Weakly nonlinear shape oscillations of a Newtonian drop

Nonlinear axisymmetric shape oscillations of a Newtonian drop in a vacuum are investigated theoretically, for fundamental interest and for their relevance in transport processes across the drop surface. The weakly nonlinear analysis is carried out for, but not limited to, the modes of initial drop deformation up to m = 4. The drop Ohnesorge number covers the range between 0.01 and 1. The weakly nonlinear approach, which is carried to third order, accounts for the coupling of different oscillation modes. With increasing surface deformation, the oscillations develop an asymmetry of the times during one period the drop spends in different states of deformation, a frequency decrease below the linear value, and quasi-periodicity of the motion. In contrast to the inviscid case [D. Zrnić and G. Brenn, “Weakly nonlinear shape oscillations of inviscid drops,” J. Fluid Mech. 923, A9 (2021)], the present analysis reveals the frequency decrease and the quasi-periodicity already in the second-order approximation. The results are positively validated against relevant literature. The theory quantifies the effects of viscosity, measured by the drop Ohnesorge number, dampening the nonlinear behavior and enhancing the coupling of different oscillation modes [E. Becker et al., “Nonlinear dynamics of viscous droplets,” J. Fluid Mech. 258, 191 (1994)]. The present theory reveals the quasi-periodicity of nonlinear viscous drop shape oscillations at strong deformation. The resultant drop motion, starting from a higher-order mode of initial deformation, for which the drop exhibits aperiodic linear behavior, may turn into damped oscillatory with ongoing time due to the coupling to lower-order modes