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?
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