Parametrically driven surface waves in surfactant-covered liquids

Vertical vibration will excite standing waves on a liquid free surface. We perform a linear-stability analysis for viscous and viscoelastic liquids of arbitrary depth to determine the role that insoluble surfactants play in the formation of these parametrically driven surface waves. We find that in order to obtain time-periodic solutions which involve Marangoni forces in a non-trivial way, it is necessary to consider the high-Peclet number limit of the surfactant transport equation. Floquet theory is applied to the linearized governing equations to obtain a recursion relation for the temporal modes of the free-surface deflection. The recursion relation is then solved numerically to obtain the critical vibration amplitude needed to excite the surface waves, and the corresponding wavenumber. The results show that the presence of surfactants raises or lowers the critical amplitude and wavenumber depending on the spatial phase shift between the surfactant-concentration variations and surface deflections.