Pattern formation of frictional fingers in a gravitational potential

Aligned finger structures, with a characteristic width, emerge during the slow drainage of a liquid-granular mixture in a tilted Hele-Shaw cell. A transition from vertical to horizontal alignment of the finger structures is observed as the tilting angle and the granular density are varied. An analytical model is presented, demonstrating that the alignment properties are the result of the competition between fluctuating granular stresses and the hydrostatic pressure. The dynamics is reproduced in simulations. We also show how the system explains patterns observed in nature, created during the early stages of a dike formation.

Figure 1

(a) Top view of the Hele-Shaw cell. The coordinate $y$ is running from the outlet towards the upper edge of the cell; $\kappa$ is the curvature (inverse of the in-plane radius of curvature $R$) along the interface (orange dashed line). The front is a region of accumulated grains along the air-liquid interface; $L$ is the thickness of this front. The cell is $20\times 30{\mathrm{cm}}^2$. (b) Side view. The cell is tilted by an angle $\alpha$. The cell gap is $h=0.5$ mm. The filling fraction $\varphi$ is the height of the initial sedimented granular layer relative to $h$.

Figure 2

(a) Air finger displacing submerged grains. The air-liquid-grain interface appears as a black line. The interface is surrounded by a compaction front, which appears darker than the undisturbed sedimented layer of grains to the right in the image. The scale bar is 1.0 mm. (b) Close-up of the interface showing grains in contact with the interface as a bright band due to reflection of illumination from the side. The scale bar is 0.5 mm; $q=0.07\mathrm{ml}/\min$, $\varphi =0.4$, and $\alpha =0$ for both images.

Figure 3

Final configuration of the experimentally observed pattern at constant tilt angle $\alpha =4^{\circ }$ with varying filling fraction $\varphi$. Residual compacted granular material appears dark and empty regions of the cell appear white. The finger alignment changes direction from vertical to horizontal as $\varphi$ increases. Each image frame is 200 mm wide.

Figure 4

Final configuration of experimental pattern at different tilt angles $\alpha$ at filling fraction $\varphi =0.05$. Each image frame is 200 mm wide.

Figure 5

Snapshots of the dynamics, experiments versus simulations. (a) The pattern is dominated by vertically aligned fingers at $\varphi =0.025$ and $\alpha =5^{\circ }$. (b) The pattern is dominated by horizontally aligned fingers at $\varphi =0.2$ and $\alpha =3^{\circ }$. See (a) SM video 1 and (b) SM video 2 [39].

Figure 6

Pairwise comparison of the final configuration of experiments (black left frames) to simulations (blue right frames) for different values of the filling fraction $\varphi$ and the tilting angle $\alpha$. The red lines indicate contours of constant $\eta$, which are estimated up to a constant factor in Eq. (9). As $\eta$ increases, the vertical alignment turns into horizontal alignment and then into no alignment. The value of $\eta$ doubles for every contour. The gravitational pull is pointing downward in every frame.

Figure 7

Histograms of the alignment angle $\theta$ sample from the finger contour for each data point in the phase diagram in Fig. 6. The sample contains 10 000 values of $\theta$ and the histograms consist of 15 equally spaced bins. Here $\theta =0$ and $\theta =\pi /2$ correspond, respectively, to vertically and horizontally aligned finger structures.

Figure 8

Statistics of the samples in Fig. 7 plotted against $ln\eta$. Circles correspond to sample medians and bars correspond to the smallest intervals of $\theta$ which contain 30% of the sample values, $\text{SI}\left(0.3\right)$. The triplets of numbers along the $ln\eta$ axis indicate the values of $(\eta ,\alpha ,\varphi )$ for each sample.

Figure 9

Feature comparison 1 between (a) the experimental observations at $\varphi =0.4$ and $\alpha =4^{\circ }$ and (b) the remaining structures on dike walls found in the Inmar formation. Fingers (red arrows) are being intercepted by a finger (green arrows) which grows perpendicular to the average flow direction. The gravitational pull is indicated by $g$. The scale bar in (b) applies to both experiment and the dike figure.

Figure 10

Feature comparison 2 between (b) the experimental observations at $\varphi =0.025$ and $\alpha =4^{\circ }$ and (a) the remaining structures on dike walls found in the Inmar formation. Aligned finger structures with tip splitting and termination are marked by blue and red triangles, respectively. The gravitational pull is indicated by $g$. The scale bar in (a) applies to both experiment and the dike figure.