Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl

TL;DR

This paper develops a novel mathematical framework to analyze steady axisymmetric water waves with vorticity and swirl, using bifurcation theory to construct solution curves without restrictions on stagnation points.

Contribution

It introduces a new formulation of the free boundary problem that allows for global bifurcation analysis of water waves with vorticity and swirl.

Findings

Construction of local and global solution curves bifurcating from laminar flows.

Overcoming coordinate singularities to apply Rabinowitz's bifurcation theorem.

No restrictions on stagnation points in the flow.

Abstract

We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. This can be formulated as an elliptic free boundary problem in terms of Stokes' stream function. A change of variables allows us to overcome the generic coordinate-induced singularities and to cast the problem in the form "identity plus compact", which is amenable to Rabinowitz' global bifurcation theorem, while no restrictions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local and global solution curves, bifurcating from laminar flows with a flat surface, are constructed.