Extending the scope of microscopic solvability: Combination of the Kruskal-Segur method with Zauderer decomposition

TL;DR

This paper extends the Kruskal-Segur method by combining it with Zauderer decomposition, enabling the solution of complex free boundary problems like dendritic growth in potential flow analytically.

Contribution

It introduces a novel combination of two analytical techniques, broadening the applicability of the Kruskal-Segur approach to nonlinear bulk equations.

Findings

First analytic solution for velocity selection in dendritic growth with flow

Extended the scope of the Kruskal-Segur method to PDE-based free boundary problems

Demonstrated effectiveness through a specific pattern formation problem

Abstract

Successful applications of the Kruskal-Segur approach to interfacial pattern formation have remained limited due to the necessity of an integral formulation of the problem. This excludes nonlinear bulk equations, rendering convection intractable. Combining the method with Zauderer's asymptotic decomposition scheme, we are able to strongly extend its scope of applicability and solve selection problems based on free boundary formulations in terms of partial differential equations alone. To demonstrate the technique, we give the first analytic solution of the problem of velocity selection for dendritic growth in a forced potential flow.