On the glug-glug of ideal bottles

Abstract

We present an experimental study of the emptying of an ideal vertical bottle under gravity $g$. The idealization reduces the bottle to a cylinder of diameter $D_{0}$, length $L$, closed at the top and open at the bottom through a circular thin-walled hole of diameter $d$, on the axis of the cylinder. The study is performed in the low-viscosity limit. The oscillatory emptying of the ‘bottle’ is referred to as the glug-glug, and is characterized by its period $T$, whereas the whole emptying process is characterized by a time $T_{e}$. Concerning the long time scale $T_{e}$, we show that: \[ \frac{T_{e}}{T_{e0}}=\left(\frac{D_{0}}{d}\right)^{5/2}, \] where $T_{e0}\,{\approx}\, 3.0 L/\sqrt{gD_{0}}$ is the emptying time of an unrestricted cylinder. On the short time scale $T$, we show that the physical origin of the oscillations lies in the compressibility of the surrounding gas. The period can be written as: \[ T\,{=}\,\frac{L}{\sqrt{\gamma P_{0}/\rho}}\Phi(\skew1\bar{z}_{i}/L), \] where $\gamma$ is the ratio of specific heats of the gas, $P_{0}$ its pressure and $\rho$ stands for the density of the liquid. The function $\Phi$ is dimensionless and changes with the relative position of the liquid interface $\skew1\bar{z}_{i}/L$. Finally, this analysis of time scales involved in the emptying of vertical cylinders is applied to other liquid–gas oscillators.