We perform a numerical study of the fluctuations of the rescaled hydrodynamic transversevelocity field during the cooling state of a homogeneous granular gas. We are interestedin the role of Molecular Chaos for the amplitude of the hydrodynamic noise and itsrelaxation in time. For this purpose we compare the results of Molecular Dynamics (MD,deterministic dynamics) with those from Direct Simulation Monte Carlo (DSMC, randomprocess), where Molecular Chaos can be directly controlled. It is seen that the large timedecay of the fluctuation’s autocorrelation is always dictated by the viscosity coefficientpredicted by granular hydrodynamics, independently of the numerical scheme (MD or DSMC).On the other side, the noise amplitude in Molecular Dynamics, which is known toviolate the equilibrium Fluctuation-Dissipation relation, is not alwaysaccurately reproduced in a DSMC scheme. The agreement between the two models improves ifthe probability of recollision (controlling Molecular Chaos) is reduced by increasing thenumber of virtual particles per cells in the DSMC. This result suggests that DSMC is notnecessarily more efficient than MD, if the real number of particles is small(~103 ± 104) and if one is interested in accurately reproducefluctuations. An open question remains about the small-times behavior of theautocorrelation function in the DSMC, which in MD and in kinetic theory predictions is nota straight exponential.

Continuum mechanics (e.g., hydrodynamics, elasticity theory) is based on the assumptionthat a small set of fields provides a closed description on large space and time scales.Conditions governing the choice for these fields are discussed in the context of granularfluids and multi-component fluids. In the first case, the relevance of temperature orenergy as a hydrodynamic field is justified. For mixtures, the use of a total temperatureand single flow velocity is compared with the use of multiple species temperatures andvelocities.

An overview of recent results pertaining to the hydrodynamic description (both Newtonianand non-Newtonian) of granular gases described by the Boltzmann equation for inelasticMaxwell models is presented. The use of this mathematical model allows us to get exactresults for different problems. First, the Navier–Stokes constitutive equations withexplicit expressions for the corresponding transport coefficients are derived by applyingthe Chapman–Enskog method to inelastic gases. Second, the non-Newtonian rheologicalproperties in the uniform shear flow (USF) are obtained in the steady state as well as inthe transient unsteady regime. Next, an exact solution for a special class of Couetteflows characterized by a uniform heat flux is worked out. This solution shares the samerheological properties as the USF and, additionally, two generalized transportcoefficients associated with the heat flux vector can be identified. Finally, the problemof small spatial perturbations of the USF is analyzed with a Chapman–Enskog-like methodand generalized (tensorial) transport coefficients are obtained.

We consider a dense granular shear flow in a two-dimensional system. Granular systems(composed of a large number of macroscopic particles) are far from equilibrium due toinelastic collisions between particles: an external driving is needed to maintain themotion of particles. Theoretical description of driven granular media is especiallychallenging for dense granular flows. This paper focuses on a gravity-driven densegranular Poiseuille flow in a channel. A special focus here is on the intriguingphenomenon of fluid-solid coexistence: a solid plug in the center of the system,surrounded by fluid layers. To find and analyze various flow regimes, a multi-scaleapproach is taken. On macro scale, granular hydrodynamics is employed. On micro scale,event-driven molecular dynamics simulations are performed. The entire phase diagram ofparameters is explored, in order to determine which flow regime occurs in various regionsin the parameter space.

The study of the fluctuations in the steady state of a heated granular system isreviewed. A Boltzmann-Langevin description can be built requiring consistency with theequations for the one- and two-particle correlation functions. From the Boltzmann-Langevinequation, Langevin equations for the total energy and the transverse velocity field arederived. The existence of a fluctuation-dissipation relation for the transverse velocityfield is also studied.

The interplay between dissipation and long-range repulsive/attractive forces inhomogeneous, dilute, mono-disperse particle systems is studied. Thepseudo-Liouville operator formalism, originallyintroduced for hard-sphere interactions, is modified such that it provides very goodpredictions for systems with weak long-range forces at low densities, with the ratio ofpotential to fluctuation kinetic energy as control parameter. By numerical simulations,the theoretical results are generalized with empirical, density dependentcorrection-functions up to moderate densities.

The main result of this study on dissipative cooling is an analytical prediction for the reduced cooling rate due torepulsive forces and for the increased rate due to attractive forces. In the latter case,surprisingly, for intermediate densities, similar cooling behavior is observed as insystems without long-range interactions. In the attractive case, in general, dissipationleads to inhomogeneities earlier and faster than in the repulsive case.

A computer aided method using symbolic computations that enables the calculation of thesource terms (Boltzmann) in Grad’s method of moments is presented. The method is extremelypowerful, easy to program and allows the derivation of balance equations to very highmoments (limited only by computer resources). For sake of demonstration the method isapplied to a simple case: the one-dimensional stationary granular gas under gravity. Themethod should find applications in the field of rarefied gases, as well. Questions ofconvergence, closure are beyond the scope of this article.

The evolution of a force-free granular gas with a constant restitution coefficient isstudied by means of granular hydrodynamics. We numerically solve the hydrodynamicequations and analyze the mechanisms of cluster formation. According to our findings, thepresently accepted mode-enslaving mechanism may not be responsible for the latterphenomenon. On the contrary, we observe that the cluster formation is mainly driven byshock-waves, which spontaneously originate and develop in the system. This agrees with apreviously suggested mechanism of formation of density singularities in one-dimensionalgranular gases.

The space missions Voyager and Cassini together with earthbound observations revealed a wealth of structures in Saturn’s rings. There are, for example, waves being excited at ring positions which are in orbital resonance with Saturn’s moons. Other structures can be assigned to embedded moons like empty gaps, moon induced wakes or S-shaped propeller features. Furthermore, irregular radial structures are observed in the range from 10 meters until kilometers. Here some of these structures will be discussed in the frame of hydrodynamical modeling of Saturn’s dense rings. For this purpose we will characterize the physical properties of the ring particle ensemble by mean field quantities and point to the special behavior of the transport coefficients. We show that unperturbed rings can become unstable and how diffusion acts in the rings. Additionally, the alternative streamline formalism is introduced to describe perturbed regions of dense rings with applications to the wake damping and the dispersion relation of the density waves.