Falling liquid films on a slippery substrate with variable fluid properties

We investigate the stability of a gravity-driven, thin, Newtonian liquid flowing on a uniformly heated slippery inclined plane. We construct a mathematical model of the flow that comprises the Navier-Stokes equation coupled with the equation of energy. While the rest of the boundary conditions are standard for thin-film problems, we apply a Navier slip boundary condition at the solid-liquid interface. We assume that the fluid thermophysical properties - density, dynamical viscosity, surface tension, and thermal diffusivity vary linearly with temperature as long as the change in temperature is small. In the analysis part, we follow the standard long-wave theory and construct a nonlinear evolution equation for the film thickness. This is followed by a linear and weakly nonlinear stability analysis. The linear analysis allows us to compute the critical Reynolds number of our problem and from this study, we conclude that the slippery substrate destabilizes the film flow. The weakly nonlinear stability analysis finds a finer description of various stable/unstable zones. Finally, we perform a numerical simulation of the evolution equation in a periodic domain using spectral methods. Our numerical results support the analytical predictions of the instability threshold using the linear and weakly nonlinear theories. The influence of the small Biot number is also investigated in presence of the slip length together with the variable fluid properties.

Automated summary

We investigated the stability of a thin Newtonian liquid flowing on a heated slippery inclined plane. By constructing a mathematical model and conducting linear and nonlinear analyses, we concluded that the slippery substrate destabilizes the flow. Our numerical simulation results supported the analytical predictions and further examined the influence of small Biot numbers considering slip length and variable fluid properties.