= Solution
Write the <complex potential of a two-dimensional potential flow> as $w=\phi+i\psi$. Its <complex velocity> is $u-iv=dw/dz$. Here
$$
\frac{dw}{dz}=U\left(1-\frac{a^2}{z^2}\right)-\frac{i\Gamma}{2\pi z}\longrightarrow U\quad(|z|\to\infty),
$$
so the far-field motion has velocity $(U,0)$ and speed $U$. In <polar coordinates> the <stream function>, shifted by a harmless constant, is
$$
\psi(r,\theta)=U\left(r-\frac{a^2}r\right)\sin\theta-\frac\Gamma{2\pi}\log(r/a).
$$
It is zero on $r=a$, proving the cylinder is a <streamline>.
<Stagnation points> are zeros of $dw/dz$, hence roots of $Uz^2-i\Gamma z/(2\pi)-Ua^2=0$. Put $\gamma=\Gamma/(4\pi Ua)\in(0,1)$. The two roots are
$$
\boxed{z_\pm=\pm a\sqrt{1-\gamma^2}+ia\gamma.}
$$
Both have modulus $a$ and are distinct. Their angles satisfy $\sin\theta=\gamma$, namely $\arcsin\gamma$ and $\pi-\arcsin\gamma$. The tangential velocity on the cylinder is $-2U\sin\theta+\Gamma/(2\pi a)$, confirming the same zeros. <Streamlines> are level curves of $\psi$; the positive <circulation> places both <stagnation points> on the upper side and increases the flow speed below the cylinder.
\Image[/past-exam-of-the-options-i-course-of-the-university-of-oxford/2005-cylinder-streamlines.png]
{title=Streamlines and the two surface stagnation points for positive subcritical circulation}
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