Rigorous Results in Existence and Selection of Saffman-Taylor Fingers by Kinetic Undercooling

TL;DR

This paper rigorously analyzes the existence and selection of symmetric Saffman-Taylor fingers in a Hele-Shaw cell influenced by small kinetic undercooling, revealing conditions under which solutions exist based on a zero Stokes multiplier.

Contribution

It provides a rigorous mathematical framework for finger selection with kinetic undercooling, extending previous surface tension results and addressing subtleties in the analysis.

Findings

Existence of symmetric finger solutions near width 1/2 for small undercooling.

Identification of conditions involving the Stokes multiplier for solution existence.

Demonstration of similarities and differences with surface tension-based finger selection.

Abstract

The selection of Saffman-Taylor fingers by surface tension has been extensively investigated. In this paper we are concerned with the existence and selection of steadily translating symmetric finger solutions in a Hele-Shaw cell by small but non-zero kinetic undercooling ($\epsilon^2 $). We rigorously conclude that for relative finger width $\lambda$ near one half, symmetric finger solutions exist in the asymptotic limit of undercooling $\epsilon^2 ~\rightarrow ~0$ if the Stokes multiplier for a relatively simple nonlinear differential equation is zero. This Stokes multiplier $S$ depends on the parameter $\alpha \equiv \frac{2 \lambda -1}{(1-\lambda)}\epsilon^{-\frac{4}{3}} $ and earlier calculations have shown this to be zero for a discrete set of values of $\alpha$. While this result is similar to that obtained previously for Saffman-Taylor fingers by surface tension, the analysis for the problem with kinetic undercooling exhibits a number of subtleties as pointed out by Chapman and King (2003) [The selection of Saffman-Taylor fingers by kinetic undercooling, Journal of Engineering Mathematics 46, 1-32]. The main subtlety is the behavior of the Stokes lines at the finger tip, where the analysis is complicated by non-analyticity of coefficients in the governing equation.