On the glug-glug of ideal bottles
Résumé
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 D0, 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 Te. Concerning the long time scale $T_{e}$, we show that: \[ \frac{T_{e}}{T_{e0}}=\left(\frac{D_{0}}{d}\right)^{5/2}, where Te0 ≈3.0L/√gD0 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 γ is the ratio of specific heats of the gas, P0 its pressure and ρ stands for the density of the liquid. The function Φ is dimensionless and changes with the relative position of the liquid interface ¯zi/L. Finally, this analysis of time scales involved in the emptying of vertical cylinders is applied to other liquid–gas oscillators.