Two-finger selection theory in the Saffman-Taylor problem

Details Two-finger selection theory in the Saffman-Taylor problem Two-finger selection theory in the Saffman-Taylor problem

1999-09-15

arxiv.org/abs/cond-mat/9909225v1

Physics

Condensed Matter

Statistical Mechanics

4 pages, 3 figures

Scientific paper

10.1103/PhysRevE.60.R5013

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.

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