Effects of velocity slip on the inertialess instability of a contaminated two-layer film flow

Abstract

This manuscript provides a theoretical stability analysis for the configuration of two-layer immiscible flow down an inclined plane with velocity slip along the incline in the limit of zero Reynolds number. Surfactants may be present at the air–liquid interface, liquid–liquid interface, or both. In addition to an Orr–Sommerfeld analysis (again at zero Reynolds number), a long wavelength stability analysis is performed and the results are shown to be consistent. The interface mode, namely the mode of instability that arises because of viscosity stratification, is examined. Stability results (growth rates as a function of slip parameter and neutral stability boundaries) for various configurations of viscosity, surfactant placement, and layer thickness are compared with those of the previous literature and found to agree. It is found that velocity slip along the inclined plane reduces the maximum growth rate of instabilities in configurations where they occur, and the range of unstable wave numbers shrinks as well, indicating that slip has a promise for stabilization. This suggests that there is a possibility of using this favourably as a control option for two-layer flows in the absence or presence of surfactants, in relevant applications by designing the substrate to be a porous substrate with small permeability or a slippery substrate or a rough substrate or a hydrophobic substrate which can be modelled as substrates with velocity slip.