Modal Coupling in Large-Amplitude Shape Oscillations of Inviscid Droplets
Year: 2026
Type: dissertation
Abstract: This study investigates the non-linear effects of inviscid droplet shape oscillations with large amplitudes, with a special focus on modal coupling. The research builds upon the geometrically non-linear theory (GNLT) for droplet shape oscillations by Plümacher et al. (2020), which allows for arbitrarily large deformations. The theory rests on the potential flow assumption and the momentum jump condition at the droplet surface including the surface tension. Using the unified transform method of Fokas (1997), the governing equations are transformed into a system of integro-differential equations defined on a unit sphere. These equations are solved with a numerical scheme developed in this work, which makes use of a Galerkin method and uses spherical harmonics as basis functions. Separate solvers are implemented for the axisymmetric and the non-axisymmetric cases. The numerical procedure is verified by spatial and temporal convergence studies for the axisymmetric case, and by the conservation of volume and energy for both cases. For the axisymmetric droplet oscillations, the center of mass, the frequency shift and the extended time spent in the prolate state within oscillation period are analyzed for different initial deformation modes, showing good agreement with existing literature. Furthermore, the results demonstrate that for large initial deformations, the modal coupling significantly influences the oscillation. 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 non-linear 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. For the non-axisymmetric case, the role of the discrete symmetries of the droplet shape and the velocity potential in mode excitation is investigated. The numerical results show that, for sufficiently small amplitudes, the oscillation frequency is independent of the azimuthal wave number. This is in agreement with classical theoretical predictions. The central result of this work is that the common discrete symmetries of the initial deformation and the initial velocity potential are preserved throughout the oscillation. Consequently, only modes compatible with these symmetries are excited. When the initial velocity potential breaks symmetries of the initial deformation mode, additional modes are excited compared to the case of a zero initial velocity. A similar result is observed when two initial deformation modes are superimposed. In this case, only those symmetries that both modes possess are preserved. If fewer symmetries remain, more modes are excited than for a single initial deformation mode.
DOI: 10.26083/tuda-8562
Author: Akbari, Schahin
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224384999