Simulation of Hierarchical Viscous Fingering Pattern in Lifting Hele-Shaw Cell

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Viscous fingers in the lifting Hele-Shaw cell form a hierarchical pattern due to competition between growing fingers in a converging geometry. If the defending fluid is visco-plastic, a permanent three-dimensional pattern is formed, which is usually approximated as quasi-two-dimensional. We present here a Monte-Carlo simulation which attempts to reproduce the real 3-dimensional pattern in rectangular geometry using a very simple algorithm. We study the finger length distribution, the rate of increase of coverage of the invading fluid, the height profile of the final pattern and observe how these change with characteristics of the fluids.