Analytic Theory of the Selection Mechanism in the Saffman-Taylor Problem

Abstract

We present an analytic approach to the problem of predicting the widths of fingers in a Hele-Shaw cell. Our analysis is based on the WKB technique developed recently for dealing with the effects of surface tension in the problem of dendritic solidification. We find that the relation between the dimensionless width $\ensuremath{\lambda}$ and the dimensionless group of parameters containing the surface tension, $\ensuremath{\nu}$, has the form $\ensuremath{\lambda}\ensuremath{-}\frac{1}{2}\ensuremath{\sim}{\ensuremath{\nu}}^{\frac{2}{3}}$ in the limit of small $\ensuremath{\nu}$.