Past exam of the fluids and waves course of the University of Oxford / 2017 / 1 / b / i / Solution

Work away from the line source, so $r>0$. In polar coordinates, $u_r=Q/(2\pi r)$ and $u_{\theta }=0$. The supplied divergence and curl formulas giveHence the field defines an incompressible flow and an irrotational flow for $r>0$. The restriction matters: the outward volume flux through any enclosing circle isThe origin is the excluded injection point. Applying the divergence theorem to a disk through that singularity would violate its smoothness hypothesis.