Short-term oscillation and falling dynamics for a water drop dripping in quiescent air

Abstract

The short-term transient falling dynamics of a dripping water drop in quiescent air has been investigated through both simulation and experiment. The focus is on the short term behavior and the time range considered covers about eight dominant second-mode oscillations of the drop after it is formed. Due to the small fluid inertia the growth of the drop is quasi-static and is well captured by the static pendant drop theory. Nevertheless, the pinching dynamics and the resulting post-formation state of the drop trigger a nonlinear oscillation when the drop falls. The initial shape of the drop when it is just formed is decomposed into spherical harmonic modes. The pinching dynamics such as interface overturning introduces small-scale variation on the drop contour, which in turn contributes to the finite amplitudes of the higher-order modes. Furthermore, the initial kinetic energy when the droplet is just formed is as important as the initial surface energy contained in the drop shape, and is found to amplify the initial oscillation amplitude and to induce a phase shift in the oscillation of all the modes. By incorporating both the initial surface and kinetic energy, the linear model for a free drop oscillation yields very good predictions for the second and third modes. The mode amplitude spectra show both the primary frequencies that are consistent with the Lamb's theory and the secondary frequencies arising from different modes due to nonlinear inter-mode coupling. The complex transient flow inside and outside the drop is induced by the interaction between the falling motion and the nonlinear oscillation. The streamlines indicate that the internal flow is substantially different from the Hill vortex for a falling drop without oscillation. The temporal evolutions of both the internal flow and the wake morphology follow the dominant second oscillation mode.