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

[Chuck88]

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?

 

[tiny-tim]

Hi Chuck88!

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

 

[Chuck88]

Hi Chuck88!


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

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?

 

[tiny-tim]

Hi Chuck88!

… 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.

 

[Chuck88]

Hi Chuck88!


Because curl grad = 0.

So if the flow F has a potential φ,

then F = ∇φ.

so ∇ x F = ∇ x ∇φ = 0.

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