Assessing one-dimensional models for axisymmetric liquid columns through analysis of drop oscillations

Abstract

The validity of one-dimensional (1D) models for liquid columns (capillary jets, liquid bridges, and other similar systems) for predicting the dynamics of their breakup into drops is studied when viscosity is not relevant. These models assume a simplified radial dependence, which in turn is supported by the premise that the system is slender enough. However, even though the latter hypothesis does not seem to be fulfilled when the system gets close to breakup, 1D models continue to describe its evolution surprisingly well. Our numerical simulations of a liquid jet confirm this good behavior, even when the liquid jet becomes a sequence of incipient drops joined by filaments. A dynamic definition of the slenderness allows computing the stage until which the liquid jet can be considered slender from the standpoint of 1D models. Beyond that point, a detailed numerical study of mass and energy reveals that the action that each incipient main drop suffers from the two adjacent filaments becomes barely significant. Thus, the drop can be considered to oscillate almost freely, while the filaments remain nearly static. To explain how the 1D models can predict the evolution of a so scarcely slender system as a droplet, we compare the linear modal analysis of the Lee, Cosserat and parabolic 1D models for an inviscid free drop with the 3D analysis in cylindrical coordinates. The frequencies of the oscillation modes with lowest index, the most relevant ones arising from the breakup, are in general well predicted. This happens when the radial dependence of both pressure and axial velocity assumed in 1D models reasonably fit the corresponding 3D exact results. The Cosserat model (and even more the parabolic one) performs better than the Lee model.