Selection of the Saffman-Taylor Finger Width in the Absence of Surface Tension: An Exact Result

Abstract

Using exact time-dependent nonsingular solutions [Mineev-Weinstein and Dawson, Phys. Rev. E 50, R24 (1994); Dawson and Mineev-Weinstein, Physica (Amsterdam) 73D, 373 (1994)], we solve the Saffman-Taylor finger selection problem in the absence of surface tension. We show that a generic interface in a Hele-Shaw cell evolves to a nonlinearly stable single uniformly advancing finger occupying one-half of the channel width. This result contradicts the generally accepted belief that surface tension is indispensable for the selection of the $\frac{1}{2}$-width finger.