Past exam of the options I course of the University of Oxford / 2004 / L

With the complex velocity convention $w^{\prime }(z)=u-iv$, differentiation gives $w^{\prime }=-i\Gamma /(2\pi z)$. Thus $u=-\Gamma y/(2\pi r^2)$ and $v=\Gamma x/(2\pi r^2)$. Resolving into polar components gives $\boxed{u_r=0,u_{\theta }=\Gamma /(2\pi r)}$.

Superposition for the opposite-signed pair givesup 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 withfor $a>0$.

For a vortex at $z_1=x_1+i{y}_1$ 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 isOn 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.