Irrotational/Rotational Flows and Velocity Potentials

[jeffbarrington]

Hi,

I am a bit confused about rotational and irrotational flows, in this way:

When I do exam questions/problems, there is often a bit at the start of a question on flows about why you can treat the velocity as the gradient of a potential. The only information it gives you is that it is incompressible and irrotational. Now, I would have said that 'irrotational' implies that the curl of the velocity is zero, which I have always been told implies that the velocity can be written as the gradient of a potential. But then what about the flow velocity u_0/r in the theta_hat direction, in cylindrical coordinates? This has zero curl, but the flow comes back round on itself so can't be written as a gradient of a potential, which throws "curl(A) = 0 => A = grad(f)" into doubt. What on Earth is going on here? Is "curl(A) = 0 => A = grad(f)" incomplete?

It seems that the condition is that there mustn't be some net circulation about some point, in which case I would have said 'irrotational' does not necessarily mean zero vorticity as many sources give, but instead means there isn't a net circulation about a point, whilst there is with the 1/r velocity field. I have a feeling this 1/r thing may be causing a breakdown of "curl(A) = 0 => A = grad(f)" because of the flow's unrealistic behaviour as r -> 0.

Thanks in advance.

 

[Charles Link]

With a quick look at it, I could not spot any flaw in your initial logic. Perhaps we're both missing something simple, but I think you may be correct, that the velocity function is not well-behaved as r ==>0 may make Stokes' theorem not valid for this case. I will take a closer look at it, but it is a bit of a puzzle. $\\$ Editing..This one is interesting. @Ray Vickson Can you take a look at this please.