Modal coupling of nonlinear inviscid axisymmetric droplet shape oscillations

We investigate the nonlinear axisymmetric shape oscillations of inviscid droplets, focusing on the modal coupling between different oscillation modes. The research builds upon the geometrically exact nonlinear theory for droplet shape oscillations by Plümacher [], which allows for large deformations and superimposed initial deformations. The theory rests on the potential flow assumption and the usual momentum jump condition at the droplet surface including the surface tension. Using the unified transform method of A. S. Fokas [], the governing equations are transformed into a system of integrodifferential equations defined on a unit sphere. These equations are then solved numerically with a Galerkin method that uses spherical harmonics and exhibits overall spectral convergence. The frequency shift and the extended time spent in the prolate state within an oscillation period are analyzed for different initial deformation modes and show good agreement with existing literature. The results demonstrate that for large initial deformations, the modal coupling significantly influences the oscillation dynamics, particularly in the interaction between even and odd modes. To study modal coupling in detail, superimposed initial deformations are investigated in which a large-amplitude even mode is superimposed with various individual small-amplitude odd modes. It is shown that the initially excited large-amplitude even mode remains nearly unaffected by the superimposed small-amplitude odd modes. The initially superimposed small-amplitude odd mode, however, is altered by the large-amplitude even mode without being significantly amplified or damped. This nonlinear modal coupling is not present in single even initial deformation modes, where it is shown that an even initial deformation mode cannot excite odd modes.