Drainage, rebound and oscillation of a meniscus in a tube

In this paper, the drainage and subsequent rebound of a liquid column in a cylindrical tube is examined experimentally and theoretically. When liquid is drawn up into a capillary and then released under gravity, inertia allows the meniscus to overshoot the equilibrium capillary rise height. The meniscus then rebounds up the tube, again overshooting the equilibrium height and undergoes oscillation. By varying both the immersion depth and radius of the tube, one can observe rich dynamical behavior, with the most dramatic being the formation of a fast liquid jet, barely visible to the naked eye but easily captured with high-speed video. In addition to the flow separation caused by the sudden expansion at the end of the tube, this jet serves as a mechanism of energy dissipation. Some qualitative differences between the works of Quere et al. ["Rebounds in a capillary tube," Langmuir 15, 3679-3682 (1999)] and Lorenceau et al. ["Gravitational oscillations of a liquid column in a pipe," Phys. Fluids 14(6), 1985-1992 (2002)] and the present experiment are observed and discussed. A critical condition for oscillatory behavior is derived theoretically and matches well with the experimental observation. Once in the oscillatory regime, both the maximum depth below and the maximum rebound height above the equilibrium level are investigated by performing a systematic sweep through the relevant parameter space, incorporating the initial meniscus height, immersion depth, tube radius, and fluid properties. Lastly, the characteristic period of oscillation, t_p, is assessed and found to be largely independent of fluid viscosity, and could be reasonably well-collapsed by a single curve whereby tp∼hi, where h_i is the tube immersion depth.