Continuous to intermittent flows in growing granular heaps

If a granular material is poured from above on a horizontal surface between two parallel, vertical plates, a sand heap grows in time. For small piles, the grains flow smoothly downhill, but after a critical pile size $X_c$, the flow becomes intermittent: sudden avalanches slide downhill from the apex to the base, followed by an “uphill front” that slowly climbs up, until a new downhill avalanche interrupts the process. By means of experiments, controlling the distance between the apex of the sandpile and the container feeding it from above, we show that $X_c$ grows linearly with the input flux, but scales as the square root of the feeding height. We explain these facts from a phenomenological model based on the experimental observation that the flowing granular phase forms a “wedge” on top of the static one, differently from the case of stationary heaps. Moreover, we demonstrate that our controlled experiments allow to predict the value of $X_c$ for the common situation in which the feeding height decreases as the pile increases in size.

Figure 1

Experimental setup. (a) Sketch of the experimental setup, where the sliding container allows to control the value of the grain feeding height $h$ (the darker color indicates the mobile grains). (b) Diagram showing three stages of the heap growth at constant $h$, corresponding to times $t_1, t_2$, and $t_3$. The upper side shows a lateral view of the Hele-Shaw cell including a vertical line of pixels that allows to construct the spatial-temporal diagram shown in the bottom panel. The diagram, which corresponds to a real experiment, illustrates the constancy of $h$.

Figure 2

Quantifying the continuous to intermittent flow transition. The gray continuous line corresponds to the temporal evolution of the average slope of the pile, while the black continuous line represents its horizontal length very near the bottom of the Hele-Shaw cell. The light gray line (dashed) is a fit following the law $x\sim t^{1/2}$. The transition occurs at $X_c\approx 10.3$ cm, $t_c\approx 20$ s, and with a repose angle of ${\theta }_s\approx 33.5^{\circ }$, as clearly seen in the zoom shown in the inset. The data represented in the graphs correspond to a fixed input flux of $F_{\text{in}}=2.68{\mathrm{cm}}^2/\mathrm{s}$ and a constant deposition height of $h\approx 1.5$ cm.

Figure 3

Dependencies of the transition length on the input flux and deposition height. Transition length as a function of (a) the input flux, for different deposition heights, and (b) the deposition height, for different input fluxes. Symbols represent experimental data while the lines the theoretical expression proposed in Eq. (3). Note that the bottom of the cell has a finite size of 28 cm, thus for certain parameter values the base of the pile reaches the open edge of the cell before transition occurs. This leads to some curves shown having fewer data points than others.

Figure 4

Quantitative comparison between experiments and model. Dependence with (a) the input flux and (b) with the deposition height of the proportionality constant in Eq. (3), $P$. Black dashed lines represent the value of $P=X_c/\left(F_{\text{in}}\sqrt{h}\right)$ used in our model to reproduce the experimental data, and the height of the gray rectangles represents its standard error. $P_{\text{model}}=3.27\pm 0.65\mathrm{s}/{\mathrm{cm}}^{3/2}$.

Figure 5

Transition time dependency on the deposition height. Transition time as a function of deposition height for different input fluxes.

Figure 6

Equivalence between fixed-$h$ and variable-$h$ experiments. The black solid lines show the deposition height for a variable-$h$ experiment versus (a) the pile length and (b) the time, while the dashed increasing lines represent (a) a quadratic fit and (b) a linear fit to the dependence between the respective magnitudes at the transition for fixed-$h$ experiments (squares are experimental points). Gray solid lines follow the law (a) $h=23.5$ cm $-0.67x$ and (b) $h=23.5$ cm $-(0.99\mathrm{cm}/{\mathrm{s}}^{-1/2})\sqrt{t}$. The insets display the time evolution of the pile horizontal length for variable-$h$ experiments. (a) Input flow of $2.43{\mathrm{cm}}^2/\mathrm{s}.$ (b) Input flow of $0.75{\mathrm{cm}}^2/\mathrm{s}$.

Figure 7

(a) Flowing layer in a typical experiment during the continuous regime. The image is obtained from the overlay of 100 images (corresponding to $1/10$ second) and then applying an edge detector. (b) Sketch used to derive the model equations. The region indicated with dashed lines represents the control volume (CV) while the solid white lines on top of them represent the control surfaces (CS) through which there is interchange of mass and momentum. The control volume moves with velocity $v_{\perp }$ to include only the fluid layer. The velocity $v_{\parallel }=\frac{1}{\delta w}{\int }_0^w{\int }_0^{\delta }{v}_{x}dzdy$.