Previous work has indicated that inelastic grains undergoing homogeneous cooling may be unstable, giving rise to the formation of velocity vortices, which may also lead to particle clustering. In this effort, molecular dynamics (MD) simulations are performed over a wide parameter space to determine the critical system size demarcating the stable and unstable regions. Specifically, a system of monodisperse, frictionless, inelastic hard spheres is simulated for restitution coefficients e ! 0.6 and solids fractions / 0.4. Simulations for each e, / pairing are then carried out over a range of system sizes to determine the critical dimensionless length scale L C =d (L is the system length and d is the particle diameter), above which velocity vortices appear (unstable system) and below which they are suppressed (stable system). The results show excellent agreement with the theoretical predictions obtained by Garzó [Phys. Rev. E 72, 021106 (2005)] using a linear stability analysis of kinetic-theory-based (continuum) equations that were derived from the Enskog equation. Finally, the time required for onset of the unstable behavior is also explored via MD and found to be a universal function of the ratio of L=d to L C =d.

An intriguing phenomenon displayed by granular flows and predicted by kinetic-theory-based models is the instability known as particle "clustering," which refers to the tendency of dissipative grains to form transient, loose regions of relatively high concentration. In this work, we assess a modified-Sonine approximation recently proposed [Garzó, Santos, and Montanero, Physica A 376, 94 (2007)] for a granular gas via an examination of system stability. In particular, we determine the critical length scale associated with the onset of two types of instabilities--vortices and clusters--via stability analyses of the Navier-Stokes-order hydrodynamic equations by using the expressions of the transport coefficients obtained from both the standard and the modified-Sonine approximations. We examine the impact of both Sonine approximations over a range of solids fraction φ<0.2 for small restitution coefficients e = 0.25-0.4, where the standard and modified theories exhibit discrepancies. The theoretical predictions for the critical length scales are compared to molecular dynamics (MD) simulations, of which a small percentage were not considered due to inelastic collapse. Results show excellent quantitative agreement between MD and the modified-Sonine theory, while the standard theory loses accuracy for this highly dissipative parameter space. The modified theory also remedies a high-dissipation qualitative mismatch between the standard theory and MD for the instability that forms more readily. Furthermore, the evolution of cluster size is briefly examined via MD, indicating that domain-size clusters may remain stable or halve in size, depending on system parameters.

A linear stability analysis of the Navier-Stokes (NS) granular hydrodynamic equations is performed to determine the critical length scale for the onset of vortices and clusters instabilities in granular dense binary mixtures. In contrast to previous attempts, our results (which are based on the solution to the inelastic Enskog equation to NS order) are not restricted to nearly elastic systems since they take into account the complete nonlinear dependence of the NS transport coefficients on the coefficients of restitution αij . The theoretical predictions for the critical length scales are compared to molecular dynamics (MD) simulations in flows of strong dissipation (αij ≥ 0.7) and moderate solid volume fractions (φ ≤ 0.2). We find excellent agreement between MD and kinetic theory for the onset of velocity vortices, indicating the applicability of NS hydrodynamics to polydisperse flows even for strong inelasticity, finite density, and particle dissimilarity.PACS numbers: 05.20. Dd, 45.70.Mg, 51.10.+y, Although hydrodynamics is frequently used to describe rapid granular flows, there are still some open questions about the domain of validity of this description [1]. As for ordinary fluids at moderate densities, the constitutive equations for the irreversible fluxes and the forms of the transport coefficients can be derived from the revised Enskog kinetic theory (RET) [2] conveniently adapted to account for the dissipative dynamics. The derivation of such fluxes from the corresponding kinetic equation assumes the existence of a hydrodynamic regime where all space and time dependence of the distribution function only occurs through the hydrodynamic fields (normal solution). The Chapman-Enskog expansion [3] around the homogeneous cooling state (HCS) provides a constructive means to obtain an approximation to such a normal solution for states in which spatial gradients are not too large. A first-order Chapman-Enskog expansion provides the Navier-Stokes (NS) hydrodynamic equations and also explicit expressions for the corresponding transport coefficients, which are defined as functions of the coefficient of restitution and other system parameters. However, there are still some concerns regarding the transition from kinetic theory to hydrodynamics beyond the quasielastic limit [1]. The reason for this concern resides in the fact that the inverse of the cooling rate, which measures the rate of energy loss due to collisional dissipation, introduces a new timescale not present for elastic collisions. The variation of the granular temperature over this new timescale is faster than over the usual hydrodynamic timescale. However, as the inelasticity increases, it is possible that the system could lack a separation of time scales between the hydrodynamic and the pure kinetic excitations such that there is no aging to hydrodynamics or, in the language of kinetic theory, there is no normal solution at finite dissipation.Strictly speaking, to definitively address the validity of hydrodynamics for dissipative systems, the complete...

In this work we quantitatively assess, via instabilities, a Navier-Stokes-order (smallKnudsen-number) continuum model based on the kinetic theory analogy and applied to inelastic spheres in a homogeneous cooling system. Dissipative collisions are known to give rise to instabilities, namely velocity vortices and particle clusters, for sufficiently large domains. We compare predictions for the critical length scales required for particle clustering obtained from transient simulations using the continuum model with molecular dynamics (MD) simulations. The agreement between continuum simulations and MD simulations is excellent, particularly given the presence of well-developed velocity vortices at the onset of clustering. More specifically, spatial mapping of the local velocity-field Knudsen numbers (Kn u ) at the time of cluster detection reveals Kn u 1 due to the presence of large velocity gradients associated with vortices. Although kinetic-theory-based continuum models are based on a smallKn (i.e. small-gradient) assumption, our findings suggest that, similar to molecular gases, Navier-Stokes-order (small-Kn) theories are surprisingly accurate outside their expected range of validity.

Flow instabilities driven by the dissipative nature of particle–particle interactions have been well documented in granular flows. The bulk of previous studies on such instabilities have considered the impact of inelastic dissipation only and shown that instabilities are enhanced with increased dissipation. The impact of frictional dissipation on the stability of grains in a homogeneous cooling system is studied in this work using molecular dynamics (MD) simulations and kinetic-theory-based predictions. Surprisingly, both MD simulations and theory indicate that high levels of friction actually attenuate instabilities relative to the frictionless case, whereas moderate levels enhance instabilities compared to frictionless systems, as expected. The mechanism responsible for this behaviour is identified as the coupling between rotational and translational motion. These results have implications not only for granular materials, but also more generally to flows with dissipative interactions between constituent particles – cohesive systems with agglomeration, multiphase flows with viscous dissipation, etc.