Stability of conducting viscous film flowing down an inclined plane with linear temperature variation in the presence of a uniform normal electric field

Weakly nonlinear stability analysis of a thin liquid film falling down a heated inclined plane with linear temperature variation in the presence of a uniform normal electric field has been investigated within the finite amplitude regime. A generalized kinematic equation for the development of free surface is derived by using long wave expansion method. A normal mode approach and the method of multiple scales are used to investigate the linear and weakly nonlinear stability analysis of film flow, respectively. It is found that both Marangoni and electric Weber numbers have destabilizing effect on the film flow. The study reveals that both supercritical stability and subcritical instability are possible for this type of film flow. It is interesting to note that both the Marangoni and electric Weber numbers have qualitatively same influence on the stability characteristics but the effect of Marangoni number is much stronger compare to the electric Weber number. Scrutinizing the effect of Marangoni and electric Weber numbers on the amplitude and speed of waves it is found that, in the supercritical region amplitude and speed of the nonlinear waves increases with the increase in Marangoni and electric Weber numbers, while in the subcritical region the threshold amplitude decreases with the increase in Marangoni and electric Weber numbers. Finally, we obtain that spatially uniform solution is side-band stable in the supercritical region for our considered parameter range.