Nonlinear Oscillations in a Capped Liquid-Air Column

Patrick Weidman

Lorenceau, et al (Phys. Fluids ,14 ,2002)  studied the gravitational oscillations of a liquid column inside a vertical tube immersed in a liquid bath.  We expand on that work to investigate the effect of air trapped above the liquid column on the oscillation amplitude and frequency.  Assuming Boyle's law ($pV$ = const.) for the air, one can derive a pair ODE's describing the upward and downward variable mass liquid motion.  The singular pressure losses incurred at the end of the immersed tubed are accounted for, but viscous and capillary losses are neglected.  This system admits two new parameters: $\alpha = (L - H)/H$ and $\beta = P_{\infty}/\rho g H$, where $L$ is the total tube length, $H$ is the immersion depth, and $P_{\infty}$ is the ambient pressure above the liquid bath. While the separate upward and downward motions with pressure losses are described by conservative equations, their combined motion is not.  Numerical solutions reveal the effect of $\alpha$ and $\beta$ on the amplitude and frequency of liquid oscillations and these results are compared with their two-term asymptotic approximations. Preliminary results obtained in a laboratory experiment will also be presented.