Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 39A / b / Solution

At infinity,soContinuity of $q_r=r^{-1}{\psi }_{\theta }$ gives $[{\psi }_{\theta }{]}_{-}^{+}=0$. Since $\nabla p=-12\mu h^{-3}\mathbf{q}$, continuity of tangential pressure derivative gives $[h^{-3}{\psi }_r{]}_{-}^{+}=0$.

The harmonic solutions regular at zero and with the prescribed far field areThe interface conditions giveIntegrating the outward flux across the appropriate semicircle gives