Past exam of the options I course of the University of Oxford / 2013 / K / 1 / Solution

Use the complex potential of a two-dimensional potential flow convention $W=\varphi +i\psi$, for which $W^{\prime }(z)=u-iv$. A positive circulation is counterclockwise. The two point vortices and the two-dimensional point sink giveBranch choices change additive constants, not the local velocity field. A vortex is transported by the nonsingular velocity due to the other vortex and the sink; its own singular term is omitted. At $z=c$ this givesThe other vortex follows $-c$ by symmetry. Put $c=r{e}^{i\theta }$, with $r(0)=a$ and $\theta (0)=0$. Multiplication by $\bar{c}$ givesFor $Q>0$, integration yieldsEquivalently, with $s=1-Qt/(\pi a^2)$,The curves obey $\theta =(\Gamma /(2Q))\log (a/r)$, so they are logarithmic spirals. At $t=\pi a^2/Q$, both vortices reach the sink at the origin. For nonzero circulation they make infinitely many turns in that finite time; for zero circulation they approach radially. The point-vortex evolution becomes singular at capture, so the formula does not define a regular continuation through the event.