On properties of vertical velocity for 2-D steady water waves

TL;DR

This paper investigates the properties of vertical velocity in 2-D steady water waves, proving the existence of inflection points, the behavior of vertical velocity along streamlines, and confirming Constantin's conjecture.

Contribution

It establishes the existence of inflection points for each streamline and confirms Constantin's conjecture on vertical velocity in Stokes waves, extending results to cases with monotonic vorticity.

Findings

Inflection points exist for each streamline.

Maximum vertical velocity occurs at inflection points.

Results extend to flows with monotonic vorticity.

Abstract

In this article, we mainly investigate the properties of vertical velocity v for two dimensional steady water waves over a flat bed. Firstly we prove the existence of the inflection point for each streamline, then we find the behavior of v along each streamline depends strictly on concavity and convexity of streamline, which contributes to complete Constantin conjecture on v in Stokes wave. And the location of maximum vertical fluid velocity is also proven to be at the inflection point. Besides, we also extend our results to the cases with monotonous vorticity.