Understanding an instability in vibrated granular monolayers

Dynamics and kinetic theory of hard spheres under strong confinement

The kinetic theory description of a low-density gas of hard spheres or disks, confined between two parallel plates separated at a distance smaller than twice the diameter of the particles, is addressed starting from the Liouville equation of the system. The associated Bogoliubov, Born, Green, Kirkwood, and Yvon hierarchy of equations for the reduced distribution functions is expanded in powers of a parameter measuring the density of the system in the appropriate dimensionless units. The Boltzmann level of description is obtained by keeping only the two lowest orders in the parameter. In particular, the one-particle distribution function obeys a couple of equations. Contrary to what happens with a Boltzmann-like kinetic equation that has been proposed for the same system on a heuristic basis, the kinetic theory formulated here admits stationary solutions that are consistent with equilibrium statistical mechanics, in both the absence and presence of external fields. In the latter case, the density profile is rather complex due to the coupling between the inhomogeneities generated by the confinement and by the external fields. The general theory formulated provides a solid basis for the study of the properties of strongly confined dilute gases.

Energy nonequipartition in a collisional model of a confined quasi-two-dimensional granular mixture

A collisional model of a confined quasi-two-dimensional granular mixture is considered to analyze homogeneous steady states. The model includes an effective mechanism to transfer the kinetic energy injected by vibration in the vertical direction to the horizontal degrees of freedom of grains. The set of Enskog kinetic equations for the velocity distribution functions of each component is derived first to analyze the homogeneous state. As in the one-component case, an exact scaling solution is found where the time dependence of the distribution functions occurs entirely through the granular temperature T. As expected, the kinetic partial temperatures T_{i} of each component are different and, hence, energy equipartition is broken down. In the steady state, explicit expressions for the temperature T and the ratio of partial kinetic temperatures T_{i}/T_{j} are obtained by considering Maxwellian distributions defined at the partial temperatures T_{i}. The (scaled) granular temperature and the temperature ratios are given in terms of the coefficients of restitution, the solid volume fraction, the (scaled) parameters of the collisional model, and the ratios of mass, concentration, and diameters. In the case of a binary mixture, the theoretical predictions are exhaustively compared with both direct simulation Monte Carlo and molecular dynamics simulations with a good agreement. The deviations are identified to be originated in the non-Gaussianity of the velocity distributions and on microsegregation patterns, which induce spatial correlations not captured in the Enskog theory.

Granular dynamics and gravity

The role of gravity on the dynamics of granular particles is examined via their velocity distributions. Acceleration due to gravity, particle number and the coefficient of restitution have all been varied. Various quantitative relationships governing the relationship between parameters involved in the vertical and horizontal velocity component distributions are derived and applied to the data, and for the horizontal velocity components' distribution in particular remarkably good agreement is obtained.

Resonant phenomena and mechanism in vibrated granular systems

We were motivated to perform this research by the investigation of Brownian motors in excited granular materials converting the chaotic motion of granules into the oriented motion of motors. We conducted experimental studies to explore the horizontal motion of granules in vertically vibrated annular granular systems, including mixed and pure granular systems with an asymmetrical periodic structure on the bottom. The variations of the horizontal granular flow caused by the height, vibrating parameters, and mixing ratio were described in detail. Our results revealed considerable changes in the horizontal flow of different granular systems. Most importantly, resonance was induced in the horizontal granular flow by the vertical vibration; that is, the horizontal flow reached its maximum at specific vibrating parameters. A collisional model of rigid objects was constructed to probe the flowing resonances in these granular systems and provided a qualitative agreement with the experimental results obtained. We conclude that when a flowing resonance occurs, the granular system oscillates horizontally with a natural frequency under periodic external excitation. The frequency matching between the external excitation and the horizontal oscillation is responsible for the flowing resonance. Our results could improve the current understanding of the dynamic properties of granular systems under external excitation.

Inhomogeneous cooling state of a strongly confined granular gas at low density

The inhomogeneous cooling state describing the hydrodynamic behavior of a freely evolving granular gas strongly confined between two parallel plates is studied, using a Boltzmann kinetic equation derived recently. By extending the idea of the homogeneous cooling state, we propose a scaling distribution in which all the time dependence occurs through the granular temperature of the system, while there is a dependence on the distance to the confining walls through the density. It is obtained that the velocity distribution is not isotropic, and it has two different granular temperature parameters associated to the motion perpendicular and parallel to the confining plates, respectively, although their cooling rates are the same. Moreover, when approaching the inhomogeneous cooling state, energy is sometimes transferred from degrees of freedom with lower granular temperature to those with a higher one, contrary to what happens in molecular systems. The cooling rate and the two partial granular temperatures are calculated by means of a Gaussian approximation. The theoretical predictions are compared with molecular dynamics simulation results and a good agreement is found.