Stability of gravity-driven free surface flow of surfactant-laden liquid film flowing down a flexible inclined plane

- Stability of surfactant-laden film flow down a flexible inclined wall is analyzed.
- The wall deformability qualitatively alters the stability behavior of film flow.
- Marangoni mode becomes unstable even in creeping flow due to wall elasticity.

We examined the linear stability of gravity-driven creeping flow of a liquid film flowing down an inclined plane when the inclined plane is coated with a deformable solid layer and the gas-liquid interface is contaminated with a monolayer of insoluble surfactant. The contaminated liquid film flowing down a rigid incline admits gas-liquid (GL) interfacial mode and surfactant-induced Marangoni mode, both of which remain stable in the creeping flow limit. The primary aim of this study is to explore the effect of the presence of a deformable wall, in place of a rigid inclined wall, on the stability behavior of Marangoni mode which originates because of the transport of surfactant at the GL interface. In presence of a deformable solid layer, two additional parameters namely, shear modulus and thickness of deformable solid layer also affects the stability behavior of falling film configuration. Our long-wave asymptotic analysis and results at finite and arbitrary wavenumbers demonstrated the destabilization of Marangoni mode solely due to the presence of deformable solid layer. Specifically, we have shown that for a given solid thickness, the Marangoni mode becomes unstable when the shear modulus of solid layer decreases below a critical value (i.e. the solid layer becomes sufficiently soft). The effect of increasing solid thickness is found to be destabilizing. The LS interfacial mode also becomes unstable at high wavenumbers below a threshold value of shear modulus, however, this value is much smaller than that required to trigger Marangoni mode instability. This implies that as the solid coating becomes more and more deformable, the Marangoni mode becomes unstable first followed by the LS interfacial mode. The GL mode was always found to be stable in creeping flow limit. Further, our long-wave analysis shows that the solid deformability has an additional stabilizing effect on GL mode. The neutral stability curves in nondimensional solid deformability parameter (or equivalently shear modulus) vs. wavenumber plane clearly depicts that the Marangoni mode is the dominant unstable mode in the creeping flow limit. Thus, the present study shows the destabilization of Marangoni mode solely due to presence of deformable solid layer and this we believe is the first example of the case where the instability of the Marangoni mode is observed when the fluid-fluid interface (here, GL interface) remains stress-free in the basic state.