= Solution
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 potential>
$$
\boxed{
\phi=-U\left(r+\frac{a^2}{r}\right)\cos\theta
}.
$$
It is <harmonic function>[harmonic] for $r>a$ and gives
$$
\boxed{
u_r=-U\left(1-\frac{a^2}{r^2}\right)\cos\theta,
\qquad
u_\theta=U\left(1+\frac{a^2}{r^2}\right)\sin\theta.}
$$
\b[Thus $u_r=0$ 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 $U\mathbf e_x$.]
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