Abstract
This paper studies mainly the evolution and linear stability of the nonlinear surface waves of a two-dimensional viscous liquid film along an uneven inclined non-uniformly heated wall. A long wave perturbation method is used to derive zero- and first-order evolutions equations of the nonlinear surface wave flowing on an uneven substrate. Based on the obtained evolution equations, the diagram of evolution progress for film surface wave on a sinusoidal corrugated substrate is drawn, the linear stability analysis is also studied, and the effect of various parameters on the flow stability of liquid membrane is analysed. Theoretical results demonstrate that the free surface of the film shows sine wave and has the same frequency as the substrate, and the film thickness will decrease gradually along the flow direction. Marangoni number gives a stabilizing effect, the stable zone increases with the increase of Marangoni number. While, Peclet number and the angle theta are unstable factors, the stable region decreases with the increase of them. Besides, the trends that Marangoni number, Peclet number and angle theta may impact on the stability of the film whice are consistent with one another, so a liquid film is easy to destabilize at the trough of the substrate.
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