 Solution

In cylindrical coordinates, the transverse velocity is $A{\mathbf{e}}_{\theta }/r$, so its contribution to the circulation is $Ad\theta$. The vertical contribution is $U(z)dz$, the differential of an antiderivative of $U$, and therefore has zero integral around any closed curve. If $k$ is the oriented winding number of the curve's projection about the origin, thenIn particular, a single positive winding gives $2\pi A$; reversing orientation reverses the sign. The nonzero circulation is compatible with irrotational flow because the domain is not simply connected. For $A\ne 0$, an enclosing curve cannot bound a surface contained entirely in the domain to which Stokes theorem would apply.