The role of higher-order viscous and interfacial effects on the onset of surfactant-covered faraday waves

Abstract Faraday waves are gravity-capillary waves that emerge on the surface of a vertically vibrated fluid when the energy injected via vibration exceeds the energy lost due to viscous dissipation. Because dissipation primarily occurs in the free surface boundary layer, it is particularly sensitive to free surface properties, including surface tension, and the elasticity and viscosity of surfactants present at the surface. We study the Faraday wave onset acceleration $a_{c}$, the minimum vibration amplitude needed to excite these waves. Our system comprises a finite-depth, infinite-breadth Newtonian fluid covered by an insoluble surfactant and subjected to vertical vibration, which produces sub-harmonic Faraday waves. By accounting for surface tension, and Marangoni and Boussinesq effects, we derive a novel expression for the onset acceleration, valid up to second order in the expansion parameter $\varUpsilon = \sqrt{\tfrac{1}{\mathcal{R}e}}$. We consider new regions of parameter space not previously studied, including a wide variety of fluid depths and driving frequencies. We find that for sufficiently shallow systems the addition of surfactant can lower the onset acceleration due to a decrease in energy dissipation instead of the expected result in which energy dissipation is enhanced. We discuss the possible use of the model to develop a low cost surface viscometer for surfactant monolayers, and the possible use of surfactants to facilitate Faraday wave induced mixing in thin fluid films.