Axisymmetric vortex

Question

An axisymmetric vortex, for which the azimuthal velocity $u_\theta$ is proportional to $r^{-\beta}$. What are the values for $\beta$ so that the circulation ($\Gamma(r))$ is finite

Answer 1

Around a circular contour of radius $r$:

$$\Gamma(r) = \int_R \nabla \times \mathbb{u}\cdot d\mathbf {A} = \oint_C\mathbf{u} \cdot d\mathbf{l} = \int_0^{2\pi}u_\theta r \, d\theta = 2\pi r u_\theta.$$

With $u_\theta$ proportional to $r^{-\beta}$, say $u_\theta = Cr^{-\beta}$, we have $\Gamma(r) = 2\pi Cr^{1-\beta}$ and the circulation is finite as $r \to \infty$ if $\beta \geqslant 1$.