Abstract
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.
Highlights
Thin liquid films flowing down an inclined plate have rich dynamics, i.e., a variety of phenomena that extends from rich wave phenomena to an extraordinary sequence of wave transitions.1–8 thin liquid films flowing down a heated substrate exhibit complex and interesting interfacial dynamics, including the formation of fingering ripples and traveling wave patterns.1,2,9–13 Such dynamics are related to the nonlinear spatiotemporal evolution of their free surface, as shown in Fig. 1, when subjected to various thermal, structural, and mechanical factors.14,15 Basically, thin liquid films can be regarded as dissipative systems with a variety of properties because they are favorable to energy flow and energy transfer within the system
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 effects are investigated under the framework of linear stability analysis
These nonuniformities induce surface tension gradients, as a result of temperature variations, giving rise to interfacial tangential (Marangoni) stresses and causing interfacial long-wave deformations that will induce a motion scitation.org/journal/adv along the interface.15. Since this interfacial displacement is associated with motion in the bulk to conserve the liquid volume, the thermocapillary and capillary effects can lead to disordered convective motions within the fluid. These free surface thermocapillary-based instabilities as a result of thermal Marangoni effects are of two types in the case of a liquid film resting on a heated plate: the short-wave modes discovered by Pearson18 and the long-wave modes discovered by Scriven and Sterling,19 whose effects on a liquid film falling down a uniformly heated plate have been explained by Goussis and Kelly
Summary
Thin liquid films flowing down an inclined plate have rich dynamics, i.e., a variety of phenomena that extends from rich wave phenomena to an extraordinary sequence of wave transitions. thin liquid films flowing down a heated substrate exhibit complex and interesting interfacial dynamics, including the formation of fingering ripples and traveling wave patterns. Such dynamics are related to the nonlinear spatiotemporal evolution of their free surface, as shown in Fig. 1, when subjected to various thermal, structural, and mechanical factors. Basically, thin liquid films can be regarded as dissipative systems with a variety of properties because they are favorable to energy flow and energy transfer within the system. Yeo et al. investigated the influence of long-wave thermocapillary instabilities on a thin layer of liquid film lying on a solid substrate subjected to both uniform and nonuniform heating conditions Their investigations regarded the linear stability properties of the nonlinear spatiotemporal evolution of the base states as applying transverse disturbances using a transient growth-type analysis and by splitting the problem to be treated into two parts: an evolution equation for the base flow and an evolution equation for the amplitude of the small perturbations. We will show that when we consider the case of a thin-liquid film flowing on a uniformly heated inclined plate, at small values of the Biot number, a reasonable approximation considering the physical context where the nonlinearity is relatively small is considered to totally balance the dispersion, and a multiplescale expansion method combined with the reductive-perturbation method will lead to a cubic complex Ginzburg–Landau (CCGL) equation..
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