Past exam of the options I course of the University of Oxford / 2011 / K / 2 / Solution

Use the convention that positive vortex strength is counterclockwise and complex velocity is $u-iv$. A complex potential of a two-dimensional potential flow for the superposed uniform flow and two point vortices isLocal choices of logarithm branch change no velocity. Differentiation givesFor nonzero vortex strength and $U>0$, its two zeros areThe number under the square root is nonzero, so these are distinct stagnation points, neither at a vortex.

A free point vortex moves with the velocity induced by the other vortex and the background, omitting its own singular field. At either vortex, the other induces vertical velocity $-\Gamma /(4\pi a)$, while the background contributes $(U,0)$. Their velocities are therefore equal. Their separation stays $(2a,0)$, making this induced velocity constant for all time. Both travel along parallel straight lines with