Instabilities in the homogeneous cooling of a granular gas: A quantitative assessment of kinetic-theory predictions

Abstract

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.

Higher-order moment theories for dilute granular gases of smooth hard spheres

“…Note that the results from the G26 theory presented above are in the dilute limit (φ → 0). One may notice from Mitrano et al (2011) that their results are also in excellent agreement with those obtained from the theoretical expression of Garzó (2005) at φ = 0.1 for 0.5 e 1, and the latter (green dashed line)-for φ = 0-again agree perfectly with those obtained from the G26 theory for 0.5 e 1. Therefore, it is stated that the G26 theory also provides the correct critical length for all values of the coefficient of restitution.…”

“…The critical length in units of the mean free path ℓ0 plotted over the coefficient of restitution e. The solid (black) line denotes the critical length computed from (6.29) and (6.27) while the dashed (green) line and the squares denote the critical lengths computed from the theoretical expressions obtained in Garzó (2005) and Brey et al (1998b), respectively. The circles depict the two-dimensional DSMC simulation results of Brey et al (1998b) while the triangles represent the molecular dynamic simulation results of Mitrano et al (2011) at solid fraction φ = 0.1.…”

“…Therefore it can be expected that the results from full three-dimensional DSMC simulation of a dilute granular gas on critical length would agree with those predicted by the G26 theory (black solid line). The triangles represent the results of molecular dynamic simulations carried out by Mitrano et al (2011) at solid fraction φ = 0.1. Note that the results from the G26 theory presented above are in the dilute limit (φ → 0).…”

Impact of roughness on the instability of a free-cooling granular gas

“…In addition, comparison between kinetic theory and computer simulations for the critical size can be considered a clean way of assessing the degree of accuracy of the former, since theoretical results are obtained in most of the cases under certain approximations (e.g., by considering the leading terms in a Sonine polynomial expansion). The accuracy of the kinetic-theory prediction of L c for a low-density monodisperse granular gas of inelastic smooth hard spheres [4,5] has been verified by means of the direct simulation Monte Carlo DSMC method [6] and, more recently, by molecular dynamics (MD) simulations for a granular fluid at moderate density [7,8]. Similar comparisons have also been made for polydisperse systems [9][10][11][12] in the low-density regime [13] and moderate densities [14].…”

A numerical study of the Navier–Stokes transport coefficients for two-dimensional granular hydrodynamics

“…Some examples include applications of the Navier-Stokes hydrodynamics to symmetry breaking and density/temperature profiles in vibrated gases [26,27], binary mixtures [45] and supersonic flow past a wedge in real experiments [4,46,47]. Another group refers to spatial perturbations of the homogeneous cooling state for an isolated system where the MD results of the critical length for the onset of vortices and clusters [48,49] are successfully compared with the predictions from linear stability analysis [50] performed on the basis of the GDL transport coefficients.…”