Surfactant-induced Marangoni stresses and surface viscosity in Faraday instability

Abstract

We investigate the influence of insoluble-surfactant elasticity and Boussinesq-Scriven surface viscosity on the parametric instability of an interface separating two viscous fluids. The two-fluid Floquet formulation of Kumar& Tuckerman (J. Fluid Mech., vol. 279, 1994, pp. 49-68) is extended to include surfactant transport, Marangoni stresses and surface viscosity. For the parameter range considered, both mechanisms increase the onset acceleration relative to the clean interface state, but through distinct responses: Marangoni elasticity produces a non-monotonic threshold, whereas surface viscosity yields a monotonic increase towards saturation. Analysis of the corresponding unforced gravity-capillary modes shows that both mechanisms modify modal damping and natural frequency. Wavenumber selection reduces the detuning from subharmonic resonance by approximately an order of magnitude relative to that evaluated at the clean interface critical wavenumber. Along the selected branches, variations in the onset acceleration closely follow those in modal damping, with residual detuning playing a secondary role. Despite their distinct interfacial stress mechanisms, both Marangoni elasticity and surface viscosity suppress tangential interfacial motion in the strong-rheology limit, approaching a common tangentially immobilized condition.