Effects of soluble and insoluble surfactant on laminar interactions of vortical flows with a free surface

Abstract

We study the two-dimensional, laminar interactions between a contaminated free surface and a vortical flow below. Two canonical vortical flows are considered: a pair of vortex tubes impinging onto the free surface; and an unstable shear wake behind a body operating on the surface. A quantitative model for free-surface viscous flows in the presence of soluble or insoluble surfactants is developed. For the low to moderate Froude numbers considered here, for which weakly nonlinear free-surface boundary conditions are valid, the surface boundary layer and vorticity production are weak for clean water and the vortical flow evolution does not differ qualitatively from that under a free-slip boundary. When even a small amount of contamination is present, the flow can be dramatically affected. The vortical flow creates gradients in the surfactant surface concentration which leads to Marangoni stresses, strong surface vorticity generation, boundary layers, and even separation. These significantly influence the underlying flow which itself affects surfactant transport in a closed-loop interaction. The resulting flow features are intermediate between but qualitatively distinct from those under either a free- or no-slip boundary. Surfactant effects are most prominent for insoluble surface contamination with likely development of surfactant shocks and associated surface features such as Reynolds ridges. For soluble surfactant with initially uniform bulk concentration, surface concentration variations are moderated by sorption kinetics between the surface and bulk phases, and the overall effects are generally diminished. For initially stratified bulk concentrations, however, the evolution dynamics becomes more varied and surfactant effects may be amplified relative to the insoluble case. The dependence of these results on the properties of the contamination is studied.