STABILITY OF AXISYMMETRICAL WAVES ON LIQUID-FILMS FLOWING DOWN A VERTICAL COLUMN TO AZIMUTHAL AND STREAMWISE DISTURBANCES

Axisymmetric film waves on the outside surface of a vertical column have been observed to be more susceptible to azimuthal disturbances as the column radius increases. We scrutinize this phenomenon here and show that, for liquids with high surface tension, infinitesimal azimuthal disturbances first become unstable when the flow rate per unit azimuthal length exceeds (5sigmaupsilon/6rhor2g) where r is the column radius and sigma is the interfacial tension. However, even beyond the onset of these azimuthal disturbances, only long, finite-amplitude, streamwise axisymmetric waves are unstable to such disturbances. We also show that, in contrast to earlier results, relatively short waves near the neutral curve are unstable to the classical streamwise Eckhaus sideband instability. There is a window of streamwise wavenumbers, including the wave number k(m) corresponding to the maximum growth rate, that is stable to both instabilities. This window is reminiscent of the ''Busse balloon'' in Rayleigh-Benard instability and we study their size and shape as a function of column radius. In all cases, finite-amplitude axisymmetric waves with the maximum growing streamwise wavenumber k(m) are stable to both sideband instabilities. This then suggests that there are two necessary conditions for the appearance of finite-amplitude azimuthally varying waves-the destabilization of the infinitesimally small azimuthal modes and the nonlinear excitation of axisymmetric waves longer than the maximum-growing ones. We suggest that subharmonic instability of the maximum-growing waves can provide the second mechanism.