Why is it right that the irrotational flow must have the velocity potential?

AI Thread Summary
Irrotational flow must have a velocity potential due to the Helmholtz theorem, which states that if a vector field is irrotational, it can be expressed as the gradient of a scalar potential. The discussion emphasizes that if the curl of a flow is non-zero, it indicates that a potential function does not exist, as the curl of a gradient is always zero. The relationship between irrotational flow and velocity potential is crucial for understanding fluid dynamics. Additional resources were shared to further clarify these concepts. The conversation highlights the foundational principles of fluid mechanics related to irrotational flow and potential functions.
Chuck88
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When I am studying the ratation of the fluid, I found one sentence: "The irrotational flow must have the velocity potential." Why? Can someone tell me the derivation of this equation?
 
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Hi Chuck88! :smile:
Chuck88 said:
When I am studying the ratation of the fluid, I found one sentence: "The irrotational flow must have the velocity potential." Why? Can someone tell me the derivation of this equation?

It's the Helmholtz theorem, see http://en.wikipedia.org/wiki/Helmholtz_decomposition#Statement_of_the_theorem

If F is irrotational, and if φ is defined as shown there, you can check that φ = F :wink:
 
tiny-tim said:
Hi Chuck88! :smile:


It's the Helmholtz theorem, see http://en.wikipedia.org/wiki/Helmholtz_decomposition#Statement_of_the_theorem

If F is irrotational, and if φ is defined as shown there, you can check that φ = F :wink:

Thanks for your reply and from the materials you provided, I have known some of the knowledge I did not know before. Another question is that suppose I have alreadly got the value of a curl, which is not equal to zero, how can I prove reversely that the potential does not exist?
 
Hi Chuck88! :smile:
Chuck88 said:
… suppose I have alreadly got the value of a curl, which is not equal to zero, how can I prove reversely that the potential does not exist?

Because curl grad = 0.

So if the flow F has a potential φ,

then F = φ.

so x F = x φ = 0. :wink:
 
tiny-tim said:
Hi Chuck88! :smile:


Because curl grad = 0.

So if the flow F has a potential φ,

then F = φ.

so x F = x φ = 0. :wink:

Thanks for your reply. I have found one link which could is also quite useful.
http://mathinsight.org/curl_gradient_zero
 
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