= Force on a circular cylinder from a held exterior vortex
Place a held <point vortex> of circulation $\Gamma$ at real position $b>a$ outside a circular cylinder $|z|=a$, and prescribe circulation $-\Gamma$ about the cylinder. An opposite image at $c=a^2/b$ gives <complex velocity> $q(z)=-i\Gamma(1/(z-b)-1/(z-c))/(2\pi)$. On the positively oriented cylinder contour, the only pole of $q^2$ inside is at $c$, with residue $-\Gamma^2/(2\pi^2(b-c))$. The <residue theorem> and <Blasius force theorem> give $F_x-iF_y=\rho\Gamma^2/(2\pi(b-c))$. Thus $F_y=0$ and $F_x=\rho\Gamma^2b/(2\pi(b^2-a^2))>0$: the force on the cylinder points toward the vortex. The specified cylinder circulation is part of the hypothesis; a zero-circulation cylinder has an additional central image term.
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