Rigorous Results in Existence and Selection of SaffmanTaylor Fingers by Kinetic Undercooling
Abstract
The selection of SaffmanTaylor 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 HeleShaw cell by small but nonzero 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 SaffmanTaylor 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 SaffmanTaylor fingers by kinetic undercooling, Journal of Engineering Mathematics 46, 132]. The main subtlety is the behavior of the Stokes lines at the finger tip, where the analysis is complicated by nonanalyticity of coefficients in the governing equation.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1610.00076
 Bibcode:
 2016arXiv161000076X
 Keywords:

 Mathematics  Classical Analysis and ODEs