Nonlinear stability of viscous film flowing down an inclined plane with linear temperature variation

Weakly non-linear stability analysis of a thin liquid film falling down a heated plane with linear temperature variation has been investigated in a finite amplitude regime. Using the long wave expansion method, a non-linear evolution equation for the development of the free surface is derived. A normal mode approach and the method of multiple scales are used to obtain the linear and non-linear stability solution for the film flow. The study reveals that both supercritical stability and subcritical instability are possible for this type of thin film flow. The influence of thermocapillary force on the span of supercritical/subcritical regions are examined. It is noticed that the unconditional stable region vanishes after a cutoff Marangoni number, whereas other regions increase with the increase in Marangoni number for fixed values of other parameters. Finally, we also scrutinize the effect of Marangoni number on the amplitude and speed of waves. In the supercritical region amplitude and speed of the non-linear waves increase with the increase in Marangoni number, while in the subcritical region the threshold amplitude decreases with the increase in Marangoni number.