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A velocity potential satisfying uniform flow at infinity and no radial flow through $r=a$ isThereforeOn the cylinder, $u_r=0$ and $u_{\theta }=-2U\sin \theta$. The steady Bernoulli equation, compared with the uniform flow at infinity, givesHencethe pressure in potential flow around a circular cylinder.

Work in the frame translating with the bubble. The bubble is the fixed circle $r=a$, while the velocity tends to $-U{\mathbf{e}}_x$ at infinity. The potential flow around a circular cylinder has velocity potentialIt is harmonic for $r>a$ and givesThus $u_r=0$ on $r=a$, satisfying the kinematic boundary condition, and $\mathbf{u}\to -U{\mathbf{e}}_x$ as $r\to \infty$. The laboratory-frame flow is obtained by adding $U{\mathbf{e}}_x$.