Strong nonlinearity of the interfacial-surfactant instability

The nonlinear stages of the recently uncovered instability due to insoluble surfactant at the interface between two fluids in a creeping plane Couette flow are investigated for the case when one of the fluids is a thin film and the other one is a much thicker layer. Numerical simulation of strongly nonlinear longwave evolution equations which couple the film thickness and the surfactant concentration reveals that in contrast to all similar instabilities of surfactant-free flows, no amount of the interfacial shear rate can lead to a small-amplitude saturation of the instability. Thus, the flow is stable when the shear is zero, but with non-zero shear rates, no matter how small or large (while remaining below an upper limit set by the assumption of creeping flow), it will reach large deviations from the base values– of the order of the latter or larger. It is conjectured that the time this evolution takes grows to infinity as the interfacial shear approaches zero. These conclusions hold when (a small) surfactant diffusivity is taken into account.