Effect of imposed shear on falling liquid films with variable fluid properties*

We present a study on the dynamics of a gravity-driven thin liquid film flow on a uniformly heated inclined plane in the presence of imposed shear stress. Based on the lubrication theory, we develop an evolution equation for the film thickness that accounts for gravity, shear stress, and temperature-dependent fluid properties. Linear stability analysis for this equation yields critical conditions for the onset of instability in long-wave perturbations. The analysis also shows the dependence of the critical Reynolds number on the direction of the imposed shear stress as well as other flow parameters. In addition, we perform a weakly nonlinear stability analysis based on the method of multiple scales and obtain a complex Ginzburg Landau equation. We observe that the film not only has supercritical stable and subcritical unstable zones, but also unconditional stable and explosive zones. Numerical simulations of the model are conducted to further investigate the spatiotemporal behavior of nonlinear waves by applying a constant shear stress in the upstream and downstream directions. Finally, we study the energy transfer from the base state to the disturbances in the presence of the imposed shear stress.