The effect of the thermal conductivity and thickness of the wall on the nonlinear instability of a thin film flowing down an incline

The nonlinear thermal instability of a thin liquid film falling down a heated wall is investigated. In particular, the heat conductivity and the thickness of the wall are taken into account. It is found that these two effects are represented by only one parameter which is the ratio of the nondimensional thickness of the wall and the nondimensional heat conductivity of the wall, that is d/Qcd/Qc. The longwave linear stability is described in a general form with respect to a wide range of values of this parameter in order to understand the behavior of the thin film. In the nonlinear case, the thin film instability is investigated in space and time for two examples of time dependent perturbations. The first one is at a perturbation frequency of 0.5 and the second one is at 2.5. The Reynolds numbers corresponding to the isothermal maximum growth rate are used and it is shown that they are located at important places of the k vs. R plane, where k is the wave number and R is the Reynolds number. It is found the important result that, for any fixed Marangoni number Ma  , the increase of the parameter d/Qcd/Qc stabilizes the flow and at the same time decreases the nonlinear amplitude of the perturbations.

► The effect of the wall thickness and thermal conductivity is taken into account.
► These effects appear in one parameter.
► The increase of this parameter stabilizes and suppresses wave modulation.
► For large Marangoni and Reynolds numbers there is no saturation above subcriticality.