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Self-similar motion of three point vortices
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One of the counter-intuitive results in the three-vortex problem is that the vortices can converge on and meet at a point in a finite time for certain sets of vortex circulations and for certain initial conditions. This result was already included in Groumlbli's thesis of 1877 and has since been elaborated by several authors. It arises from an investigation of motions where the vortex triangle retains its shape for all time, but not its size. We revisit these self-similar motions, develop a new derivation of the initial conditions that lead to them, and derive a number of formulae pertaining to the rate of expansion or collapse and the angular frequency of rotation, some of which appear to be new. We also pursue the problem of linear stability of these motions in detail and, again, provide a number of formulae, some of which are new. In particular, we determine all eigenmodes analytically.