On the Faraday instability in a surfactant-covered liquid

The formation of standing waves at the surfactant-covered free surface of a vertically-vibrated liquid is analyzed in this work. Assuming that the surfactants are insoluble and the effects of lateral boundaries are negligible, linear stability analysis and Floquet theory are applied to the governing equations. A recursion relation involving the temporal modes of the free-surface deflection and surfactant concentration variation results, and is solved to determine the critical vibration amplitude needed to excite the standing waves and the corresponding critical wave number. It is found that the critical vibration amplitude shows a minimum with respect to the Marangoni number, meaning that surfactants can potentially lower the value of the critical amplitude relative to its value for an uncontaminated free surface. The critical wave number, however, is found to be an increasing function of the Marangoni number. Analysis of the phase-angle difference between the free-surface deflection and the surfactant concentration variation suggests that the minimum in the critical amplitude arises because the Marangoni flows help produce a velocity field near the free surface similar to that which would be present if the liquid were inviscid.