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 alpha 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 thinner layer is the less viscous one, there is a nonzero critical Marangoni number Mc for the onset of instability; this (long-wave) threshold Mc 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.
Frenkel
Alexander L
AL
Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487, USA.
Halpern
David
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