimodal shape expected from the Navier–Stokes (NS) equations. However, from a practical point of view, the interest of this non-Newtonian behavior in molecular gases is rather academic since it requires values of gravity extremely higher than the terrestrial one. On the other hand, gravity plays a relevant role in the case of granular gases due to the mesoscopic nature of the grains. In this paper we consider a dilute gas of inelastic hard spheres enclosed in a slab under the action of gravity along the longitudinal direction. In addition, the gas is subject to a white-noise stochastic force that mimics the effect of external vibrations customarily used in experiments to compensate for the collisional cooling. The system is described by means of a kinetic model of the inelastic Boltzmann equation and its steady-state solution is derived through second order in gravity. This solution differs from the NS description in that the hydrostatic pressure is not uniform, normal stress differences are present, a component of the heat flux normal to the thermal gradient exists, and the temperature profile includes a positive quadratic term. As in the elastic case, this new term is responsible for the bimodal shape of the temperature profile. The results show that, except for high inelasticities, the effect of inelasticity on the profiles is to slightly decrease the quantitative deviations from the NS results.

On the Boltzmann Equation for Diffusively Excited Granular Media

We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative $L^2$ function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the high-velocity tails of both the stationary and time-dependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjecturedby previous authors, and we prove pointwise lower estimates for the solutions.

On the first Sonine correction for granular gases

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 distribution. We finally discuss the ambiguities inherent to a linear approximation method in a 2.

Transport coefficients of a heated granular gas

The Navier–Stokes transport coefficients of a granular gas are obtained from the Chapman–Enskog solution to the Boltzmann equation. The granular gas is heated by the action of an external driving force (thermostat) which does work to compensate for the collisional loss of energy. Two types of thermostats are considered: (a) a deterministic force proportional to the particle velocity (Gaussian thermostat), and (b) a random external force (stochastic thermostat). As happens in the free cooling case, the transport coefficients are determined from linear integral equations which can be approximately solved by means of a Sonine polynomial expansion. In the leading order, we get those coefficients as explicit functions of the restitution coefficient α. The results are compared with those obtained in the free cooling case, indicating that the above thermostat forces do not play a neutral role in the transport. The kinetic theory results are also compared with those obtained from Monte Carlo simulations of the Boltzmann equation for the shear viscosity. The comparison shows an excellent agreement between theory and simulation over a wide range of values of the restitution coefficient. Finally, the expressions of the transport coefficients for a gas of inelastic hard spheres are extended to the revised Enskog theory for a description at higher densities.

Steady state of a fluidized granular medium between two walls at the same temperature

The steady state of a low-density gas of inelastic hard spheres confined between two parallel walls at the same temperature is studied. Because of the dissipation in collisions, the state is not uniform but highly inhomogeneous with a nonlinear temperature profile. Direct Monte Carlo simulations show that in the nearly elastic limit the pressure is uniform, but the state exhibits anisotropy of the diagonal terms of the pressure tensor, contrary to the predictions of the Navier-Stokes equations. For larger inelasticity, the pressure becomes nonuniform. These rheological effects, peculiar to granular systems, are explained by means of a model kinetic equation based on the Boltzmann equation. The equation is solved by constructing a systematic perturbative expansion in the square root of the degree of inelasticity. The theoretical predictions compare well with the simulation results for small inelasticity, but they are in conflict for larger values of the degree of inelasticity. The analysis provides strong evidence that this is due to the asymptotic but divergent character of the expansion, similarly to what happens when the usual Chapman-Enskog expansion is applied to molecular fluids.