Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

The effect of inertia on the Yih-Marangoni instability of the interface between two liquid layers in the presence of an insoluble surfactant is assessed for shear-driven channel flow by a normal-mode linear stability analysis. The Orr-Sommerfeld equation describing the growth of small perturbations is solved numerically subject to interfacial conditions that allow for the Marangoni traction. For general Reynolds numbers and arbitrary wave numbers, the surfactant is found to either provoke instability or significantly lower the rate of decay of infinitesimal perturbations, while inertial effects act to widen the range of unstable wave numbers. The nonlinear evolution of growing interfacial waves consisting of a special pair of normal modes yielding an initially flat interface is analysed numerically by a finite-difference method. The results of the simulations are consistent with the predictions of the linear theory and reveal that the interfacial waves steepen and eventually overturn under the influence of the shear flow.