A discussion on the validity domain of the weighted residuals model including the Marangoni effect for a thin film flowing down a uniformly heated plate

Abstract

ABSTRACTIn 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 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 () or makes the wave in the spatial numerical simulation considerably out of phase if the Marangoni number is large (). 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.