Effect of inertia on the insoluble-surfactant instability of a shear flow

We study, for the case of the two-layer plane Couette flow, the effects of inertia on the recently found instability due to insoluble surfactants. The insoluble-surfactant instability takes place even when inertia is absent provided an interface or a free surface under a nonzero shear is laden with an insoluble surfactant. Considering a normal mode of the streamwise wave number alpha, the perturbation theory we construct is good for any a provided the Reynolds number is correspondingly small. Inertia is responsible for some notable effects, including the appearance of new regions of instability and stability. For long-and only for long-waves, the following growth-rate additivity property for the inertia and interfacial instabilities holds: the growth. rate corresponding to some nonzero values of the Marangoni number M and the Reynolds number Re is the sum of two contributions, one corresponding to the same value of M but zero Re, and the other corresponding to the same (nonzero) Re but zero M. This violation of the additivity property is in contrast to the case of a surfactantless Couette flow where this property holds for all wave numbers. Thus these results provide a counterexample to a conjecture that this additivity property is a universal principle. Among other results, when the thiner layer is the less viscous one, there is a nonzero critical Marangoni number M-c for the onset of instability; this (long-wave) threshold M, grows from zero with the Reynolds number. Also, varying the ratio of viscosities through certain characteristic values leads to changes in the topology of marginal-stability curves.