## Gravity, surfactants, and instabilities of two-layer shear flows

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A linear stability analysis of a two-layer plane Couette-Poiseuille flow of two immiscible fluid layers with different densities, viscosities and thicknesses, bounded by two infinite parallel plates moving at a constant relative velocity to each other, with an insoluble surfactant monolayer along the interface and in the presence of gravity is carried out. A normal modes approach is applied to the continuity and the Navier-Stokes equations that govern the fluid motion in the two layers, yielding two Orr-Sommerfeld equations for the perturbed vertical components of velocity in the two layers. These equations together with boundary conditions at the plates and the interface form a linear eigenvalue problem. When inertia is neglected the eigenfunctions can be determined analytically, and a dispersion equation for the increment, that is the complex growth rate, is obtained where coefficients depend on the aspect ratio, the viscosity ratio, the basic velocity shear, the Marangoni number that measures the effects of surfactant, and the Bond number that measures the influence of gravity. An extensive investigation is carried out that examines the stabilizing or destabilizing effects of these parameters on the flow within the two layers. Since the dispersion equation is quadratic in the growth rate, there are two branches: a robust branch that exists even when there is no surfactant, and a surfactant branch that vanishes when Ma = 0. Although Bo > 0 has a stabilizing effect, the results show that for certain parameters the small-amplitude long-wave disturbances may grow due to the destabilizing effects of surfactant, no matter how large the magnitude of Bo. When Bo < 0 gravity is destabilizing but surfactants can be either stabilizing or destabilizing depending on the parameters.