Nonlocal Effects Reflect the Jamming Criticality in Frictionless Granular Flows Down Inclines

The jamming transition is accompanied by a rich phenomenology such as hysteresis or nonlocal effects that is still not well understood. Here, we experimentally investigate a model frictionless granular layer flowing down an inclined plane as a way to disentangle generic collective effects from those arising from frictional interactions. We find that thin frictionless granular layers are devoid of hysteresis of the avalanche angle, yet the layer stability increases as it gets thinner. Steady rheological laws obtained for different layer thicknesses can be collapsed into a unique master curve, supporting the idea that nonlocal effects are the consequence of the usual finite-size effects associated with the presence of a critical point. This collapse indicates that the so-called isostatic length $l^{*}$, the scale on which pinning a boundary freezes all remaining floppy modes, governs the effect of boundaries on flow and rules out other propositions made in the past.

Figure 1

(a) Top: Picture of the silica particles. Bottom: Schematic of the electrostatic repulsive forces preventing frictional particle contacts. (b) Schematic of the experimental setup. (c) Successive steps to prepare a uniform and presheared layer of grains. (d) Image from Camera 1 giving access to the particle surface velocity $v_s$ using particle image velocimetry. (e) Image from Camera 2 used to measure the pile thickness $h$ from the laser sheet position and the pile stability from the time correlation of the laser speckle.

Figure 2

Stability of a frictionless granular layer. (a) Interval halving procedure: controlled inclination angle $\theta$ (in blue) and resulting speckle correlation $C$ (in red) versus time for a layer of thickness $h/d\simeq 22$. Red dashed line: $C_{\mathrm{th}}=0.96$ delimiting flow $(C<C_{\mathrm{th}})$ from arrest $(C>C_{\mathrm{th}})$. Blue dashed line: critical stability angle ${\theta }_c(h/d)$. (b) Stability diagram: granular layer thickness $h/d$ versus critical stability angle ${\theta }_c$. Different open markers indicate different runs. Crosses correspond to the quasistatic friction coefficient ${\mu }_c(h/d)=\mu (J\to 0,h/d)$ obtained from Fig. 3. Black line: best fit with $h/d={[(\tan {\theta }_c-\tan {\theta }_c^{\infty })/a]}^{-\alpha }$ yielding $\alpha =-0.9\pm 0.1$, $a=0.15\pm 0.03$, and ${\theta }_c^{\infty }=5.0°\pm 0.1°$.

Figure 3

(a) Macroscopic friction coefficient $\mu$ versus viscous number $J$ for different layer thicknesses $h/d$. Solid lines: Best fits with $\mu ={\mu }_c+b^{\prime }{J}^{{\beta }^{\prime }}$. The values of ${\mu }_c(h/d)$ (crosses) are reported in Fig. 2 using the relation ${\mu }_c(h/d)=\tan {\theta }_c(h/d)$. Inset: Same data plotting the reduced macroscopic friction $\mu -{\mu }_c(h/d)$ versus $J$. (b) Rescaled flow rule plotting $(h/d{)}^{1/\alpha }(\mu -{\mu }_c^{\infty })$ versus $(h/d{)}^{1/\alpha \beta }J$ with $\alpha =0.9$ and $\beta =0.35$. Black line: $g(x)=a+(c{x}^{\gamma })(b{x}^{\beta })/(b{x}^{\beta }+c{x}^{\gamma })$, with $a=0.15$, $b=2.0\pm 0.1$, $c=29\pm 5$, and $\gamma =0.73\pm 0.05$.