Third-order resonance effects and the nonlinear stability of drop oscillations

Abstract

The three-dimensional nonlinear oscillations of an isolated, inviscid drop with surface tension are studied by a multiple timescale analysis and pre-averaging applied to the variational principle for the appropriate Lagrangian. Amplitude equations are derived which describe the generic cubic resonance caused by the spatial degeneracy of the eigenfrequencies of the linear normal modes. This resonant coupling leads to the instability of the finite amplitude axisymmetric oscillations to small non-axisymmetric perturbations, as is demonstrated here for the three-and four-lobed normal modes. Solutions to the interaction equations that describe finite amplitude, non-axisymmetric travelling-wave solutions are also obtained and their stability is investigated. A non-generic cubic resonance between the two-lobed and four-lobed oscillatory modes leads to quasi-periodic motions.