From weakly to strongly nonlinear viscous drop shape oscillations: An analytical and numerical study

Nonlinear axisymmetric shape oscillations of a Newtonian drop in a vacuum are investigated using two different theoretical methods, for fundamental interest and for the significance of the oscillations in transport processes across the drop surface. The extended discontinuous Galerkin method is contrasted to the weakly nonlinear theory. While the former allows large drop surface deformation amplitudes to be analyzed with high precision and drop volume errors below 0.11% even at the largest deformations, the latter provides analytical insight into the origin of quasiperiodic time behavior of the oscillations and reveals the oscillation modes coupled in the nonlinear motion. Results from both methods for moderate initial deformation amplitudes at modes of initial drop deformation 𝑚=2, 3, and 4 are in excellent agreement, showing the time asymmetry of the oscillation and the decrease of the oscillation frequency with increasing deformation amplitude. The Fourier power spectra for the first oscillation period exhibit decreased dominant frequencies as compared to the linear results as well as the mode coupling as nonlinear effects. The numerical method is used to compute the oscillatory and damping behavior of viscous drops, as well as the interconversion of kinetic and surface energies during the oscillations at strong initial deformations