# I Doubt about irrotational flows

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1. Aug 13, 2017

### Rodrigo Schmidt

So, i just started an elementar study on hidrodynamics and i'm stuck with something.
We have that the circulation in a closed path Γ is given by:
$C_Γ=\oint_Γ \vec v⋅\vec {dl}$
And that, in a irrotational flow, $C_Γ = 0$ for any given Γ.
But if we have an irrotational vortex wouldn't $\oint_Γ \vec v⋅ \vec {dl} \neq 0$ ?
This seems contradictory, and i really can't notice how to handle with this.
Sorry if this seems too basic, but i'm really just starting with this topic.

2. Aug 13, 2017

I think the answer is that there is no such thing as an irrotational vortex. Am I missing something? In general a vortex has $\oint \vec{v} \cdot d \vec{l} \neq 0$, and the more energetic it is, the larger this integral. Irrotational (non-rotational) means there are no vortexes (vortices) present. $\\$ Irrotational has $\nabla \times \vec{v}=0$, which means by Stokes theorem $\oint \vec{v} \cdot d \vec{l}=0$ everywhere.

Last edited: Aug 13, 2017
3. Aug 13, 2017

### Rodrigo Schmidt

That's exactly what i tought, but the thing is that in my book and in some internet sources there are mentions of supposedly irrotational vortexes (which are also called potential vortexes), which seems very strange and confusing to me.

4. Aug 13, 2017

The mathematical stack exchange gives a brief explanation of what you are referring to. This is the first time I have seen this kind of thing, but their explanation with an animated diagram gives an illustration of it. https://math.stackexchange.com/questions/428839/irrotational-vortices

5. Aug 13, 2017

### Rodrigo Schmidt

Thank you very much! It seems strange, but it's more clear to me now!

6. Aug 14, 2017

### vanhees71

Well, the socalled "irrotational vortex" or "potential vortex" is among the nicest examples for the importance of topology in "classical vector analysis". One should demonstrate it with great care to any student when disucssing Poincare's lemma. I've done this already in these forums