__Nonlinear
Oscillations in a Capped Liquid-Air Column__

__Patrick Weidman__

University of Colorado at Boulder

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.