# How do the Navier-Stokes equations predict vortex creation in a vorticity free fluid?

For this question I was looking at the Navier-Stokes equations after you get the curl, this gives you the equation: $$\frac{\partial \vec{\Omega}}{\partial t}-\nabla \times (\vec{v} \times \vec{\Omega})=\frac{\eta}{\rho} \nabla ^2 \vec{\Omega}$$ Most of the sources I've read talk about how this is very different from taking the curl of the euler-equation due to the extra term. I understand that but there is something I don't understand, if you start with $$\Omega=0$$ this equation would seem to imply that: $$\frac{\partial \vec{\Omega}}{\partial t}=0$$ Which means that at any point in space you would expect no new vortices to emerge and so you expect vorticity to stay $$0$$ throughout all of space so vortexes can't be created from an initially vortex free flow. This seems to break from normal intuition in how you could increase velocity in a pipe and the flow will eventually go from laminar to turbulent, or other examples where changing only velocity in a straight line can cause the emergence of vortexes. Is this because these equations no longer apply (i.e. this happens when a non-conservative force is introduced)? Or in real life do these equations still apply but you can never get a truly vortex free fluid and so the effects can build up and create more vorticity?

• Very interesting and smart question! I think that what you are writing is not the most general vorticity equation, see: en.wikipedia.org/wiki/Vorticity_equation The term with the gradients and pressure can generate vorticity if the fluid is NOT barotropic. Sep 13, 2022 at 6:42
• You can also see this answer: in your setting vorticity can only be produced at the boudary physics.stackexchange.com/a/82229/226902 Sep 13, 2022 at 7:00

Good question! The fundamental behavior of a with Navier-Stokes modeled Madelung's fluid is turbelence forming when resonating on certain energy levels called eigenvalues. Your cited transport equation is the first step to understand this "spooky" process though you will have to continue curling the navier-stokes d.e. until you get the diffusion equation.

Solving the geodesics out of this equation (after inserting it back into the original and dimensionless Navier-Stokes d. e. ) has to be done with a Schrodinger like energy equation called Calogero-Moser Hamitonian. Please do see also Citation for details.

R. Meulens , "A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation", AIP Advances 12, 015308 (2022) https://doi.org/10.1063/5.0074083

The solutions then suffice both to a higher order diffusion equation and the 3rd order Weierstrass d. e. or an ordinary Kdv Flow hierarchy eq, what gives it its name, N-solitons and are spirals, circles or lines depending on the situational parameters like input velocity, reynolds number, (dynamic) viscosity and the used length scale. Those situational parameters may determine also relevant bifurcation process points.

• This is your third post, and each time you used it to promote this article. Please stop doing this, and respect that SE is Q&A site. Provide self-contained answers as much as possible. Nov 24, 2022 at 7:27
• Believe it or not that is the sincere answer to the open question of turbelence. Alas, it is a complex subject that needs a level of elaboration to explain it with credibility. It is somewhat "spooky". Kind regards
– Reng
Nov 24, 2022 at 10:00
• Nevertheless, your answer, as it is, isn't up to SE standards because it contains very little useful information. Simply providing a link is ground to get the answer deleted. Also, "spooky" isn't a scientifically-defined term, and doesn't help in any way to make your answer useful for anyone. Nov 24, 2022 at 10:14
• I referred to the lately Mr. Einstein "spooky" comment regarding the aspect quantum mechanics crossing-over to fluid dynamics (real-life). You might be right, I did not use mathematical symbols, but all details are put in plain English. When I am better acquainted with the used Latex script version, I will upload the formulas. Kind regards
– Reng
Nov 24, 2022 at 10:19