Instability of an oscillatory fluid layer with insoluble surfactants

The linear stability of an infinite fluid layer with a deformable free surface covered by an insoluble surfactant and bounded below by a horizontal rigid plate oscillating in its own plane is studied based on the Floquet theory. The differential system governing the stability problem for perturbations of arbitrary wavenumbers is solved numerically by a Chebyshev collocation method. Stability boundaries are obtained in a wide range of amplitude and frequency of the modulation as well as surfactant elasticity. Results show that the presence of the surfactant may significantly stabilize (destabilize) the flow by raising (lowering) the critical Reynolds number associated with the onset of instability. The effect of the surfactant plays a stabilizing role for small surfactant elasticity and a destabilizing one for relatively large surfactant elasticity. The destabilizing effect of the surfactant on the stability of flows with a zero-shear surface is found for the first time. The disturbance modes in the form of travelling waves may be induced by the surfactant and dominate the instability of the flow.