= Solution
The incident <uniform flow> $U\mathbf e_z$ has spherical components $u_r=U\cos\theta$, $u_\theta=-U\sin\theta$. Integrating the <Stokes streamfunction> formulas, with the additive constant chosen to vanish on the axis, gives
$$
\Psi=\frac12Ur^2\sin^2\theta.
$$
It is regular at the origin. Applying the <sphere theorem for a Stokes streamfunction> yields
$$
\boxed{\widetilde\Psi(r,\theta)=\frac U2\left(r^2-\frac{a^3}r\right)\sin^2\theta.}
$$
The resulting <velocity> components are
$$
\widetilde u_r=U\left(1-\frac{a^3}{r^3}\right)\cos\theta,
\qquad
\widetilde u_\theta=-U\left(1+\frac{a^3}{2r^3}\right)\sin\theta.
$$
To find the <velocity potential>, integrate $\phi_r=\widetilde u_r$ and verify $r^{-1}\phi_\theta=\widetilde u_\theta$. This gives
$$
\boxed{\phi(r,\theta)=U\left(r+\frac{a^3}{2r^2}\right)\cos\theta+\text{constant}.}
$$
The first term is the incident <uniform flow>, and the second is a <potential dipole> located inside the excluded <sphere>. At $r=a$, the radial <velocity> vanishes, while the far-field <velocity> tends to $U\mathbf e_z$. The poles are <stagnation points>; the surface tangential speed reaches $3U/2$ at the equator. These formulas describe <uniform potential flow around a fixed sphere>.
\Image[/past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2015-sphere-streamlines.png]
{title=Streamlines of a uniform potential flow around a fixed sphere}
{description=A meridional section with $U=a=1$. The solid sphere is impermeable, and the flow approaches a uniform stream in the positive $z$ direction. The two marked surface points are stagnation points.}
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