Non-linear instability of a thin film flowing down a cooled wavy thick wall of finite thermal conductivity

• The flow instability of the cooled thin film flow down a wavy wall is investigated.
• The thickness and thermal conductivity of the wall are taken into account.
• It is found that a negative Marangoni number can destabilize the free surface response to the wall deformations.
• The spatial resonance is more effective than cooling to stabilize the free surface time-dependent perturbations.

The flow of a thin film down a vertical cold thick wall with finite thermal conductivity is investigated under the lubrication approximation. It is shown that, despite the cooling from the wall, it is possible to find a new flow instability. That is, the free surface response to the wall deformation increases its amplitude with the negative Marangoni number. This amplitude growth is independent from the evolution of the time-dependent perturbations imposed on the free surface which, in contrast, are stabilized by cooling from the wall. However it is demonstrated that, even in this case, spatial resonance (see Dávalos-Orozco, 2007, 2008) is more effective to stabilize the time-dependent perturbations. From the results it is evident that these effects are possible only when the magnitudes of the thicknesses ratio and the thermal conductivities ratio are small.