 Irrotational circulation in an exterior domain

In smooth barotropic inviscid flow with conservative body force, Kelvin's circulation theorem preserves circulation of material curves. Initially zero vorticity therefore gives zero circulation on contractible loops for all times. A loop winding around an excluded obstacle can have a nonzero constant circulation: the planar field $\mathbf{u}=\Gamma {\mathbf{e}}_{\theta }/(2\pi r)$ is irrotational outside the origin but has circulation $\Gamma$ on a positively oriented enclosing loop.