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A linear stability analysis of a two-layer plane Couette flow of twoimmiscible fluid layers with different densities, viscosities and thicknesses,bounded by two infinite parallel plates moving at a constant relative velocityto each other, with an insoluble surfactant along the interface and in thepresence of gravity is carried out. The normal modes approach is applied to theequations governing flow disturbances. These equations, together with boundaryconditions at the plates and the interface, yield a linear eigenvalue problem.When inertia is neglected velocity amplitudes are linear combinations ofhyperbolic functions, and a quadratic dispersion equation for the complexgrowth rate is obtained where coefficients depend on the aspect ratio, theviscosity ratio, the basic velocity shear, the Marangoni number Ma thatmeasures the effects of surfactant, and the Bond number Bo that measures theinfluence of gravity. An extensive investigation is carried out that examinesthe stabilizing or destabilizing influences of these parameters. There are twocontinuous branches of the normal modes: a robust branch that exists even withno surfactant, and a surfactant branch that vanishes when Ma $\downarrow 0$.Due to the availability of the explicit forms for the growth rates, in manyinstances the numerical results are corroborated with analytical asymptotics.For the less unstable branch, a mid-wave interval of unstable wavenumbers(Halpern and Frenkel (2003)) sometimes co-exists with a long-wave one. We studythe instability landscape, determined by the threshold curve of the long-waveinstability and the critical curve of the mid-wave instability in the (Ma,Bo)-plane. The changes of the extremal points of the critical curves with thevariation of the other parameters, such as the viscosity ratio, and the extremabifurcation points are investigated.

Surfactant and gravity dependent instability of two-layer channel flows: Linear theory covering all wave lengths