Influence of an External Perpendicular Oscillation on Stability of a Vertical Falling Liquid Film

A vertical falling Newtonian liquid film flow is inherently unstable to surficial long-wave disturbances. Imposing external oscillation can stabilize the long-wave instability, but also triggers additional parametric instabilities. The effect of oscillation frequency on the stability is subtle. By using the “viscosity-gravity” scaling, the effect of oscillation frequency on the stability can be investigated exhaustively by separating it from other control parameters. In this paper, the effects of external perpendicular oscillation on the stability of a vertical falling liquid film are then investigated by a combination of linear stability analyses based on Floquet theory and numerical simulations with an unsteady weighted residual model (WRM). The linear analyses show that, increasing oscillation amplitude always has a stabilizing effect on the long-wave instability. On the other hand, increasing or decreasing oscillation frequency can suppress the long-wave instability, depending on whether the oscillation amplitude or the acceleration is fixed. The effect of varying oscillation frequency on the long-wave instability is opposite to that on the parametric instabilities. The long-wave and parametric instabilities compete with each other as the oscillation amplitude and frequency are varied with the Reynolds number fixed. A weakness of the long-wave instability always accompanies enhancements of the parametric instabilities, and vice versa. As a contrast, an increase of Reynolds number always results in more unstable long-wave and parametric instabilities. The numerical simulations with the WRM show that the wave amplitudes and the minimal local thickness of film are proportional to the unstable wavenumbers range rather than the growth rate of the instability. For a given oscillation frequency and Reynolds number, there exist a critical oscillation amplitude above which externally imposed oscillations perpendicular to the transversal direction of the film can also trigger a chaotic behavior in the film, just like what happens in the case where the oscillation is parallel to the stream-wise direction of the film.