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 unequal fingers advancing with the same velocity but with different relative widths ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ and different tip positions. For vanishingly small dimensionless surface tension ${d}_{0},$ an infinite discrete set of values of the total filling fraction $\ensuremath{\lambda}={\ensuremath{\lambda}}_{1}+{\ensuremath{\lambda}}_{2}$ and of the relative individual finger width $p={\ensuremath{\lambda}}_{1}/\ensuremath{\lambda}$ are selected out of a two-parameter continuous degeneracy. They scale as $\ensuremath{\lambda}\ensuremath{-}1/2\ensuremath{\sim}{d}_{0}^{2/3}$ and $|p\ensuremath{-}1/2|\ensuremath{\sim}{d}_{0}^{1/3}.$ The selected values of $\ensuremath{\lambda}$ differ from those of the single finger case. Explicit approximate expressions for both spectra are given.