Laplacian Growth I: Finger Competition and Formation of a Single Saffman-Taylor Finger without Surface Tension: An Exact Result

摘要

We study the exact non-singular zero-surface tension solutions of theSaffman-Taylor problem for all times. We show that all moving logarithmicsingularities a_k(t) in the complex plane \omega = e^{i\phi}, where \phi is thestream function, are repelled from the origin, attracted to the unit circle andeventually coalesce. This pole evolution describes essentially all thedynamical features of viscous fingering in the Hele-Shaw cell observed bySaffman and Taylor [Proc. R. Soc. A 245, 312 (1958)], namely tip-splitting,multi-finger competition, inverse cascade, and subsequent formation of a singleSaffman-Taylor finger.