A thin liquid film falling on a uniformly heated horizontal plate spreads into fingering ripples that can display a complex dynamics ranging from continuous waves, nonlinear spatially localized periodic wave patterns (i.e. rivulet structures) to modulated nonlinear wavetrain structures. Some of these structures have been observed experimentally, however conditions under which they form are still not well understood. In this work we examine profiles of nonlinear wave patterns formed by a thin liquid film falling on a uniformly heated horizonal plate. In this purpose, the Benney model is considered assuming a uniform temperature distribution along the film propagation on the horizontal surface. It is shown that for strong surface tension but relatively small Biot number, spatially localized periodic-wave structures can be analytically obtained by solving the governing equation under appropriate conditions. In the regime of weak nonlinearity, a multiple-scale expansion combined with the reductive perturbation method leads to a complex Ginzburg-Landau equation, the solutions of which are modulated periodic pulse trains which amplitude, width and period are expressed in terms of characteristic parameters of the model. a) Electronic mail: dikande.alain@ubuea.cm; http://laramans.blogspot.com/.
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.
In the present work, we focus on the longitudinal model of microtubules (MTs) proposed by Satar i c~̵́ et al. [Phy. Rev. E 48, 89 (1993)], that consider cell MTs to have ferroelectric properties, i.e., a displacive ferro-distortive system of dimers and usually referred to as u-model of MTs. It has been shown that during the hydrolysis of GTP into GDP, the energy released is transferred along the MTs trough kink-like solitons. Substantially, we propose to theoretically investigate the dynamic of MTs by intrinsically taking into account the effect of the oriented molecules of polarized cytoplasmic water and enzymes surrounding the MT. In this regards, we introduce a cubic nonlinear term in the electric potential characterizing the polyelectrolyte features of MTs and show that in addition to the kink and antikink solitons, asymmetrical bright and dark solitons, and discrete modes can also propagate along the MTs. Theses results are supported by numerical analysis. The investigation shows us that the nonlinear dynamics of MTs is strongly impacted by the intrinsic electric field, the polyelectrolyte and the viscosity effects. Moreover, new solitons and discrete solitary modes may help to find new phenomena occurring in the microtubulin systems.
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