Stability of soluble surfactant-laden falling film over a hydrophobic incline in the presence of external shear

The hydrodynamic stability analysis of gravity-driven, soluble surfactant-laden fluid streaming down a slippery, slanted plane in the presence of external shear force is being explored in this article. The Navier–Stokes equations are considered for the fluid flow along with the appropriate advection–diffusion equations for the concentration of different surfactant species. The monomers considered here are anticipated to dissolve in the bulk flow and can be adsorbed at the interface of air-liquid as well. Also, the adsorption–desorption kinetics of the surfactants at the free surface is taken into consideration. In order to control the instability of the soluble surfactant-laden flow, the wall slip is considered, which changes the shear rate near the wall, and an externally imposed shear stress, which directly impacts the shear-layered flow near the free surface. The Orr–Sommerfeld eigenvalue system is obtained and is solved numerically in the arbitrary wave regime using the Chebyshev spectral collocation method. An analytical solution using the long-wave approximation method in the small wavenumber regime (k≪1) is also discussed. The numerical analysis reveals the existence of temporal surface and surfactant modes. Moreover, in the high modified Reynolds number regime, for low inclination angles, another temporal mode named shear mode is observed. The external shear force initially destabilizes the surface mode, but as inertial effects grow, the shear-induced normal stress acts to stabilize it. However, the surfactant mode is a damped mode. The surfactant parameter βa, and solubility parameter Ra have destabilization effect on the surfactant mode, but for the higher values of these parameters, the flow system turns into an insoluble surfactant-laden flow. The slip parameter stabilizes the surfactant mode, but the external shear destabilizes it. Finally, the external shear triggers the shear mode instability. Although the slip parameter has exactly the opposite effect on shear mode compared to the external shear.