Effect of soluble surfactant on thermocapillary instability in falling film

We conduct a study on linear thermocapillary instability for a two-dimensional gravitydriven viscous film flowing over a uniformly heated impermeable plane, even though the film is contaminated with a soluble surfactant. Specifically, we revisit the recent study soluble surfactant plays in different unstable temporal modes accountable for thermocapillary instabilities. In the present study, we have identified a total of four dominant temporal modes, namely, the H-mode (surface mode), shear mode, and two additional thermocapillary S- and P-modes. From the long-wave stability analysis, we can report that the surfactant Marangoni number stabilizes, whereas the thermal Marangoni number destabilizes the H-mode instability. As a result, we establish a mathematical connection between the surfactant and thermal Marangoni numbers, demonstrating that the onset of H-mode instability is independent of the effects of heat transfer and soluble surfactant. Moreover, we see that the nondimensional base surface surfactant concentration FE has a dual role in the H-mode instability in its permissible range. However, the numerical results based on the Chebyshev spectral collocation method in the arbitrary wave-number regime reveal that the measure of surfactant solubility parameter s stabilizes the H-mode, S-mode, P-mode, and shear-mode instabilities by diminishing their unstable zones as long as it increases. However, the parameter FE exhibits both stabilizing and destabilizing roles in the H-mode, S-mode, P-mode, and shear-mode instabilities. In particular, increasing FE has a stabilizing impact, but eventually, it becomes destabilizing as it approaches very close to FE approximate to 1. Finally, a modal coalescence happens between the H-mode and the S-mode for a specific set of parameter values.