The Onset Acceleration for Surfactant Covered Faraday Waves

TL;DR

This paper derives an analytical expression for the onset acceleration of Faraday waves on surfactant-covered fluids, revealing how surfactants can either increase or decrease the threshold for wave formation depending on parameters.

Contribution

It provides a new analytical model for Faraday wave onset that incorporates surfactant effects and extends previous numerical results to new parameter regimes.

Findings

Surfactants can lower the onset acceleration under certain conditions.

The model matches previous numerical results with modified parameters.

New parameter regimes for wave onset are identified.

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 this dissipation primarily occurs in the free surface boundary layer, their emergence is particularly sensitive to free surface properties including the surface tension, elasticity, and viscosity of surfactants present at the free surface. We study this sensitivity by considering a Newtonian fluid bath covered by an insoluble surfactant subject to vertical vibrations which produce sub-harmonic Faraday waves. By assuming a finite-depth, infinite-breadth, low-viscosity bulk fluid and accounting for surface tension, Marangoni, and Boussinesq effects, we derive an expression for the onset acceleration up to second order in the expansion parameter $\Upsilon = \sqrt{\tfrac{1}{\mathcal{R}e}}$. We recover the results of previous numerical investigations, but only by modifying the Marangoni and Boussinesq numbers to account for the low-viscosity limit. The analytic expression allows us to consider a range of parameters not previously studied, including a wide variety of fluid depths and driving frequencies. In addition, we uncover regions of parameter space for which our model predicts that the addition of surfactant would lower, rather than elevate, the onset acceleration. We discuss the possible use of this model in developing a surface viscometer for surfactant monolayers.