Existence of nonsymmetric vortex and bubble solutions

Determine whether solutions of the coefficient relations governing the hollow-vortex and capillary-bubble conformal mappings exist without inversion symmetry.

Background

The numerical construction in the paper exploits the vanishing of all odd conformal-mapping coefficients, which yields inversion symmetry of the hollow-vortex and bubble boundaries. The authors note that this property follows when the relevant solution is unique, but do not establish whether other solutions lacking this symmetry can occur.

The unresolved issue concerns the existence of free-boundary solutions for the third-order straining-flow vortex problem and the analogous surface-tension bubble problem that do not possess the inversion symmetry $(x,y)\mapsto(-x,-y)$.