Doubt about irrotational flows

In summary, the conversation discusses the concept of circulation in closed paths and its relationship to irrotational flow. The questioner is confused about the possibility of an irrotational vortex, and the respondent explains that such a vortex is not possible and provides a source for further information. They also mention the relevance of topology in understanding this concept and provide a link to a forum discussion about it.
  • #1
Rodrigo Schmidt
14
5
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.
 
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  • #2
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.
 
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  • #3
Charles Link said:
Irrotational has ## \nabla \times \vec{v}=0## , which means by Stokes theorem ## \oint \vec{v} \cdot d \vec{l}=0 ## everywhere.
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
Rodrigo Schmidt said:
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.
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
 
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  • #5
Charles Link said:
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
Thank you very much! It seems strange, but it's more clear to me now!
 
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  • #6
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

https://www.physicsforums.com/threads/struggling-with-ab-effect.872156/#post-5477281

For your purposes, you can ignore that it is in a discussion about the Aharonov-Bohm effect, although it's one of the most fascinating applications of these kind of topological arguments in physics.
 
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1. What is an irrotational flow?

An irrotational flow is a type of fluid flow in which the fluid particles move in a smooth and steady manner without any rotation. This means that the curl of the velocity vector is equal to zero, indicating that there is no net angular momentum or circulation within the flow.

2. What are some examples of irrotational flows?

Some common examples of irrotational flows include laminar flow in a pipe, flow around a sphere, and flow over an airfoil. These flows are characterized by smooth and organized motion of fluid particles without any swirling or eddying patterns.

3. How does an irrotational flow differ from a rotational flow?

In an irrotational flow, the fluid particles move in a smooth and orderly manner without any rotation. On the other hand, in a rotational flow, the fluid particles have a swirling or circular motion around a central axis. This results in a non-zero curl of the velocity vector, indicating the presence of angular momentum and circulation.

4. What are the applications of studying irrotational flows?

Understanding irrotational flows is crucial in many fields of science and engineering, such as aerodynamics, hydrodynamics, and fluid mechanics. It is also relevant in studying weather patterns, ocean currents, and even blood flow in the human body. Additionally, it can help in designing more efficient and streamlined structures and devices.

5. How do scientists and engineers analyze irrotational flows?

There are various mathematical and computational techniques used to analyze irrotational flows, such as Bernoulli's equation, the stream function, and potential flow theory. Experimental methods, such as flow visualization and particle image velocimetry, can also be used to study and measure the characteristics of irrotational flows.

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