Potential flows and Helmholtz decomposition

[pigna]

Hi. I'm studying fluid dynamics and in particular potential flows. I know that for an irrotational flow the velocity field is a conservative field and it can be rapresented by the gradient of a scalar field v=-∇Φ. In this case the explicit form of Φ is something like a line integral between a reference point where Φ=0 and a generic point of the domain.

This can be obtained using the stokes theorem and the domain has to be simply connected. Moreover i know that a generic vectorial field ( without any assuption about the fact it is irrotational or not, solenoidal or not) can be decomposed using the helmholtz theorem in the form v=- ∇Φ +∇×Ψ where Φ is a scalar potential while ψ is a vectorial potential. In this case the explicit form of the two potential require a non local integration over the volume and over the boundaries and greens functions are used to find out this results ( l have found the explicit formulations reporter on wikipedia as on other sources and I have also find them out by myself).

I'm a little confused because I thought that imposing the curl of velocity equal to zero in the explicit formulation of the helmholtz decomposition it should reduce, in some ways, to a potential form as the one obtained previously considering directly the flow as irrotational and using the stokes theorem. I have struggled a lot with this issue, but I haven't obtained any results. Can someone give me a tip or a reference or tell me where I'm wrong...
Thanks...

 

[Orodruin]

The Helmholtz decomposition is not unique. In general there will be several different combinations of scalar and vector potentials that will give you the same field. However, in the case of an irrotational flow, it is possible to put the vector potential to zero.

 

[pigna]

It is exactly what I'm not able to do. I have tried to do that or to rewrite the curl term as a gradient but I haven't reached any result... do you know some book or internet content that address this issue in a comprehesible way?

 

[Orodruin]

Could you show your work?

 

[pigna]

 

[pigna]

Ok l have posted two time the same images... sorry... as you can see confusion and no conclusions. Probably when i say that I ' m not sure if an integral goes or not to zero it actually goes because in different case I don't even use the hypothesis of irrotational flows. And as you can see I'm not able to set ∇×Ψ=0. There are a lot of calculations and so I send pictures, I Hope they are understandable... thanks you very much if you take a look...