Two-finger selection theory in the Saffman-Taylor problem

TL;DR

This paper investigates the selection of stationary solutions with two unequal fingers in the Saffman-Taylor problem, revealing how surface tension influences their widths and positions, and providing explicit spectral expressions.

Contribution

It introduces a solvability theory that predicts discrete sets of two-finger solutions with unequal widths, scaling laws, and explicit spectra, extending understanding beyond single-finger solutions.

Findings

Discrete sets of two-finger solutions are selected by surface tension.

Selected widths scale as $d_0^{2/3}$ and $d_0^{1/3}$.

Explicit approximate spectral expressions are provided.

Abstract

We find that solvability theory selects a set of stationary solutions of the Saffman-Taylor problem with coexistence of two \it unequal \rm fingers advancing with the same velocity but with different relative widths $\lambda_1$ and $\lambda_2$ and different tip positions. For vanishingly small dimensionless surface tension $d_0$, an infinite discrete set of values of the total filling fraction $\lambda = \lambda_1 + \lambda_2$ and of the relative individual finger width $p=\lambda_1/\lambda_2$ are selected out of a two-parameter continuous degeneracy. They scale as $\lambda-1/2 \sim d_0^{2/3}$ and $|p-1/2| \sim d_0^{1/3}$. The selected values of $\lambda$ differ from those of the single finger case. Explicit approximate expressions for both spectra are given.