Extension of a mathematical analogy between vortical flows without surface tension and irrotational capillary flows

Abstract

In a recent paper, a mathematical analogy was found between two physically distinct classes of two-dimensional free boundary problems in inviscid fluid mechanics. One class involves free surface flows with constant vorticity in the bulk but without boundary surface tension, the other comprises free surface flows without vorticity in the bulk but with non-zero boundary surface tension (capillarity). Several examples were given of pairs of physically-distinct free boundary problems that amount to solving the same mathematical problem, albeit with the possibility of a single constraint arising for the capillary flows. The present work shows that this analogy extends to another pair of physically-distinct problems: a hollow vortex equilibrium in a shear flow with a third order irrotational strain and a steady bubble with surface tension in a simple straining flow. For both problems, we construct an isolated exact solution when a parameter $γ=0$ and provide numerical solutions for non-zero $γ$. An interesting feature is that non-trivial conformal mapping identities emerge from the class of problems with capillarity that apply equally for the hollow vortex problems when the parameters are appropriately constrained.