Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 39B / ii / Solution

The outer circle obeys $x^2+(y-2a{)}^2=(2a{)}^2$, and henceIts counterclockwise tangent is again $(\cos 2\theta ,\sin 2\theta )$. Its angular velocity is $-\Omega /4$ and radius is $2a$, so its signed tangential speed is $-a\Omega /2$. The boundary conditions areThe first follows both from impermeability and from the common streamfunction value forced by the contact point.