Force on a cylinder from an external dipole ($F_x=\frac{4\pi \rho {\mu }^2{a}^{2}b}{(b^2-a^2{)}^3},F_y=0$)

The Milne-Thomson circle theorem places an image dipole inside a cylinder at the inverse point of an external dipole. In the residue of the squared complex velocity, only the cross term between the analytic external velocity and the image double pole contributes. The product of their conjugate dipole strengths is independent of orientation, so the cylinder force is radial toward the external dipole.