Numerical approach to frictional fingers

Experiments on confined two-phase flow systems, involving air and a dense suspension, have revealed a diverse set of flow morphologies. As the air displaces the suspension, the beads that make up the suspension can accumulate along the interface. The dynamics can generate “frictional fingers” of air coated by densely packed grains. We present here a simplified model for the dynamics together with a new numerical strategy for simulating the frictional finger behavior. The model is based on the yield stress criterion of the interface. The discretization scheme allows for simulating a larger range of structures than previous approaches. We further make theoretical predictions for the characteristic width associated with the frictional fingers, based on the yield stress criterion, and compare these to experimental results. The agreement between theory and experiments validates our model and allows us to estimate the unknown parameter in the yield stress criterion, which we use in the simulations.

Figure 1

Experimental setup. (a) Hele-Shaw cell dimensions. (b) The system is fixed horizontally, filled with a fluid and sedimented beads, and driven either by air injection or by withdrawal of the liquid through a syringe pump connected at the nozzle at the sealed short side channel. (c) The advancing gas phase accumulates a front of grains.

Figure 2

(a) Bead size distribution. (b) Cone of granular material poured through a funnel; angle of repose, $\approx {27}^{\circ }\mathrm{C}$. Microscopy images of beads (Malvern; Morphology G3) show thir approximately spherical shape.

Figure 3

The pattern formation process is documented over a 10-h period, with 2-h intervals between the individual images. Air is injected through the inlet at the bottom side. The cell is $20\times 30$ cm.

Figure 4

Closeup of the frictional finger pattern where the compacted front is visible as the dark band surrounding the air fingers. Inset: The front thickness $L$ and the radius of curvature $R$, which are local parameters along the interface. The scale bar is 20 mm long.

Figure 5

Finger formation for increasing values of the filling fraction $\varphi$. Liquid is drained from the bottom side. The cell is $20\times 30$ cm.

Figure 6

Sketch of the menisci around the beads at the air-liquid interface. The beads are wetting, resulting in concave menisci in the interspace between the beads.

Figure 7

Discretization procedure. (a) Hele-Shaw cell seen from above. The air phase is on the left-hand side. Adjacent to the air interface is the front, which is an accumulated region of beads. We can assign a front thickness, $L$, at every mobile point along the interface. (b) The interface is discretized as a chain of nodes. The beads are discretized into a two-dimensional concentration field, which takes the value 1 in the front, and the initial filling fraction $\varphi$ in the regions of not yet accumulated/sedimented beads. The size of the grid cells and the node spacing are exaggerated for the purposes of the illustration. $L_i$ is the shortest distance from node $i$, through the accumulated regions, to a point in the bead field below 1. The grid-cell candidates are limited to the shaded circular section of ${90}^{\circ }$, centered around the direction perpendicular to the interface.

Figure 8

Schematic of the moving segment. Seven nodes are moving. The old configuration is represented by circles on the dashed line; the new configuration, by pentagons on the solid line. The spline which determines the updated positions is based on the two neighboring nodes on each side of the interval (hexagons) and the center node moved a distance ${\delta }_{\text{move}}$ in the direction normal to the chain. The positions of the new nodes are distributed along the spline function. Dimensions in the figure are exaggerated.

Figure 9

Examples of the evolution of the numerical scheme with a central, circular injection point. The size of the geometry is $10\times 10$ cm. Each row represents a time series of the evolution. Top row, $\varphi =0.1$; middle row, $\varphi =0.3$; and bottom row, $\varphi =0.5$. There is an additional noise field in the initial bead configuration, limited to $\varphi \to \varphi \pm 0.05$.

Figure 10

Schematic of a steadily growing finger. $\mathrm{\Lambda }$ is half the finger width, $L_s$ is the front thickness at the sides of the finger, and $L_t$ is the front thickness of the fingertip. The annulus section at the fingertip identifies a small region of the front at the fingertip, bounded by an angle $\theta$. The maximum curvature of the fingertip is $\kappa =1/R$.

Figure 11

The characteristic length $\mathrm{\Lambda }$ is the ratio between the area of the finger structures and the finger structure circumference. Error bars correspond to one standard deviation; data points without error bars correspond to single observations. The dashed line corresponds to the best fit of the theoretical prediction in Eq. (15). The slope of $0.38$ cm corresponds to the numerical value prefactor $2\sqrt{\gamma \xi /{\sigma }_{\xi }}$, which is used to infer the numerical value of $\xi \gamma /{\sigma }_{\xi }$.