Long wave instability of thin film flowing down an inclined plane with linear variation of thermophysical properties for very small Biot number

We investigated interfacial instability of a thin liquid film flowing down an inclined plane, considering the linear variation of fluid properties such as density, dynamical viscosity, surface tension and thermal diffusivity, for the small variation of temperature. Using long wave expansion method and considering order analysis specially for very small Biot number (Bi) we obtained a single surface equation in terms of the free surface h(x,t). Considering sinusoidal perturbation method we carried out linear stability analysis and obtained the critical Reynolds number (Rec) and linear phase speed (cr), both of which depend on Kμ,Kρ but independent of Kσ,Kκ. Using the method of multiple scales, weakly nonlinear stability analysis is carried out. We demarcated subcritical, supercritical, unconditional and explosive zones and their variations for the variation of Kμ,Kρ and Kσ. Also we discussed the variations of threshold amplitude in the subcritical as well as in the supercritical zones for the variation of Kμ,Kρ and Kσ. Finally we discussed the variation of nonlinear wave speed Ncr for the variation of Kμ,Kρ and Kσ.