In multivariate calculus classes you learn a theorem that says that "A vector field is the gradient of a potential function on a domain $D$ if and only if it's curl-free on $D$."
When I try to apply my intuition to this theorem, I get somewhat confused. In the context of a 2D fluid flow:
On the one hand, the velocity field of a vortex looks like it should have non-zero curl.
On the other hand, I also expect the velocity field of any fluid flow to have some kind of conservation property.
The 'curl-free vector field' theorem seems to imply that the velocity field of a general fluid flow can't be characterised by a potential function.
Does this mean flows with vortices are not energy conservative somehow?
Otherwise, there some other conserved quantity that we can use to characterise the velocity field of a fluid? For example, can we use a pointwise kinetic energy function to characterize a velocity field?
(Disclaimer: I haven't had much formal physics education so I apologize if I've abused and/or confused fundamental concepts.)