Scaling Behavior of Granular Particles in a Vibrating Box

TL;DR

This study investigates how granular particles in a vibrating box behave under different vibration intensities using simulations and analytical models, revealing scaling laws and their limitations.

Contribution

The paper introduces a combined numerical and analytical approach to understand granular behavior, highlighting the scaling in system quantities and the conditions where theoretical models break down.

Findings

Quantities obey scaling in variable x = Af.

Simulation and theory agree for small x.

Deviations occur for large x due to quasi-incompressibility breakdown.

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 Navier-Stokes 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 $x$.