Past exam of the fluids and waves course of the University of Oxford / 2017 / 2 / b / ii / Solution

Multiply the complex velocity by the complex displacement:The real term is the circulation integrand. With a counterclockwise curve, $\mathbf{n}ds=(dy,-dx)$ is outward, so the imaginary term is the outward volume flux integrand. Integrating givesThis also fixes the sign of $M$: a positive line source has $w^{\prime }=Q/(2\pi z)$ and the integral is $iQ$, so its outward volume flux is $Q$.