Linear stability of a contaminated two-layer films flowing down a slippery inclined plane: Parametric study and physical mechanism

Abstract

A linear stability analysis of isothermal two-layer surfactant-laden films flowing down a slippery inclined plane is carried out, with a focus on the shear mode instabilities. The intention is to extend our previous study [Bhat and Samanta, “Linear stability for surfactant-laden two-layer film flows down a slippery inclined plane,” Chem. Eng. Sci. 220, 115611 (2020)] in the high Reynolds number regime. The stability analysis is performed under the umbrella of the Orr–Sommerfeld-type boundary value problem. The method of energy budget is employed to figure out the proper physical mechanisms responsible for the growth of the shear modes under the influence of various flow parameters. Numerical results reveal that the flow configuration under consideration is susceptible to two distinct shear modes in the high Reynolds number regime. The shear mode associated with the lower fluid layer is stabilized if the viscosity ratio, wall slip, and interfacial surfactant concentration are increased. However, increasing density ratio has a destabilizing impact on the lower-layer shear mode. On the other hand, the shear mode associated with the upper fluid layer is destabilized if the viscosity ratio and interfacial surfactant concentration are increased. However, the density ratio exhibits a non-monotonic impact, but the wall slip exhibits a stabilizing influence on the upper-layer shear mode. The energy budget analysis predicts that the primary energy source terms for the growth of the shear mode instabilities are base shear stresses, which transfer energy to the disturbance through the Reynolds stress term.