New solutions for capillary waves on curved sheets of fluid

Stationary nonlinear capillary waves on an annular liquid sheet are studied with the aim of computing antisymmetric wave profiles, whose existence was conjectured by Crowdy (2001, Eur. J. Appl. Math., 12, 689–708). The waves are computed using a complex variable approach, by seeking a Fourier series representation of an analytic function whose coefficients are determined numerically subject to the boundary condition constraints. Typical profiles are presented with both equal and different surface tensions prevailing at the two free surfaces. Symmetric solutions similar to those reported by Crowdy are shown together with new antisymmetric configurations. Annuli with inner boundary possessing a higher degree of rotational symmetry than the outer boundary are shown to occur as bifurcations from a main branch where the symmetry is the same on both surfaces.