A boundary-integral method for two-phase displacement in Hele-Shaw cells

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Abstract
We develop a time-dependent numerical algorithm, using a boundary-integral approach, to investigate fingering in Hele-Shaw cells. Starting from a sinusoidal variation in the initial interface, stable fingers quickly form for a wide range of the dimensionless surface-tension parameter. For very low values of the parameter, the incipient finger bifurcates. The stable fingers are clearly the same as those obtained by McLean & Saffman (1981) using a steady-state algorithm. These steady-state solutions were found to be linearly unstable. We resolve this apparent discrepancy regarding stability by tracing the fate of small disturbances placed on and about the finger tip. We show that some small disturbances do, indeed, grow initially; however, they reach a maximum amplitude and decay as they convect backward from the tip of the finger to regions where stabilizing surface tension is the major physical force. Relatively large imposed disturbances, on the other hand, cause a finger to bifurcate; the critical disturbance amplitude decreases as the surface tension is reduced

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