[Physics] Difference between vorticity and circulation

Question

The definition of vorticity is $\boldsymbol{\omega} = \nabla \times \mathbf{v}$, 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, $\boldsymbol{\omega} = \mathbf{0}$.

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:
$$C = \int_{\text{circular path}}{\mathbf{v}\cdot d\mathbf{l} = 2\pi ru_{\theta}},$$
a finite constant. But from the Stoke's theorem I know that:
$$\int_{\text{enclosed area}}{\left(\nabla \times \mathbf{v}\right)\cdot d \mathbf{A}} = \int_{\text{enclosing curve}}{\mathbf{v}\cdot d\mathbf{l}},$$
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?

Answer 1

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).