The free boundary of steady axisymmetric inviscid flow with vorticity I: near the degenerate point

TL;DR

This paper classifies the types of singularities that occur near degenerate points on the free boundary in steady axisymmetric inviscid flows with vorticity, revealing the possible wave profiles and their geometric features.

Contribution

It provides a detailed classification of singularities near degenerate points in axisymmetric flows with vorticity, expanding understanding of free boundary behaviors in such fluid dynamics problems.

Findings

Stokes corner, horizontal cusp, and flatness profiles at stagnation points

Cusp profiles on the symmetric axis (excluding the origin)

Garabedian pointed bubble, cusp, or flatness at the origin

Abstract

In this paper, we investigate the singularity near the degenerate points of the steady axisymmetric flow with general vorticity of an inviscid incompressible fluid acted on by gravity and with a free surface. We called the points on the free boundary at which the gradient of the stream function vanishes as the degenerate points. The main results in this paper give the different classifications of the singularity near the degenerate points on the free surface. More precisely, we obtained that at the stagnation points, the possible profiles must be a Stokes corner, or a horizontal cusp, or a horizontal flatness. At the degenerate points on the symmetric axis except the origin, the wave profile must be a cusp. At the origin, the possible wave profiles must be a Garabedian pointed bubble, or a horizontal cusp, or a horizontal flatness.