Oscillating Taylor bubble during the emptying of a partially filled water bottle

Abstract

The oscillatory behavior observed during the emptying of a vertical cylinder partially filled with water has been studied for large neck-to-bottle diameter ratios, d^{*}. For large apertures (d^{*}>0.8), a Taylor bubble invades the cylinder from the bottom, and its rising speed exhibits periodic oscillations coupled to periodic motions of the free surface limiting the top air buffer initially present in the bottle. We introduce an elementary model where the vertical oscillation of the free surface is represented by a variable mass oscillator exciting the oscillatory dynamics of the Taylor bubble. In this system, the top-air buffer acts as a spring, whose stiffness is related to its compressibility. The variable mass is the mass of the liquid in the cylinder that decreases as the Taylor bubble progresses during the emptying. The motion of the bubble is solved assuming that the unsteady flow generated by the free-surface motion is potential in the vicinity of the apex of the bubble. A comparison with experimental results obtained at the laboratory shows that the model agrees well with the data if it takes into account dissipation. This study shows that a viscous damping, proportional to the velocity, with a constant damping coefficient is able to accurately represent the dissipative processes such as the effect of viscous Stokes boundary layers at the walls.