Viscous-fingering experiments with periodic boundary conditions

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Abstract
Experimental realization of a Hele-Shaw cell with periodic boundary conditions (PBC’s) is achieved by building an azimuthal cell with two coaxial cylinders separated by a small gap. The development of viscous-fingering patterns formed by a critical binary liquid mixture at very low viscosity contrast has been observed and recorded from the onset of instability to very late stages. Comparison with the experimental results measured in cells which have sidewalls shows that PBC’s yield few differences of results. At the early stage there is no sidewall disturbance so the Fourier transforms show less noise in the low-wave-number modes. Because the annular cell is larger than the earlier cells, it was possible to follow the flow to a very late nonlinear stage where, instead of showing steadily lengthening and broadening fingers, the necks of the longer fingers, crowded by fattening bulbs of the tips of the less-long fingers, constrict until their width is no longer negligible in comparison to the cell gap, at which point the pattern breaks up into a rich variety of bubbles.

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