Thermocapillary instability on a film falling down a non-uniformly heated slippery incline. (July 2021) - Chattopadhyay, Souradip; Mukhopadhyay, Anandamoy; Barua, Amlan K.; Gaonkar, Amar K.
Abstract: A gravity-driven, thin, incompressible liquid film flow on a non-uniformly heated, slippery inclined plane is considered within the framework of the long-wave approximation method. A mathematical model incorporating variation in surface tension with temperature has been formulated by coupling the Navier–Stokes equation, governing the flow, with the equation of energy. For the slippery substrate, the Navier slip boundary condition is used at the solid–liquid interface. An evolution equation is formed in terms of the free surface, which includes the effects of inertia, thermocapillary as well as slip length. Using the normal mode approach, linear stability analysis is carried out and a critical Reynolds number is obtained, which reflects its dependence on the Marangoni number M n as well as slip length δ . This depicts that δ and M n both have the destabilization effect on the flow field. The linear study also reveals that the inertia force has a negligible effect compare to the thermocapillary or slip. In addition, the study highlights a critical Marangoni number at which the instability sets in when the thermocapillary stress attains a critical value. The method of multiple scales is used to investigate the weakly nonlinear stability analysis of the flow. The study interprets that the variation of M n and δ have substantial effects on different stable/unstable zones. It also shows that within a considered parametric domain, the unconditional stable zone completely vanishes for any value of M n, when the slip length δ attains a critical value. The study also divulges that in the subcritical unstable (supercritical stable) zone the threshold amplitude ( ζ a ) decreases (increases) with the increment of M n and δ . Further, we discussed the spatial uniform solution of the complex Ginzburg–Landau equation for sideband disturbances. Employing the Crank–Nicolson method, the nonlinear evolution equation of the free surface is solved numerically in a periodic domain, considering the sinusoidal initial perturbation of small amplitude. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The evolution of the maximum h max and minimum h min thickness amplifies, for small change of M n and δ . Further, it shows that the influence of the thermocapillary force amplifies the destabilizing nature of δ . The traveling wave solution confirms the existence of a fixed point for the considered parametric domain, chosen from the experimental data. Finally, the Hopf bifurcation of the fixed point exhibits that the nonlinear wave speed at the transcritical point increases as δ increases but decreases as M n increases. The noteworthy result which arises from the study is that a transcritical Hopf bifurcation exists if the slip length δ > max 1 6 M n − 1 3, 1 2 M n − 2 3 − M n . Highlights: The dynamics of thin film flows along a non-uniformly heated slippery incline is investigated. The slippery wall promotes the instability. When the wall is heated, the instability is reinforced. Flow control is possible through the Marangoni and slip effects. A transcritical Hopf bifurcation exists provided the slip length satisfies a certain condition. The nonlinear wave speed at the transcritical point increases (decreases) as the slip (thermocapillary) increases.