Past exam of the fluids and waves course of the University of Oxford / 2014 / 2 / b

A uniform flow with physical velocity $(U\cos \alpha ,U\sin \alpha )$ has complex velocity $U{e}^{-i\alpha }$ and incident potential $f(\zeta )=U{e}^{-i\alpha }\zeta$. Apply the Milne-Thomson circle theorem with $R=1$:Indeed, on $\zeta =e^{i\theta }$ it is $W=2U\cos (\theta -\alpha )$, which is real, so the boundary is a streamline. At infinity, $W^{\prime }(\zeta )\to U{e}^{-i\alpha }$, verifying the specified incoming uniform flow.