Nonlinear Hydrodynamic Instability Control on a Shear-Imposed Two-Layer Film Flow by Means of Velocity Slip and Insoluble Surfactant

Abstract

The impact of an insoluble surfactant on the free surface of a two-layer fluid flowing down an inclined slippery plane is analyzed under external shear stress. The flow dynamics in both layers are governed by the Navier-Stokes equations, with shear stress applied to the upper layer. Using perturbation and normal-mode analysis, the governing equations and boundary conditions are transformed into the Orr-Sommerfeld system and solved via the spectral collocation method. The study reveals the presence of an unstable Yih mode due to the free surface. This instability is suppressed by the Marangoni effect, which arises from the surfactant-induced surface tension gradient. The Marangoni mode stabilizes the interface by counteracting perturbations. However, as the external shear stress increases, the stabilizing effect of the Marangoni mode weakens. Beyond a certain threshold, the Marangoni mode destabilizes, leading to interfacial disturbances. This study highlights the interplay between surfactants, shear stress, and hydrodynamic stability. The findings have applications in coating processes, microfluidics, and geophysical fluid dynamics.