Linear stability for surfactant-laden two-layer film flows down a slippery inclined plane

Abstract

Consider a two-layer film flows down a slippery inclined plane where both interface and free surface are contaminated by insoluble surfactants. A detailed linear stability analysis is performed in the presence of several flow parameters. Further, a coupled system of Orr-Sommerfeld equations is derived for the two-layer film flows with a free surface. The analytical calculation is accomplished based on the long-wave asymptotic expansion, while the numerical simulation is accomplished based on the Chebyshev spectral collocation method. Four modes, so-called surface mode, interface mode, surface surfactant mode and interface surfactant mode are identified in the long-wave regime. It is found that the surface surfactant mode is always stable, but the interface surfactant mode can be unstable if the Péclet number Pe2 corresponding to the interfacial surfactant exceeds its critical value and mr>1, where m and r respectively stand for the viscosity ratio and the density ratio. Further, in the long-wave regime, the interface mode can be stabilized, but the surface mode can be destabilized by introducing the effect of wall slip when m<1. However, the effect of wall slip on the interface and surface modes is completely opposite as soon as m>1. Furthermore, the viscosity ratio provides a dual role in the primary instability generated by the surface mode, i.e., it exhibits a stabilizing effect close to the criticality but exhibits a destabilizing effect far away from the criticality. However, the above results regarding the surface mode are fully converse if the density ratio, or, the thickness ratio varies rather than the viscosity ratio. Moreover, the interface surfactant mode can be stabilized by increasing the magnitude of density ratio, viscosity ratio and thickness ratio. In addition, the shear modes appear in the numerical simulation when the Reynolds number is very large and the inclination angle is very small. The shear mode associated with the lower fluid layer can be stabilized, but the shear mode associated with the upper fluid layer can be destabilized by increasing value of the viscosity ratio.