In this paper, the validity of the Weighted Residuals Model (WRM), including the Marangoni effect, is investigated through linear stability analyses and two-dimensional nonlinear numerical simulations. The linear stability analyses with the WRM show that the model's accuracy decreases nearly linearly with the Marangoni number for each Reynolds number and achieves a maximum at approximately Re = 4 for each Marangoni number. This is quite different from the isothermal case, for which the error increases monotonically with the Reynolds number and remains small for small-to-moderate Reynolds numbers. This is very important for application of the WRM but has yet to be reported or investigated. The effects of the Reynolds number and Marangoni number on the nonlinear evolution of film layers are then investigated through numerical simulations. At small Reynolds number, it is found that the error caused by the Marangoni effect in predicting the phase speed can be ignored if the Marangoni number is small (Ma = 5) or makes the wave in the spatial numerical simulation considerably out of phase if the Marangoni number is large (Ma = 50). On the other hand, the saturation states can be generated by the WRM no matter whether the Marangoni number is small or large. When the Reynolds number is increased to a moderate value and the Marangoni number is taken as zero, it is found that the saturation wave produced by the WRM is very similar to the experimental one, except for the amplitude of the wave being somewhat larger and the wave speed as well as the wavelength being slightly smaller. Hence, it can be inferred that the WRM predicts the saturation waves well for small-to-moderate Reynolds numbers if the Marangoni numbers are limited to a small range depending on the Reynolds numbers. ARTICLE HISTORY

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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.

The Kelvin-Helmholtz instability is believed to be the dominant instability mechanism for free shear flows at large Reynolds numbers. At small Reynolds numbers, a new instability mode is identified when the temporal instability of parallel viscous two fluid mixing layers is extended to current-fluid mud systems by considering a composite error function velocity profile. The new mode is caused by the large viscosity difference between the two fluids. This interfacial mode exists when the fluid mud boundary layer is sufficiently thin. Its performance is different from that of the KelvinHelmholtz mode. This mode has not yet been reported for interface instability problems with large viscosity contrasts. These results are essential for further stability analysis of flows relevant to the breaking up of this type of interface.

Fluid mud often exists in coastal areas with an interface separating it from its upper water layer. When a surface wave propagates over a bed covered with water and fluid mud, it will cause an interfacial wave of the mud-water interface, which damps the surface wave and results in mass transport of fluid mud. Most researches about wave attenuation and mass transport of fluid mud are based on the assumption that the mud-water interface is unbroken. This assumption excludes the breaking interfacial waves that are known as an important mechanism responsible for mass and momentum transport between the two fluids. When the surface wave is long, its velocity field, which also serves as basic flows, may be susceptible to the Kelvin-Helmholtz (K-H) instability if the shears at the interface are strong enough. In the present paper, the critical conditions for the K-H instability to occur for the mud-water interface is investigated via linear stability analysis and numerical simulation. It is found that, for a K-H instability to occur, the Stokes boundary layer thickness induced by a surface wave must be large enough to penetrate the fluid mud layer and produce a strong shear at the interface. Meanwhile, a critical condition is found for a long surface wave to cause breakup of mud-water interface through K-H instability. This is practically instructive for waterway and harbor construction and protection because it predicts that a thicker mud layer is harder to be taken away by a surface wave.

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