Modulational instability in thin liquid film flowing down an inclined uniformly heated plate

The modulational instability properties regarding the evolution of interfacial disturbances of the flow of a thin liquid film down an inclined uniformly heated plate subject to thermal Marangoni (thermocapillary) effects are investigated under the framework of linear stability analysis. The investigation has been performed both analytically and numerically using a complex cubic Ginzburg–Landau equation without the driving term to provide comprehensive pictures of the influence of nonlinearity, dissipation, and dispersion on interfacial disturbance generation and evolution. It is shown that when the interplay between linear and nonlinear effects is relatively important, the disturbances evolve as a superposition of groups of traveling periodic waves with different amplitudes, and the interfacial disturbances evolve as smooth modulations. Furthermore, the dynamic modes of these disturbances become aperiodic. Conversely, when the evolution of instabilities is influenced by strong nonlinearity, the flow saturates, and different situations lead to different possible modulated wavy structures, caused by the interplay between nonlinear and linear dispersive and dissipative effects. Moreover, the appearance and the spatial and temporal evolution of the modulated disturbance profiles are influenced by both the amplitude of the disturbances and the linear dissipative term. Here, based on our investigation, two cases are highlighted. In the first case, which corresponds to very small amplitude of the disturbances, the dynamic modes of the disturbances evolve from periodic traveling waves to spatial and temporal modulated periodic solitary wave patterns. In the second case, by increasing the amplitude of the disturbances, the appearance of modulational modes is rapid, and therefore, we can observe the development of modulationally marginal-like stable patterns or spatial and temporal modulated patterns with nonuniform profiles.