Bernoulli Equation for Rotational Flow

[soothsayer]

Hi PF! I've been working on a research problem involving fluid dynamics, and I'm currently looking at a "bathtub flow". This is where water is draining through a hole, and we have a vortex. In a paper I have found dealing with this flow, the velocity potential was written as:
$\psi = Alnr + B\phi$

which gives a velocity profile of:

$\vec{v} = \frac{A}{r} \hat{r} + \frac{B}{r} \hat{\phi}$

But this doesn't make a lot of sense to me, because this gives $\vec{\bigtriangledown} \times \vec{v} = 0$, but the flow should naturally have a vorticity, which means the curl should not be zero.

Ultimately, I want to be able to simplify the Euler equation into something like the Bernoulli Equation, but I've never seen a Bernoulli Equation for rotational flow. Can anyone point me in the right direction? I know for irrotational flow, $\vec{\bigtriangledown} \times \vec{v} = 0$, we get:

$\frac{\partial \psi}{\partial t} + \frac{1}{2}(\vec{\bigtriangledown} \psi )^2 + \frac{p}{\rho} + gz = f(t)$

Which I don't feel like I should be able to use for this situation...Thanks for any help!

 

[boneh3ad]

Considering your original equation was a velocity potential, the flow field would have to be irrotational to make any sense at all. This comes from the fact that
$$\nabla\times\vec{v} = \nabla\times\nabla\phi \equiv 0.$$
Don't confuse the concepts of vorticity and rotationality, with that of circulation. They are related but different. There would certainly circulation in this case but not necessarily vorticity.

Given that this model is irrotational and therefore conservative and is also steady, you could absolutely use Bernoulli's equation.

 

[Chestermiller]

To expand on what boneh3ad said, think about how the fluid parcels can move along a circular path without rotating about their own axes.

 

[soothsayer]

Ohh, I see. Thanks guys, I was kinda wondering whether I had misunderstood the definition of vorticity. I guess I kinda assumed since the fluid model had a vortex, there would necessarily be vorticity XD

I'll use Bernoulli's equation then. Thank you!

 

[boneh3ad]

There would be vorticity right at the center of that vortex, but that is a singularity in your equation anyway so isn't valid.