2
$\begingroup$

The definition of vorticity is $\mathbf{\omega} = \nabla \times \mathbf{v}$, where $\mathbf{v}$ is the velocity vector field.

Now, if I look at a rotating flow in cylindrical coordinates I find that: $$\nabla \times \mathbf{v} = \frac{1}{r}\frac{\partial (r v_{\theta})}{\partial r},$$ in case of a free vortex I also know that $v_{theta} \propto 1/r$ and therefore the derivative in the above equation vanishes for a vortex centered at $r=0$. In other words, the vorticity is zero everywhere, $\mathbf{\omega} = \mathbf{0}$.

I can also look at the situation globally, and instead of the localised curl I take a line integral of the speed along a circular path distance $r$ from the centre. In this case I find that: $$C = \int_{\textit{circular path}}{\mathbf{v}\cdot d\mathbf{l} = 2\pi ru_{\theta}},$$ a finite constant. But from the Stoke's theorem I know that: $$\int_{\textit{enclosed area}}{\left(\nabla \times \mathbf{v}\right)\cdot d \mathbf{A}} = \int_{\textit{enclosing curve}}{\mathbf{v}\cdot d\mathbf{l}},$$ but if the circulation is a finite non-zero constant, the curl must be also non-zero somewhere within the enclosed area! Thus the vorticity is non-zero somewhere in the velocity vector field!

These two findings seemingly contradict each other, where am I making a mistake?

$\endgroup$
2
$\begingroup$

There is a singularity at the origin: a delta function in the vorticity field. The vorticity is zero (irrotational flow) everywhere but at the origin, where it is infinite. The circulation around any path not enclosing the origin is zero. The circulation around any path enclosing the origin is a constant (non-zero).

$\endgroup$
  • $\begingroup$ I did not notice the simple pole at the origin. Is it then that only non-physical flows can have zero vorticity everywhere with non-zero circulation? i.e. that if I find something similar happening, the field must have a pole somewhere? $\endgroup$ – Akerai Jan 20 '18 at 3:34
  • $\begingroup$ I would say yes: Stoke's theorem works. But there are physical flows that approximate the vortex you described. Water draining out of a basin has all its vorticity concentrated right at the center of the vortex. It's just a large vorticity over a small area rather than an infinite one at a point. $\endgroup$ – Ben51 Jan 20 '18 at 3:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.