Resonance of a liquid column in a capillary tube

Abstract

A liquid column with height H, trapped in a capillary tube of radius R, possesses a mechanical eigenfrequency. The axis of the capillary tube is oriented parallel to the gravitational force, so that the lower meniscus is fixed at the lower end of the tube, if the density \( \rho \) of the liquid is higher than that of the surrounding fluid. Basic assumptions of the linear theory are a pinned contact line, the spherical cap approximation and \( H \gg R \). The shape of the susceptibility function, the dimensionless ratio of the mean liquid displacement to the driving pressure gradient plotted versus frequency, depends on the parameter \( X_0 = \omega_0/\omega_c \) only, where \( \omega_0 \) denotes the eigenfrequency of the undamped system and \( \omega_c = \eta /(\rho R^2) \) stands for the characteristic frequency with the viscosity \( \eta \). For \( X_0 > \sqrt{24} \) the system is underdamped and resonance occurs, while for \( X_0 < \sqrt{24} \) the system is overdamped and resonance cannot be observed. If one interpretes capillary tubes as a model of a porous medium, the present mechanism contributes to the damping of sound waves in porous media saturated by two immiscible fluids, one of them being trapped.