Abstract
We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the relative velocity field is a Beltrami field, meaning that the vorticity is collinear to the velocity. The existence theory is based on multi-parameter bifurcation theory.
Highlights
This paper is concerned with three-dimensional steady water waves driven by gravity and surface tension
In a frame of reference moving with the wave, the fluid motion is governed by the stationary Euler equations (u · ∇)u = −∇ p − ge3 ∇·u=0 in η, in η, with a kinematic boundary condition on the top and bottom boundaries: u·n=0 on ∂ η, and a dynamic boundary condition on the free surface: p = patm − 2σ K M on z = η(x )
We can turn this into a solution of the stationary Euler equations in the threedimensional domain by letting η and u equal the two-dimensional solution for every y = x · e⊥, where e⊥ = e3 × e
Summary
This paper is concerned with three-dimensional steady water waves driven by gravity and surface tension. In a frame of reference moving with the wave, the fluid motion is governed by the stationary Euler equations (u · ∇)u = −∇ p − ge3 ∇·u=0 in η, in η, with a kinematic boundary condition on the top and bottom boundaries: u·n=0 on ∂ η, and a dynamic boundary condition on the free surface:. 1 + |∇η|2 while σ > 0 is the coefficient of surface tension and patm the constant atmospheric pressure. Any divergence-free Beltrami field generates a solution to the stationary Euler equations with pressure p = C − |u|2 − gz. One interesting feature of Beltrami flows is that they include laminar flows whose direction varies with depth (see Section 1.2.1 and Figure 1) This could potentially be of interest when considering a wind-induced surface current interacting with a subsurface current in a different direction
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
