Quadratic resonance in the three-dimensional oscillations of inviscid drops with surface tension

Abstract

The moderate-amplitude, three-dimensional oscillations of an inviscid drop are described in terms of spherical harmonics. Specific oscillation modes are resonantly coupled by quadratic nonlinearities caused by inertia, capillarity, and drop deformation. The equations describing the interactions of these modes are derived from the variational principle for the appropriate Lagrangian by expressing the modal amplitudes to be functions of a slow time scale and by preaveraging the Lagrangian over the time scale of the primary oscillations. The equations describing the motion of axisymmetric resonant modes are integrable. Stochastic motions are predicted for nonaxisymmetric deformations starting from most initial conditions, even those arbitrarily close to the axisymmetric shapes. The stochasticity is characterized by a redistribution of the energy contained in the initial deformation over all the degrees of freedom of the interacting modes.