Long-wave instability of flow with temperature dependent fluid properties down a heated incline

We present an analytical study which investigates the effect of temperature dependence in fluid properties on the interfacial instability of flow down a heated incline. Along with temperature variation in surface tension we consider variable mass density, viscosity, thermal conductivity and specific heat. A linear stability analysis is carried out which yields the critical conditions for the onset of instability in long-wave perturbations. Results are obtained for the particular case when there is variation only in surface tension, density and specific heat, and in the case with negligible and high rate of heat transfer across the free surface. For the general case, asymptotic expansions are implemented based on the assumed smallness of the variation with temperature in viscosity and thermal conductivity, or on weak heat transfer across the free surface.