Potential flow around a circular cylinder

Example

Consider a two-dimensional, steady, incompressible and irrotational flow of an inviscid fluid of density $\rho$ outside the disc $\abs{z} \le a$ in the plane $z = x + iy$. Far from the disc the flow tends to the uniform stream of speed $U > 0$ in the $x$-direction. The complex potential of this potential flow is $$w(z) = U\left(z + \frac{a^2}{z}\right), \qquad \abs{z} \ge a.$$ In polar coordinates $z = r e^{i\theta}$ the stream function is $\psi = \operatorname{Im} w = U\left(r - \frac{a^2}{r}\right)\sin\theta$. It vanishes on $r = a$, so the circle is a streamline and no fluid crosses it. The complex velocity is $w'(z) = U(1 - a^2/z^2)$, which tends to $U$ as $\abs{z} \to \infty$. On the circle the speed is $\abs{w'(a e^{i\theta})} = 2U\abs{\sin\theta}$. So there are stagnation points at $\theta = 0$ and $\theta = \pi$, and the speed reaches its maximum $2U$ at the top and bottom.