Labyrinth Patterns in Confined Granular-Fluid Systems

A random, labyrinthine pattern emerges during slow drainage of a granular-fluid system in two-dimensional confinement. Compacted grains are pushed ahead of the fluid-air interface, which becomes unstable due to a competition between capillary forces and the frictional stress mobilized by grain-grain contact networks. We reproduce the pattern formation process in numerical simulations and present an analytical treatment that predicts the characteristic length scale of the labyrinth structure. The pattern length scale decreases with increasing volume fraction of grains in the system and increases with the system thickness.

Figure 1

(a) Experiments. Pictures i–iv show a time sequence of fingers of air invading the initially circular granular-fluid system. Picture v shows the resulting branching structure of close-packed grains (white cluster). The Hele-Shaw cell plate spacing was 0.4 mm and the volume fraction of grains 20% [14]. (b) Simulations. The pattern forming process has been reproduced by modeling the competing capillary and frictional forces that govern the instability. The white areas represent air, the black compact mass, and gray the fluid with sedimented grains [15].

Figure 2

Characteristic length scale as a function of normalized volume fraction. (a) Experiments. The volume fraction of grains $\phi$ in the mixture increases from left to right as labeled in the pictures. All experiments were conducted with a plate spacing of 0.4 mm. The dimensions of each picture frame are $40\times 40\mathrm{cm}$. (b) Simulations. The observed decrease in the length scale of the pattern with increased volume fraction is reproduced in the simulations.

Figure 3

Sketch of the interface. (a) Top view of an invading finger of air with built-up mass along the front. (b) Layer of compacted grains of width $L$ ahead of the interface.

Figure 4

Characteristic length scale as a function of volume fraction of granular material. The measured wavelengths are plotted for experiments (filled squares) and simulations (open circles). The line represents characteristic wavelengths predicted by the analytic scaling law. The inset shows pattern length scale as a function of plate spacing with $\phi$ kept constant at 20%.