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

Question

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?

Answer 1

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.