I'm working on my assignment for fluid mechanics and I'm a bit lost as to how to respond to a question. I believe I can get part a tomorrow when I sit down to work it out, but I'm not sure exactly how to respond to part b. Any ideas?
(Yes, we know the two-thirds is a misprint, we had an email correction to say one-third)
The velocity components are related to the stream function by
$$u_x = -\frac{\partial \psi}{\partial y}, \,\,\, u_y = \frac{\partial \psi}{\partial x} \quad \text{(rectangular coordinates)} \\ u_r = -\frac{1}{r}\frac{\partial \psi}{\partial \theta}, \,\,\, u_\theta = \frac{\partial \psi}{\partial r} \quad \text{ (polar coordinates)}$$
The stream function here is continuously differentiable in the region above the boundary since $r \neq 0$. This means that the continuity equation is satisfied automatically (and it is not necessary to check) since
$$\nabla \cdot \mathbf{u} = \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = -\frac{\partial^2 \psi}{\partial x \partial y} + \frac{\partial^2 \psi}{\partial y \partial x} = 0$$
To finish part (b) you would check that the velocity field obtained from this stream function satisfies (1) the boundary conditions at the solid surface, and (2) the Euler equations for steady, inviscid, incompressible flow.
(1) At a solid (impermeable) surface in inviscid flow the velocity component normal to the surface must be $0$. (The no-slip condition where both normal and tangential velocity components vanish can be enforced only for viscous flow.) This is equivalent to the condition that the boundary is itself a streamline where the stream function is constant. This is clearly true since on the three sections of the boundary we have
$$\psi(r,\theta) = \frac{U}{3}\left(\frac{a^2}{r}- r \right)\cos(3 \theta) = \begin{cases}0,& r \geqslant a, \,\,\,\,\theta = \frac{\pi}{6}\\0, & r = a, \,\, \,\,\frac{\pi}{6}\leqslant \theta \leqslant \frac{5 \pi}{6} \\0,& r \geqslant a, \,\,\,\,\theta = \frac{5\pi}{6}\end{cases}$$
Alternatively, you can compute the velocity components and check that the zero normal velocity condition holds directly -- and this is a more labor intensive exercise.
I will leave the verication part (2) to you.