Hydrodynamics and instabilities of falling liquid film over a non-uniformly heated inclined wavy bottom

Hydrodynamics and instabilities of a thin viscous liquid film flowing down an undulated inclined plate with linear temperature variation have been investigated. Using the long-wave expansion method, a non-linear evolution equation for the development of the free surface is derived under the assumption that the bottom undulations are of moderate steepness. A normal mode approach has been considered to take into account the linear stability of the film to investigate both the spatial and temporal instabilities, while the method of multiple scales is used to obtain the Ginzburg-Landau-type worldly equation for studying the weakly non-linear stability solutions. The numerical study has been carried out in python with a newly developed library Scikit-FDif. The entire investigation is done for a general bottom profile followed by a case study with a sinusoidal topography. The case study reveals that the Marangoni effect destabilizes the film flow throughout the domain, whereas the bottom steepness zeta gives a dual effect for the linear stability. In the "uphill" portion, an increase in zeta stabilizes the flow, and in the "downhill" portion, an increase in zeta gives a destabilizing effect. Furthermore, a weakly non-linear study shows that both supercritical and subcritical solutions are possible for the system. It is noted that the unconditional stable region decreases and all the other region increases in the "downhill" portion in comparison with the "uphill" portion for a fixed set of parameters. The stability analysis of a truncated bimodal system is investigated. The spatial uniform solution of the complex Ginzburg-Landau equation for sideband disturbances has also been discussed. Numerical simulation indicates that a different kind of finite-amplitude permanent wave exists. The amplitudes and the phase speeds of the wave are dependent on thermocapillary as well as the bottom steepness.