Scaling behavior of granular particles in a vibrating box

Abstract

Using numerical and analytic methods, we study the behavior of granular particles contained in a vibrating box. We measure, by molecular dynamics (MD) simulation, several quantities which characterize the system. These quantities-the density and the granular temperature fields, and the vertical expansion-obey scaling in the variable x = Af . Here, A and f are the amplitude and the frequency of the vibration. The behavior of these quantities is qualitatively different for small and large values of x. We also study the system using NavierStokes type equations developed by Haff. We develop a boundary condition for moving boundaries, and solve for the density and the temperature fields of the steady state in the quasi-incompressible limit, where the average separation between the particles is much smaller than the average diameter of the particles. The fields obtained from Haff's equations show the same scaling as those from the simulations. The origin of the scaling can be easily understood. The behavior of the fields from the theory is consistent with the simulation data for small x, but they deviate significantly for large x. We argue that the deviation is due to the breakdown of the quasi-incompressibility condition for large x.

Experimental Study of a Granular Gas Fluidized by Vibrations

“…6 that the dependence of the "granular temperature", T , on the piston velocity, V , is of the form T ∝ V θ , where θ is a decreasing function of N . Finally, we discuss our results in the light of previous works [2,3,4,5,6,7,8,9,10] and give our conclusions in Sec. 7.…”

“…It is known that for a fixed number of granular layers at rest, one has in the low-density limit P Ω ∝ T [20], where P is the mean pressure, Ω the volume and T the "granular temperature". However, the dependence of T on the vibration amplitude, A, and frequency, f , of the piston and on the number of particles, n, is still a matter of debate [7,8,9,10]. Kinetic theory [2,9] or hydrodynamic models [7] show T ∝ V 2 n −1 , whereas numerical simulations [3,4,5,6] or experiments [2,3] give T ∝ V α n −β , with 1.3 ≤ α ≤ 2 and 0.3 ≤ β ≤ 1, where V = 2πf A is the maximum velocity of the piston.…”

“…However, the dependence of T on the vibration amplitude, A, and frequency, f , of the piston and on the number of particles, n, is still a matter of debate [7,8,9,10]. Kinetic theory [2,9] or hydrodynamic models [7] show T ∝ V 2 n −1 , whereas numerical simulations [3,4,5,6] or experiments [2,3] give T ∝ V α n −β , with 1.3 ≤ α ≤ 2 and 0.3 ≤ β ≤ 1, where V = 2πf A is the maximum velocity of the piston. These previous conflicting results may be explained by the V -dependence of T that we have found from our measurements: T ∝ V θ(n) , with θ continuously varying from θ = 2 when n → 0, as expected from kinetic theory, to θ ≃ 0 for large n. We emphasize that we have only considered the dependence of the granular temperature on the vibration velocity V .…”

“…However, granular gases basically differ from ordinary ones mostly due to the inelasticity of collisions, i.e., nonconservation of energy. While over the years many attempts based on kinetic theory [1] have been made to describe such dissipative granular gases, no agreement has been found so far both with experiments [2,3] and numerical simulations [3,4,5,6], for the dependence of the "granular temperature", i.e., the mean kinetic energy per particle, on the parameters of vibration [7,8,9,10]. The aim of this study is to guess possible gas-like state equations for such a dissipative granular gas and to observe new kinetic behaviors which trace back to the inelasticity of collisions, e.g., the tendency of such media to form clusters.…”

Vibrated Granular Media as Experimentally Realizable Granular Gases

“…This case is much closer to experiments, because it is relatively simple to put particles in a box and shake them, or drive particles with a vibrating piston. Indeed, there are many experimental [4][5][6], numerical [6][7][8], and theoretical [4,[9][10][11] studies of this system. However, numerous questions remain about the link between experiments on one hand, and theory and simulations on the other.…”