Linear Stability of Thin Liquid Film Flows over a Uniformly Heated Slippery Substrate under Heat Flux Boundary Condition

We investigate the stability of gravity-driven, Newtonian, thin liquid film falling down a uniformly heated slippery rigid inclined wall. All the previous authors considered Specified Temperature (ST) boundary condition to study the effects of slip length. But ST boundary condition does not include the effects of heat fluxes at wall-air and wall-liquid interfaces and so fails to incorporate the real situation. Consequently, we consider Heat Flux (HF)/mixed-type boundary condition as the thermal boundary condition on the rigid plate. This boundary condition involves the heat flux from the rigid plate to the surrounding liquid and the heat losses from the wall to the ambient air. Using long-wave expansion method we construct a highly nonlinear evolution equation in terms of the film thickness at any instant. Using normal mode approach the linear study reveals the stabilizing (destabilizing) behaviour of the wall film Biot number (dimensionless slip length). In absence of slip length, weakly nonlinear study transforms the evolution equation to the famous Kuramoto Sivashinsky (KS) equation. Finally, the numerical simulation of the full evolution equation is performed using Crank–Nicolson scheme over a periodic domain. It confirms the results obtained by the linear study.