Two-finger selection theory in the Saffman-Taylor problem

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 $λ_1$ and $λ_2$ and different tip positions. For vanishingly small dimensionless surface tension $d_0$, an infinite discrete set of values of the total filling fraction $λ= λ_1 + λ_2$ and of the relative individual finger width $p=λ_1/λ_2$ are selected out of a two-parameter continuous degeneracy. They scale as $λ-1/2 \sim d_0^{2/3}$ and $|p-1/2| \sim d_0^{1/3}$. The selected values of $λ$ differ from those of the single finger case. Explicit approximate expressions for both spectra are given.