Lorenceau, {\it et al.} ({\it Phys. Fluids}, {\bf 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.
Converted from pw.txt (plain text).