Wire perturbations in the Saffman-Taylor problem: Pattern selection of asymmetric fingers.

Abstract

Solvability conditions are derived for the asymmetric Saffman-Taylor finger. In the absence of surface tension the solution contains two free parameters \ensuremath{\lambda} and ${y}_{0}$, where \ensuremath{\lambda} is the dimensionless finger width and ${y}_{0}$ is the degree of asymmetry. In the presence of surface tension, it is shown that solvability conditions are not satisfied, signaling the appearance of a cusp on the finger boundary. The mismatch angle, \ensuremath{\Delta}\ensuremath{\theta}, due to this cusp is shown to be concave. An earlier postulate made in connection with the wire experiment is clarified here, which states that \ensuremath{\Delta}\ensuremath{\theta} mainly depends on the tangential slope of the finger profile. The remaining portion of this paper is devoted to understanding recent experiments by G. Zocchi and co-workers and M. Rabaud and co-workers [Phys. Rev. A 35, 1894 (1987); 37, 935 (1988)], where fingers were perturbed by wires. Assuming the contact angle created by the wire to be concave, we first show within linear approximation that, for ${y}_{0}$\ensuremath{\ll}1, the absolute magnitudes of the mismatch angles created by the wire and the bubble at the center are the same, establishing that all the results obtained by the bubble perturbations can be carried over to a wire experiment. We then present a theoretical prediction for \ensuremath{\delta}, the distance between the tip of the asymmetric finger and the wire at the center, as a function of external parameters. In the limit of small surface tension, the results are in fair agreement with the available data.