Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 5C (Fluid Dynamics)

Put $r^2=x^2+y^2$. On the specified domain, $r>0$, and direct differentiation gives the divergence and vorticityFor example, the third component of the curl is $A(r^2-2x^2)/r^4+A(r^2-2y^2)/r^4=0$; the other two components vanish because $U$ depends only on $z$ and the transverse velocity is independent of $z$. Hence the flow is both incompressible and irrotational exactly whenThe exclusion of the axis matters: these are ordinary derivatives on the domain, with no assertion of regularity on the removed axis.

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.