Gravitational oscillations of a capped liquid-air column

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We study the gravitational oscillations of liquid confined in a vertical tube, immersed in a liquid bath, with air trapped above the liquid column. The trapped air acts as a nonlinear spring, in addition to the gravitational restoring force. The presence of the trapped air introduces two new parameters in the problem. One parameter denoted α governs the effect of compression and expansion and the other parameter denoted β governs the effect of initial pressure of the trapped air. In the absence of viscosity, the ordinary differential equations governing the upward and downward motion of the liquid in the tube are different and, considered separately, are conservative. Together, however, they describe the damped oscillations due to irreversible energy losses at the end of the tube in the bath. Numerical simulations show the effects of α and β on the oscillations. A Shanks approximate provides a surprisingly good estimation of the nonlinear damping. Taking into account viscous losses, a first order asymptotic analysis provides a prediction for the frequency of linear oscillations, in very good agreement with experiment. © 2014 AIP Publishing LLC.

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Physics of Fluids