Spreading of a granular droplet under horizontal vibrations

By means of three-dimensional discrete element simulations, we study the spreading of a granular droplet on a horizontally vibrated plate. Apart from a short transient with a parabolic shape, the droplet adopts a triangular profile during the spreading. The dynamics of the spreading is governed by two distinct regimes: a superdiffusive regime in the early stages driven by surface flow followed by a second one which is subdiffusive and governed by bulk flow. The plate bumpiness is found to alter only the spreading rate but plays a minor role on the shape of the granular droplet and on the scaling laws of the spreading. Importantly, we show that in the subdiffusive regime, the effective friction between the plate and the granular droplet can be interpreted in the framework of the $\mu \left(I\right)$-rheology.

Figure 1

Sketch of the DEM simulated granular droplet laid on a rough base driven by horizontal vibrations and mimicking the experiment in Ref. [8].

Figure 2

Profiles of a granular droplet while spreading at acceleration $\mathrm{\Gamma }=1.10$ on a substrate of bumpiness $\lambda =1$. The chronological sequences have been recorded at times represented in units of the vibration period: $t/T=1/8$ (top), $t/T=3$ (middle), and $t/T=12$ (bottom). The color gradient codes the $z$ position of the grains in the droplet. These simulation snapshots have been rendered by the open visualization tool Ovito [18].

Figure 3

Mean two-dimensional profiles (open symbols) of a granular droplet at different instants along with their corresponding best parabolic fits (solid lines) for $\mathrm{\Gamma }=0.39$ and substrate bumpiness $\lambda =1$. Note the large discrepancy between the horizontal and the vertical scales.

Figure 4

The granular droplet profiles of Fig. 3 are normalized here by their instantaneous maximum width $W\left(t\right)$ and height $H\left(t\right)$ (open symbols). The black solid line represents a normalized parabolic function $z\left(x\right)=1-{\left(2x\right)}^2$ added to the plot as a guide to the eye.

Figure 5

Time evolution of a granular droplet height $H\left(t\right)$ and its width $W\left(t\right)$ while horizontally vibrating a rough substrate ($\lambda =1$) at different acceleration rates $\mathrm{\Gamma }$.

Figure 6

Spreading dynamics of a granular droplet on a rough substrate ($\lambda =1$) following the model proposed in Ref. [8] [Fig. 3 therein] at different acceleration rates $\mathrm{\Gamma }$.

Figure 7

A log-log plot of the granular droplet contact width on a rough substrate ($\lambda =1$) for acceleration rates $\mathrm{\Gamma }=0.39$ and 1.10. Two different spreading regimes are evidenced.

Figure 8

Evolution of the mean packing fraction $\varphi$ averaged over the granular droplet width in the case of rough substrate $\lambda =1$ at different acceleration rates $\mathrm{\Gamma }$.

Figure 9

Time evolution of $W$ relatively to its initial value $W_0$ for different substrate bumpiness $\lambda$ at the same acceleration rate $\mathrm{\Gamma }=1.10$ (upper part). The height ratio $H/H_0$ evolution is also represented (lower part).

Figure 10

Normalized solid packing fraction comparison for different substrate bumpiness at the same acceleration rate $\mathrm{\Gamma }=1.10$. ${\varphi }_0$ is the packing fraction before activating the vibrations.

Figure 11

Normalized width and height comparison for different accelerations at the same bumpiness $\lambda =0$. $W_0$ and $H_0$ are, respectively, the initial width and height.

Figure 12

Profiles of the coarse-grained vector and scalar velocity fields averaged over two vibration periods. All the velocities are normalized by the maximum velocity in both configuration $t/T=5$ (top) and at $t/T=30$ (bottom). The substrate bumpiness is $\lambda =1$ and acceleration $\mathrm{\Gamma }=1.10$. We show only the half of the droplet since the spreading is symmetrical.

Figure 13

Vertical profiles of the horizontal and vertical velocity ($v_x$ and $v_z$) calculated at $t=5T$ (a, c) and $t=30T$ (b, d) respectively.

Figure 14

Time evolution of the effective friction coefficient $\mu$ for different acceleration rates $\mathrm{\Gamma }$ at constant bumpiness $\lambda =1$.

Figure 15

Effective friction coefficient $\mu$ for different bumpiness values $\lambda$. Straight lines are the best fits.

Figure 16

Mean packing fraction for different bumpiness values $\lambda$. Straight lines are the best fits.