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

For the same counterclockwise line vortex strength $\kappa$, its local velocity potential and streamfunction are $\varphi =\kappa \theta /(2\pi )$ and $\psi =-\kappa \log r/(2\pi )$. ConsequentlySubstitution givesin agreement with the direct circulation and volume flux calculation. The sign of the imaginary coefficient is essential for a positive counterclockwise circulation.