Rankine vortex

Example

Fix a core radius $a > 0$ and a circulation $\Gamma > 0$. In plane polar coordinates $(r, \theta)$, the Rankine vortex is the planar flow with zero radial velocity and the following azimuthal velocity. $$u_\theta(r) = \begin{cases} \dfrac{\Gamma r}{2 \pi a^2}, & r \le a, \\[2mm] \dfrac{\Gamma}{2 \pi r}, & r > a. \end{cases}$$ The vorticity of a circular flow is $\zeta = \frac{1}{r} \frac{d}{dr}(r u_\theta)$. Inside the core, $\zeta = \frac{1}{r} \cdot \frac{2 \Gamma r}{2 \pi a^2} = \Gamma / (\pi a^2)$, a rigid rotation. Outside, $r u_\theta = \Gamma / (2\pi)$ is constant, so $\zeta = 0$: the flow is irrotational there. The circulation around any circle of radius $r \ge a$ is $2 \pi r \, u_\theta(r) = \Gamma$, which is the total vorticity $\pi a^2 \cdot \Gamma / (\pi a^2)$ in the core. With the pressure $p(r)$ given by $dp/dr = \rho \, u_\theta^2 / r$, the flow is a steady solution of the Euler equations.