The Marangoni effect on small-amplitude capillary waves in viscous fluids

TL;DR

This paper develops a mathematical model describing how the Marangoni effect influences the damping and behavior of small capillary waves on viscous fluid surfaces, considering surfactant transport and deriving corrections to critical wavelengths.

Contribution

It introduces a new integrodifferential equation capturing the transient dynamics of capillary waves with Marangoni effects and provides analytical corrections for surfactant concentration impacts.

Findings

Marangoni effect increases wave damping.

Derived correction to critical wavelength based on surfactant concentration.

Model applies to insoluble surfactants below cmc near critical damping.

Abstract

We derive a general integrodifferential equation for the transient behaviour of small-amplitude capillary waves on the planar surface of a viscous fluid in the presence of the Marangoni effect. The equation is solved for an insoluble surfactant solution in concentration below the critical micelle concentration (cmc) undergoing convective-diffusive surface transport. The special case of a diffusion-driven surfactant is considered near the the critical damping wavelength. The Marangoni effect is shown to contribute to the overall damping mechanism and a first-order term correction to the critical wavelength with respect to the surfactant concentration difference and the Schmidt number is proposed.