 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 $F_x-i{F}_y=\rho {\Gamma }^2/(2\pi (b-c))$. Thus $F_y=0$ and $F_x=\rho {\Gamma }^{2}b/(2\pi (b^2-a^2))>0$: 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.