A nonlinear stability analysis has been carried out for plane liquid sheets moving
in a gas medium at rest by a perturbation expansion technique with the initial
amplitude of the disturbance as the perturbation parameter. The first, second and
third order governing equations have been derived along with appropriate initial and
boundary conditions which describe the characteristics of the fundamental, and the
first and second harmonics. The results indicate that for an initially sinusoidal sinuous
surface disturbance, the thinning and subsequent breakup of the liquid sheet is due
to nonlinear effects with the generation of higher harmonics as well as feedback into
the fundamental. In particular, the first harmonic of the fundamental sinuous mode is
varicose, which causes the eventual breakup of the liquid sheet at the half-wavelength
interval of the fundamental wave. The breakup time (or length) of the liquid sheet is
calculated, and the effect of the various flow parameters is investigated. It is found
that the breakup time (or length) is reduced by an increase in the initial amplitude
of disturbance, the Weber number and the gas-to-liquid density ratio, and it becomes
asymptotically insensitive to the variations of the Weber number and the density
ratio when their values become very large. It is also found that the breakup time (or
length) is a very weak function of the wavenumber unless it is close to the cut-off
wavenumbers.
Asymptotic Analysis and Stability of Inviscid Liquid Sheets
