The Effect of Finiteness in the Saffman–Taylor Viscous Fingering Problem


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Abstract

We derive a family of exact time-evolving solutions for the the evolution of a finite blob of fluid confined to a channel in a Hele–Shaw cell. We show rigorously that, for large fluid volume, there are solutions for which one of the interfaces approaches the steady Saffman–Taylor finger solution of arbitrary width Λ∈(0, 1). On the basis of this, we argue that the far-field effects of a displaced second interface do not provide a selection mechanism for the formation of a width-½ finger when surface tension, or any other regularization, is ignored.

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