Suppressing falling film instabilities by Marangoni forces

The linear stability of a thin liquid layer falling down an inclined wall heated by a downstream linearly increasing temperature distribution is investigated. It is shown that hydrodynamic and Marangoni instabilities yield two types of transverse instabilities: long surface waves and convective rolls, and longitudinal convective rolls, much like in the case of a uniformly heated wall. However, in contrast to the problem of a uniformly heated wall, where the thermocapillary forces have a destabilizing influence on all instability modes, here they can either destabilize or stabilize the flow. For liquids with sufficiently large Prandtl numbers, increasing the temperature gradient first destabilizes the flow and then stabilizes it. On the other hand, for small Prandtl numbers, increasing the temperature gradient leads to a monotonic stabilization of all instability modes.