Manipulation and control of instabilities for surfactant-laden liquid film flowing down an inclined plane using a deformable solid layer

Abstract

We analyzed the linear stability of surfactant-laden liquid film with a free surface flowing down an inclined plane under the action of gravity when the inclined plane is coated with a deformable solid layer. For a flow past a rigid incline and in the presence of inertia, the gas-liquid (GL) interface is prone to the free surface instability and the presence of surfactant is known to stabilize the free surface mode when the Marangoni number increases above a critical value. The rigid surface configuration also admits a surfactant induced Marangoni mode which remains stable for film flows with a free surface. This Marangoni mode was observed to become unstable for a surfactant covered film flow past a flexible inclined plane in a creeping flow limit when the wall is made sufficiently deformable. In view of these observations, we investigate the following two aspects. First, what is the effect of inertia on Marangoni mode instability induced by wall deformability? Second, and more importantly, whether it is possible to use a deformable solid coating to obtain stable flow for the surfactant covered film for cases when the Marangoni number is below the critical value required for stabilization of free surface instability. In order to explore the first question, we continued the growth rates for the Marangoni mode from the creeping flow limit to finite Reynolds numbers (Re) and observed that while the increase in Reynolds number has a small stabilizing effect on growth rates, the Marangoni mode still remains unstable for finite Reynolds numbers as long as the wall is sufficiently deformable. The Marangoni mode remains the dominant mode for zero and small Reynolds numbers until the GL mode also becomes unstable with the increase in Re. Thus, for a given set of parameters and beyond a critical Re, there is an exchange of dominant mode of instability from the Marangoni to free surface GL mode. With respect to the second important aspect, our results clearly demonstrate that for cases when the stabilizing contribution of surfactant is not sufficient for suppressing GL mode instability, a deformable solid coating could be employed to suppress free surface instability without triggering Marangoni or liquid-solid interfacial modes. Specifically, we have shown that for a given solid thickness, as the shear modulus of the solid layer decreases (i.e., the solid becomes more deformable) the GL mode instability is suppressed. With further decrease in shear modulus, the Marangoni and liquid-solid interfacial modes become unstable. Thus, there exists a stability window in terms of shear modulus where the surfactant-laden film flow remains stable even when the Marangoni number is below the critical value required for free surface instability suppression. Further, when the Marangoni number is greater than the critical value so that the GL mode remains stable in the rigid limit or with the deformable wall, the increase in wall deformability or solid thickness triggers Marangoni mode instability and, thus, renders a stable flow configuration into an unstable one. Thus, we show that the soft solid layer can be used to manipulate and control the stability of surfactant-laden film flows.