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 Tₑ . Concerning the long time scale Tₑ , we show that: \[ {Tₑ}{Tₑ₀}=({D₀}{d})5/2, \] where Tₑ₀\,≈\, 3.0 L/√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: \[ T\,{=}\,{L}{√{γ P₀/ρ}}Φ(1z̄ᵢ/L), \] 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 1z̄ᵢ/L . Finally, this analysis of time scales involved in the emptying of vertical cylinders is applied to other liquid–gas oscillators.