A mathematical analogy between constant vorticity flows without surface tension and irrotational flows with surface tension

A mathematical analogy is elucidated between two physically distinct classes of two-dimensional-free boundary problems in inviscid fluid mechanics. One class involves free surface flows with constant vorticity but without surface tension; the other comprises free surface flows without vorticity but with surface tension. The results provide a rationale for several surprising observations in the recent literature that the same free surface profiles can arise in problems that are physically distinct. It is shown here, in several contexts, how two classes of physical problems amount to solving the same mathematical problem. Solutions to those mathematical problems are examined in detail. The analogy has the general implication that theoretical results on the common mathematical problem are immediately relevant to both physical problems, leading to a host of new theoretical results including theorems regarding existence, uniqueness, analytic structure and non-trivial identities involving conformal maps. There are, however, some interesting caveats that are also examined here.