Coupled Leidenfrost states as a monodisperse granular clock

Using an event-driven molecular dynamics simulation, we show that simple monodisperse granular beads confined in coupled columns may oscillate as a different type of granular clock. To trigger this oscillation, the system needs to be driven against gravity into a density-inverted state, with a high-density clustering phase supported from below by a gaslike low-density phase (Leidenfrost effect) in each column. Our analysis reveals that the density-inverted structure and the relaxation dynamics between the phases can amplify any small asymmetry between the columns, and lead to a giant oscillation. The oscillation occurs only for an intermediate range of the coupling strength, and the corresponding phase diagram can be universally described with a characteristic height of the density-inverted structure. A minimal two-phase model is proposed and a linear stability analysis shows that the triggering mechanism of the oscillation can be explained as a switchable two-parameter Andronov-Hopf bifurcation. Numerical solutions of the model also reproduce similar oscillatory dynamics to the simulation results.

Figure 1

Simulation results for $N=500$ and $v_b=24$: (a) Uncoupled granular columns with $h_w=0$; (b) the height of the center of mass ($y_{1\mathrm{com}}$) and the number of beads ($N_1$) for the left column in (a); (c) two coupled granular columns with $h_w=32$; (d) $y_{1\mathrm{com}}$ and $N_1$ for the left column in (c).

Figure 2

Coupled dynamics for $N=500, v_b=30$: $N_1\left(t\right)$ for weak coupling at $h_w=12$ (a), moderate coupling at $h_w=58$ (b), and strong coupling at $h_w=90$ (c); (d) the first positive peak value $C_1$ of the autocorrelation function of $s_1\left(t\right)$; the averaged period (e) and amplitude (f) of $s_1\left(t\right)$ for different $h_w$, respectively, with relative error bars in arbitrary units.

Figure 3

Oscillation phase diagram: (a) Oscillatory regimes with lower boundaries shown in open circles and upper boundaries in open squares, different $N$ is distinguished by colors, and no oscillation is observed for $N=700, v_b<12$ and $N=300, v_b>26$ (marked by vertical solid lines). (b) Collapsed phase diagram in the $h_w-h_{\text{inv}}$ plane.

Figure 4

Time-sequential snapshots with density analysis for one oscillation under $N=500, v_b=24$, and $h_w=32$: (a) $t=290$; (b) $t=300$; (c) $t=320$; (d) $t=330$; instantaneous density profiles are shown in horizontal bars on the left (column 1) or right (column 2) side of each granular column, and the gray dashed line indicates the height of the window.

Figure 5

Theoretical model and numerical solutions for $N=500$ and $v_b=24$: (a) The simplified density profile (red line) for a granular column; (b) the net flows between phases considered in our model; (c) $N_1\left(t\right)$ for the steady state at $h_w=20$ and the oscillatory state at $h_w=30$; (d) oscillation amplitude and period of $N_1\left(t\right)$ for different $h_w$.