Weakly Nonlinear Stability Analysis of a Non-Uniformly Heated Non-Newtonian Falling Film

A thin liquid layer of a non-Newtonian film falling down an inclined plane that is subjected to non-uniform heating has been considered. The temperature of the inclined plane is assumed to be linearly distributed and the case when the temperature gradient is positive or negative is investigated. The film flow is influenced by gravity, mean surface-tension and thermocapillary force acting along the free surface. The coupling of thermocapillary instability and surface-wave instabilities is studied for two-dimensional disturbances. A non-linear evolution equation is derived by applying the long-wave theory and the equation governs the evolution of a power-law film flowing down an inclined plane. The linear stability analysis shows that the film flow system is stable when the plate temperature is decreasing in the downstream direction while it is less stable for increasing temperature along the plate. Weakly non-linear stability analysis using the method of multiple scales has been investigated and this leads to a secular equation of the Ginzburg-Landau type. The analysis shows that both supercritical stability and subcritical instability are possible for the film flow system. The results indicate the existence of finite-amplitude waves and the threshold amplitude and non-linear speed of these waves are influenced by thermocapillarity. The results for the dilatant as well as pseudoplastic fluids are obtained and it is observed that the result for the Newtonian model agrees with the available literature report. The influence of non-uniform heating of the film flow system on the stability of the system is compared with the stability of the corresponding uniformly heated film flow system.