Numerical experiments on Hele Shaw flow with a sharp interface

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Abstract
The fingering instability of an interface between two immiscible fluids in a Hele Shaw cell is simulated numerically. The algorithm used is based on a transcription of the equations of motion for the interface in which it formally becomes a generalized vortex sheet. The evolution of this sheet is computed using a variant of the vortex-in-cell method. The resulting scheme and code make it possible to follow the collective behaviour of many competing and interacting fingers well into the nonlinear, large-amplitude regime. It is shown that in this regime the evolution is controlled essentially by just one dimensionless parameter, the ratio of fluid viscosities. The effects of varying this parameter are studied and the results compared with experimental investigations. Scaling properties of the average density profile across the evolving mixed layer between the two homogeneous fluid phases are investigated. Many phenomena are observed that must be characterized as collective interactions and thus cannot be understood in terms of flows with just a single finger.

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