Surface waves and vorticity in 2D

[hunt_mat]

TL;DR

How do surface waves change in the presence of vorticity?

The classical free surface profile for the solitary wave for irrotational and incompressible fluids for small amplitude and long wavelength is the classical Korteweg-deVries(KdV) equation given by:$$\frac{\partial\eta}{\partial t}+\frac{\partial \eta}{\partial x}+\eta\frac{\partial\eta}{\partial x}+\frac{\partial^{3}\eta}{\partial x^{3}}=0$$
I'm interested in removing the irrrotational aspect of the initial assumptions. The vorticity equation in 2D can be written as:$$\frac{D\omega}{Dt}=0,\quad\omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$$, ``Combining'' these two equations yields:$$\frac{D}{Dt}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)=0$$
The full equation is:$$\frac{\partial}{\partial t}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)+u\frac{\partial}{\partial x}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)+v\frac{\partial}{\partial y}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)=0$$
From here I can do the usual approximations to get a similar equation to the KdV? Where am I going wrong here?

 

[jambaugh]

I've not dived deep into your question but my impression is that vorticity can't propagate without viscosity in the medium. If you effectively are assuming zero viscosity then ... hmmm... I would have to deep dive to speculate further.

 

[Arjan82]

That's not true actually. Euler equations (i.e. Navier-Stokes without viscosity) can also contain viscosity. But other than that I don't have an answer here.

 

[hunt_mat]

I've not dived deep into your question but my impression is that vorticity can't propagate without viscosity in the medium. If you effectively are assuming zero viscosity then ... hmmm... I would have to deep dive to speculate further.

What you have is that on a given streamline, vorticity is conserved. There are plenty of vorticity solutions in 2D without viscosity.

 

[hunt_mat]

That's not true actually. Euler equations (i.e. Navier-Stokes without viscosity) can also contain viscosity. But other than that I don't have an answer here.

How can Euler's equations contain viscosity?

 

[Arjan82]

They don't. But they can contain vorticity (my bad.. I see the typo now...)