Transition by Intermittency in Granular Matter: From Discontinuous Avalanches to Continuous Flow

We investigate, in the rotating drum configuration, the transition from the regime of discontinuous avalanches observed at low angular velocity to the regime of continuous flow observed at higher velocity. Instead of the hysteretic transition reported previously by Rajchenbach [Phys. Rev. Lett. 65, 2221 (1990)], with an apparent bistability of the two flow regimes in a range of drum velocities, we observe intermittency with spontaneous erratic switches from one regime to the other. Both scenarios of transition are recovered by a model dynamic equation for the avalanche flow with two sources of stochasticity: a Langevin noise during the avalanche flow and a distributed maximal stability angle at which avalanches start.

Figure 1

(a) Time evolution of the mean slope angle $\theta$ of the pile for $\Omega =1.02\mathrm{deg}/\mathrm{s}$. (b) Zoom of $\theta (t)$ on the time window delimited by the two vertical dotted lines in (a). (c) Time derivative of the slope angle $\dot{\theta }$ in the zoomed time window and (d) corresponding phase portrait ($\theta$, $\dot{\theta }$). The DA regime is plotted in red (light gray) and the CF regime in blue (dark gray).

Figure 2

Distribution of life time $T$ of (a) DA and (b) CF regime for $\Omega =0.99\mathrm{deg}/\mathrm{s}$. (c) Corresponding survival function $p(T>t)$. The solid lines correspond to exponential fits $\exp (-T/\overline{T})$ with (a) ${\overline{T}}_{\mathrm{DA}}=48\mathrm{s}$ and (b) ${\overline{T}}_{\mathrm{CF}}=33\mathrm{s}$.

Figure 3

Mean life time $\overline{T}$ of DA regime (▽) and CF regime (▲) as a function of $\Omega$. (-) Exponential fits $\overline{T}=T_c\exp [\beta (\Omega -{\Omega }_c)/{\Omega }_c]$ with $T_c=30\mathrm{s}$, ${\Omega }_c=1\mathrm{deg}/\mathrm{s}$, ${\beta }_{\mathrm{DA}}=-7.9$ and ${\beta }_{\mathrm{CF}}=8.9$.

Figure 4

Mean time fraction $\varphi$ of the CF regime as a function of the drum rotation velocity $\Omega$. Experimental data points (◯) and fit by a tanh curve (-).

Figure 5

Numerical phase portrait ($\theta -{\theta }_n$, $\dot{\theta }$) from Eq. (2) for $\Omega =1\mathrm{deg}/\mathrm{s}$ ($A=8{\mathrm{s}}^{-2}$, $B=0.45{deg}^{-1}$). (a) Calculated trajectory for $\delta ϵ=0$ starting from the initial values (1.4,1) or (0.8,1). (b) Calculated CF trajectory for $\delta ϵ=8\mathrm{deg}/{\mathrm{s}}^{-2}$ starting from the fixed point. The horizontal dashed line corresponds to $\dot{\theta }=\Omega$ and the curved dashed line corresponds to the separatrix between the DA and CF regimes starting from (${\theta }_s$, $\Omega$) and that reaches a maximum value of $\dot{\theta }$ when $\dot{\theta }=\Omega$.

Figure 6

Sketch of the state diagram given by model Eq. (2) with critical velocities ${\Omega }_1$ and ${\Omega }_2$ as functions of the noise level $\delta ϵ$ for a given observation time $T_{\mathrm{obs}}$.