The rich behavior of the Boltzmann equation for dissipative gases

Within the framework of the homogeneous non-linear Boltzmann equation, we present a new analytic method, without the intrinsic limitations of existing methods, for obtaining asymptotic solutions. This method permits extension of existing results for Maxwell molecules and hard spheres to large classes of particle interactions, from very hard spheres to softer than Maxwell molecules, as well as to more general forcing mechanisms, beyond free cooling and white noise driving. By combining this method with numerical solutions, obtained from the Direct Simulation Monte Carlo (DSMC) method, we study a broad class of models relevant for the dynamics of dissipative fluids, including granular gases. We establish a criterion connecting the stability of the non-equilibrium steady state to an exponentially bound form for the velocity distribution $F$, which varies depending on the forcing mechanism. Power laws arise in marginal stability cases, of which several new cases are reported. Our results provide a minimal framework for interpreting large classes of experiments on driven granular gases

Velocity fluctuations in cooling granular gases

We study the formation and the dynamics of correlations in the velocity field for 1D and 2D cooling granular gases with the assumption of negligible density fluctuations (``Homogeneous Velocity-correlated Cooling State'', HVCS). It is shown that the predictions of mean field models fail when velocity fluctuations become important. The study of correlations is done by means of molecular dynamics and introducing an Inelastic Lattice Maxwell Models. This lattice model is able to reproduce all the properties of the Homogeneous Cooling State and several features of the HVCS. Moreover it allows very precise measurements of structure functions and other crucial statistical indicators. The study suggests that both the 1D and the 2D dynamics of the velocity field are compatible with a diffusive dynamics at large scale with a more complex behavior at small scale. In 2D the issue of scale separation, which is of interest in the context of kinetic theories, is addressed.Comment: 24 pages, 16 figures, conference proceedin

Hydrodynamics of inelastic Maxwell models

An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier--Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman--Enskog method to inelastic gases. Second, the non-Newtonian rheological properties in the uniform shear flow (USF) are obtained in the steady state as well as in the transient unsteady regime. Next, an exact solution for a special class of Couette flows characterized by a uniform heat flux is worked out. This solution shares the same rheological properties as the USF and, additionally, two generalized transport coefficients associated with the heat flux vector can be identified. Finally, the problem of small spatial perturbations of the USF is analyzed with a Chapman--Enskog-like method and generalized (tensorial) transport coefficients are obtained.Comment: 40 pages, 10 figures; v2: final version published in a special issue devoted to "Granular hydrodynamics

Instability in Shocked Granular Gases

Shocks in granular media, such as vertically oscillated beds, have been shown to develop instabilities. Similar jet formation has been observed in explosively dispersed granular media. Our previous work addressed this instability by performing discrete-particle simulations of inelastic media undergoing shock compression. By allowing finite dissipation within the shock wave, instability manifests itself as distinctive high density non-uniformities and convective rolls within the shock structure. In the present study we have extended this work to investigate this instability at the continuum level. We modeled the Euler equations for granular gases with a modified cooling rate to include an impact velocity threshold necessary for inelastic collisions. Our results showed a fair agreement between the continuum and discrete-particle models. Discrepancies, such as higher frequency instabilities in our continuum results may be attributed to the absence of higher order effects.Comment: 6 pages, prepared for the proceedings of the APS Topical Group on Shock Compression of Condensed Matter in Seattle, Washington, from July 7th through July 12th, 201

Towards dense, realistic granular media in 2D

The development of an applicable theory for granular matter - with both qualitative and quantitative value - is a challenging prospect, given the multitude of states, phases and (industrial) situations it has to cover. Given the general balance equations for mass, momentum and energy, the limiting case of dilute and almost elastic granular gases, where kinetic theory works perfectly well, is the starting point.\ud \ud In most systems, low density co-exists with very high density, where the latter is an open problem for kinetic theory. Furthermore, many additional nonlinear phenomena and material properties are important in realistic granular media, involving, e.g.:\ud \ud (i) multi-particle interactions and elasticity\ud (ii) strong dissipation,\ud (iii) friction,\ud (iv) long-range forces and wet contacts,\ud (v) wide particle size distributions and\ud (vi) various particle shapes.\ud \ud \ud Note that, while some of these issues are more relevant for high density, others are important for both low and high densities; some of them can be dealt with by means of kinetic theory, some cannot.\ud \ud This paper is a review of recent progress towards more realistic models for dense granular media in 2D, even though most of the observations, conclusions and corrections given are qualitatively true also in 3D.\ud \ud Starting from an elastic, frictionless and monodisperse hard sphere gas, the (continuum) balance equations of mass, momentum and energy are given. The equation of state, the (Navier–Stokes level) transport coefficients and the energy-density dissipation rate are considered. Several corrections are applied to those constitutive material laws - one by one - in order to account for the realistic physical effects and properties listed above

Self-diffusion in granular gases: Green-Kubo versus Chapman-Enskog

We study the diffusion of tracers (self-diffusion) in a homogeneously cooling gas of dissipative particles, using the Green-Kubo relation and the Chapman-Enskog approach. The dissipative particle collisions are described by the coefficient of restitution $\epsilon$ which for realistic material properties depends on the impact velocity. First, we consider self-diffusion using a constant coefficient of restitution, $\epsilon=$const, as frequently used to simplify the analysis. Second, self-diffusion is studied for a simplified (stepwise) dependence of $\epsilon$ on the impact velocity. Finally, diffusion is considered for gases of realistic viscoelastic particles. We find that for $\epsilon=$const both methods lead to the same result for the self-diffusion coefficient. For the case of impact-velocity dependent coefficients of restitution, the Green-Kubo method is, however, either restrictive or too complicated for practical application, therefore we compute the diffusion coefficient using the Chapman-Enskog method. We conclude that in application to granular gases, the Chapman-Enskog approach is preferable for deriving kinetic coefficients.Comment: 15 pages, 1 figur

Free Cooling Phase-Diagram of Hard-Spheres with Short- and Long-Range Interactions

We study the stability, the clustering and the phase-diagram of free cooling granular gases. The systems consist of mono-disperse particles with additional non-contact (long-range) interactions, and are simulated here by the event-driven molecular dynamics algorithm with discrete (short-range shoulders or wells) potentials (in both 2D and 3D). Astonishingly good agreement is found with a mean field theory, where only the energy dissipation term is modified to account for both repulsive or attractive non-contact interactions. Attractive potentials enhance cooling and structure formation (clustering), whereas repulsive potentials reduce it, as intuition suggests. The system evolution is controlled by a single parameter: the non-contact potential strength scaled by the fluctuation kinetic energy (granular temperature). When this is small, as expected, the classical homogeneous cooling state is found. However, if the effective dissipation is strong enough, structure formation proceeds, before (in the repulsive case) non-contact forces get strong enough to undo the clustering (due to the ongoing dissipation of granular temperature). For both repulsive and attractive potentials, in the homogeneous regime, the cooling shows a universal behaviour when the (inverse) control parameter is used as evolution variable instead of time. The transition to a non-homogeneous regime, as predicted by stability analysis, is affected by both dissipation and potential strength. This can be cast into a phase diagram where the system changes with time, which leaves open many challenges for future research.Comment: 22 pages, 15 figure