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The linear instability of a surfactant-laden two-layer falling film over aninclined slippery wall is analyzed under an influence of external shear whichis imposed on the top surface of the flow. The free surface of the flow as wellas the interface among the fluids are contaminated by insoluble surfactants.Dynamics of both the layers are governed by the Navier--Stokes equations, andthe surfactant transport equation regulates the motion of the insolublesurfactants at the interface and free surface. Instability mechanisms arecompared by imposing the external shear along and opposite to the flowdirection. A coupled Orr--Sommerfeld system of equations for the consideredproblem is derived using the perturbation technique and normal mode analysis.The eigenmodes corresponding to the Orr--Sommerfeld eigenvalue problem areobtained by employing the spectral collocation method. The numerical resultsimply that the stronger external shear destabilizes the interface modeinstability. However, a stabilizing impact of the external shear on the surfacemode is noticed if the shear is imposed in the flow direction, which is incontrast to the role of imposed external shear on the surface mode for asurfactant laden single layer falling film. Moreover, the impression of shearmode on the primary instability is analyzed in the high Reynolds number regimewith sufficiently low inclination angle. Under such configuration, dominance ofthe shear mode over the surface mode is observed due to the weaker impact ofthe gravitation force on the surface instability. The shear mode can also bestabilized by applying the external shear in the counter direction of thestreamwise flow. Conclusively, the extra imposed shear on the stratifiedtwo-layer falling film plays an active role to control the attitude of theinstabilities.

Effect of imposed shear on the dynamics of a contaminated two-layer film flow down a slippery incline