On the glug-glug of ideal bottles

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 T_e. Concerning the long time scale Te, we show that: T_e/T_e0 = (D₀/d) ^5/2, where T_e0 ≈ 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 = L/√γP₀/ρ Φ (z̄_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 z̄_i/L. Finally, this analysis of time scales involved in the emptying of vertical cylinders is applied to other liquid-gas oscillators.