# Rankine vortex

A swirling flow in a viscous fluid can be characterized by a forced vortex in its central core, surrounded by a free vortex. In an inviscid fluid, on the other hand, a swirling flow consists entirely of the free vortex with a singularity at its center point instead of the forced vortex core. The tangential velocity[1] of a Rankine vortex with circulation $\Gamma$ and radius $R$ is
$u_\theta(r) = \begin{cases} \Gamma r/(2 \pi R^2) & r \le R, \\ \Gamma/(2 \pi r) & r > R. \end{cases}$
The remainder of the velocity components are identically zero, so that the total velocity field is $\mathbf{u} = u_\theta\ \mathbf{e_\theta}$.