The velocity potential $\phi(r, \theta)$ for inviscid flow in two dimensions satisfies the Laplace equation

$$\Delta \phi=\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right] \phi(r, \theta)=0$$

(a) Using separation of variables, derive the general solution to the equation above that is single-valued and finite in each of the domains (i) $0 \leqslant r \leqslant a$; (ii) $a \leqslant r<\infty$.

(b) Assuming $\phi$ is single-valued, solve the Laplace equation subject to the boundary conditions $\frac{\partial \phi}{\partial r}=0$ at $r=a$, and $\frac{\partial \phi}{\partial r} \rightarrow U \cos \theta$ as $r \rightarrow \infty$. Sketch the lines of constant potential.