= Solution
With the <complex velocity> convention $w'(z)=u-iv$, differentiation gives $w'=-i\Gamma/(2\pi z)$. Thus $u=-\Gamma y/(2\pi r^2)$ and $v=\Gamma x/(2\pi r^2)$. Resolving into polar components gives $\boxed{u_r=0,\quad u_\theta=\Gamma/(2\pi r)}$.
Superposition for the opposite-signed pair gives
$$
\boxed{w(z)=-\frac{i\Gamma}{2\pi}\log\frac{z-ia}{z+ia},}
$$
up to a constant and branch choices. A <vortex> moves with the regular velocity generated by the other <vortex>, excluding its own singular field. At $z=ia$, the other contribution is $i\Gamma/[2\pi(2ia)]=\Gamma/(4\pi a)$; at $z=-ia$ it is the same real number. Hence the pair keeps its separation and translates parallel to the real axis with
$$
\boxed{u_{\rm pair}=\Gamma/(4\pi a),\qquad\text{speed}=|\Gamma|/(4\pi a)}
$$
for $a>0$.
For a <vortex> at $z_1=x_1+iy_1$ in the first quadrant, reflect it across either wall with opposite strength, then across both walls with its original strength. Thus the images are $\bar z_1$ and $-\bar z_1$ of strength $-\Gamma$, and $-z_1$ of strength $+\Gamma$. The potential is
$$
\boxed{w(z)=-\frac{i\Gamma}{2\pi}\log
\frac{(z-z_1)(z+z_1)}{(z-\bar z_1)(z+\bar z_1)}.}
$$
On either real or imaginary axis, $z^2$ is real and the numerator and denominator of $(z^2-z_1^2)/(z^2-\bar z_1^2)$ are conjugates. Its modulus is one, making the <stream function> constant on both walls and enforcing no normal flow. A regular entire background flow could be added only with additional far-field data; the displayed image solution is the one generated by the specified <vortex> and its walls.
Back to article page
