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On the glug-glug of ideal bottles

Published online by Cambridge University Press:  23 June 2004

CHRISTOPHE CLANET
Affiliation:
Institut de Recherche sur les Phénomènes Hors Equilibre, UMR 6594, 49 rue F. Joliot Curie, BP 146, 13384 Marseille, France
GEOFFREY SEARBY
Affiliation:
Institut de Recherche sur les Phénomènes Hors Equilibre, UMR 6594, 49 rue F. Joliot Curie, BP 146, 13384 Marseille, France

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.

Type
Papers
Copyright
© 2004 Cambridge University Press

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