Gaussian kinetic model for granular gases

Anomalous transport of impurities in inelastic Maxwell gases

A mixture of dissipative hard grains generically exhibits a breakdown of kinetic energy equipartition. The undriven and thus freely cooling binary problem, in the tracer limit where the density of one species becomes minute, may exhibit an extreme form of this breakdown, with the minority species carrying a finite fraction of the total kinetic energy of the system. We investigate the fingerprint of this non-equilibrium phase transition, akin to an ordering process, on transport properties. The analysis, performed by solving the Boltzmann kinetic equation from a combination of analytical and Monte Carlo techniques, hints at the possible failure of hydrodynamics in the ordered region. As a relevant byproduct of the study, the behaviour of the second- and fourth-degree velocity moments is also worked out.

Analogous viscosity equations of granular powders based on Eyring's rate process theory and free volume concept

Granular powders can be successfully treated with kinetic theory and statistical mechanics, though the granular powders are athermal systems and the conventional environmental temperature is too weak to drive particles to move.

Impurity in a sheared inelastic Maxwell gas

The Boltzmann equation for inelastic Maxwell models is considered in order to investigate the dynamics of an impurity (or intruder) immersed in a granular gas driven by a uniform shear flow. The analysis is based on an exact solution of the Boltzmann equation for a granular binary mixture. It applies for conditions arbitrarily far from equilibrium (arbitrary values of the shear rate a) and for arbitrary values of the parameters of the mixture (particle masses m(i), mole fractions x(i), and coefficients of restitution α(ij)). In the tracer limit where the mole fraction of the intruder species vanishes, a nonequilibrium phase transition takes place. We thereby identify ordered phases where the intruder bears a finite contribution to the properties of the mixture, in a region of parameter space that is worked out in detail. These findings extend previous results obtained for ordinary Maxwell gases, and further show that dissipation leads to new ordered phases.

Inelastic collisional effect on a dilute granular shock layer with a heated wall

The inelastic collisional effect on a shock layer of a dilute granular gas with a heated wall is numerically studied. To investigate the inelastic collisional effect via the gain term in the inelastic Boltzmann equation on the shock layer, an inelastic Bhatnagar-Gross-Krook (BGK) type equation, whose loss term is equivalent to that in the inelastic Boltzmann equation, is formulated on the basis of the kinetic theory of the granular gas. The inelastic BGK-type equation formulated for a hard-sphere particle is generalized to that for an inverse power law (IPL) molecule. Numerical results in a weakly inelastic regime confirm the nonequilirium contribution to the cooling rate, when the collision frequency depends on the particle velocity. The profile of the negative high-velocity tail of the distribution function in the generation regime of the shock wave obtained by the Direct Simulation Monte Carlo method is higher than that obtained by the proposed BGK-type equation when the collision frequency depends on the particle velocity because of the inelastic collisional effect via the gain term in the inelastic Boltzmann equation, which is not included in the proposed BGK-type equation.

Granular mixtures modeled as elastic hard spheres subject to a drag force

Granular gaseous mixtures under rapid flow conditions are usually modeled as a multicomponent system of smooth inelastic hard disks (two dimensions) or spheres (three dimensions) with constant coefficients of normal restitution alpha{ij}. In the low density regime an adequate framework is provided by the set of coupled inelastic Boltzmann equations. Due to the intricacy of the inelastic Boltzmann collision operator, in this paper we propose a simpler model of elastic hard disks or spheres subject to the action of an effective drag force, which mimics the effect of dissipation present in the original granular gas. For each collision term ij, the model has two parameters: a dimensionless factor beta{ij} modifying the collision rate of the elastic hard spheres, and the drag coefficient zeta{ij}. Both parameters are determined by requiring that the model reproduces the collisional transfers of momentum and energy of the true inelastic Boltzmann operator, yielding beta{ij}=(1+alpha{ij})2 and zeta{ij} proportional, variant1-alpha{ij}/{2}, where the proportionality constant is a function of the partial densities, velocities, and temperatures of species i and j. The Navier-Stokes transport coefficients for a binary mixture are obtained from the model by application of the Chapman-Enskog method. The three coefficients associated with the mass flux are the same as those obtained from the inelastic Boltzmann equation, while the remaining four transport coefficients show a general good agreement, especially in the case of the thermal conductivity. The discrepancies between both descriptions are seen to be similar to those found for monocomponent gases. Finally, the approximate decomposition of the inelastic Boltzmann collision operator is exploited to construct a model kinetic equation for granular mixtures as a direct extension of a known kinetic model for elastic collisions.

Forcing independent velocity distributions in an experimental granular fluid

We present experimental results on the velocity statistics of a granular fluid with an effective stochastic thermostat, in a quasi-two-dimensional configuration. We find the base state, as measured by the single particle velocity distribution P(c) in the central high-probability regions, to be well described by P(c)=f{MB}[1+a2S2(c2)] : It deviates from a Maxwell-Boltzmann f{MB} by a second order Sonine polynomial S2(c2) with a single adjustable parameter a2. We find a2 to be a function of the filling fraction and independent of the driving over a wide range of frequencies and accelerations. Moreover, there is a consistent overpopulation in the distribution's tails, which scale as P proportional, variant exp(-A x c{3/2}) . To our knowledge, this is the first time that Sonine deviations have been measured in an experimental system.

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.

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.

Maxwell and very-hard-particle models for probabilistic ballistic annihilation: hydrodynamic description

The hydrodynamic description of probabilistic ballistic annihilation, for which no conservation laws hold, is an intricate problem with hard spherelike dynamics for which no exact solution exists. We consequently focus on simplified approaches, the Maxwell and very-hard-particle (VHP) models, which allows us to compute analytically upper and lower bounds for several quantities. The purpose is to test the possibility of describing such a far from equilibrium dynamics with simplified kinetic models. The motivation is also in turn to assess the relevance of some singular features appearing within the original model and the approximations invoked to study it. The scaling exponents are first obtained from the (simplified) Boltzmann equation, and are confronted against direct Monte Carlo simulations. Then, the Chapman-Enskog method is used to obtain constitutive relations and transport coefficients. The corresponding Navier-Stokes equations for the hydrodynamic fields are derived for both Maxwell and VHP models. We finally perform a linear stability analysis around the homogeneous solution, which illustrates the importance of dissipation in the possible development of spatial inhomogeneities.

Hydrodynamic modes for a granular gas from kinetic theory

Small perturbations of the homogeneous cooling state for a low density granular gas are described by means of the linearized Boltzmann equation. The spectrum of the generator for this dynamics is shown to contain points corresponding to hydrodynamic excitations. The corresponding eigenvectors and eigenvalues are calculated to Navier-Stokes order and shown to agree with those obtained by the Chapman-Enskog method. The conditions for the hydrodynamic excitations to dominate all other excitations are discussed.

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.