Onset of Convection in Strongly Shaken Granular Matter

Strongly vertically shaken granular matter can display a density inversion: A high-density cluster of beads is elevated by a dilute gaslike layer of fast beads underneath (“granular Leidenfrost effect”). For even stronger shaking the granular Leidenfrost state becomes unstable and granular convection rolls emerge. This transition resembles the classical onset of convection in fluid heated from below at some critical Rayleigh number. The same transition is seen in molecular dynamics (MD) simulations of the shaken granular material. The critical shaking strength for the onset of granular convection can be calculated from a linear stability analysis of a hydrodynamiclike model of the granular flow. Experiment, MD simulations, and theory quantitatively agree.

Figure 1

Onset of convection in experiment and in MD simulations. (a)–(d) Snapshots of $F=11.1$ layers of $d=1.0\mathrm{mm}$ steel beads in a container of length $L/d=101$ shaken at an amplitude of $a=3.0\mathrm{mm}$. The frequency was linearly increased from $f=42\mathrm{Hz}$ to $f=48\mathrm{Hz}$ at a rate of $90\mathrm{Hz}/\min$. The transition from the steady Leidenfrost state to fully developed convection took place at $f=45\mathrm{Hz}$ [between frames (a) and (b)]. (e)–(h) Snapshot from the corresponding MD simulations. Again, $F=11.1$, shaking amplitude $a=3.0\mathrm{mm}$, and linearly increased frequency (at the rate of $90\mathrm{Hz}/\min$) with $f=44\mathrm{Hz}$ in (e). The onset of convection takes place at a frequency of $f=45\mathrm{Hz}$ ($S=75$, frame $f$). The light-colored particles are moving upward and the dark ones downward. Respective movies of both experiments and numerical simulations can be seen in Ref. 39.

Figure 2

Convection threshold in the $(S,F)$ phase diagram from experiments, MD simulations, and theory. The experiments and simulations are performed with $d=1\mathrm{mm}$ glass beads, with shaking amplitude $a=2.0\mathrm{mm}$ (blue dots), $a=3.0\mathrm{mm}$ (red squares), and $a=4.0\mathrm{mm}$ (black triangles). The shaking strength $S$ is increased by increasing the frequency $f$. The open symbols correspond to the onset of convection in the experiments and the solid ones to the onset in the MD simulations. The solid line is the result of the linear stability analysis of the Leidenfrost state ($\mathrm{Pr}=1.7$).