The effects of slightly soluble surfactants on the flow around a spherical bubble

Abstract

This paper reports the results of a numerical investigation of the transient evolution of the flow around a spherical bubble rising in a liquid contaminated by a weakly soluble surfactant. For that purpose the full Navier-Stokes equations are solved together with the bulk and interfacial surfactant concentration equations, using values of the physical-chemical constants of a typical surfactant characterized by a simple surface kinetics. The whole system is strongly coupled by nonlinear boundary conditions linking the diffusion flux and the interfacial shear stress to the interfacial surfactant concentration and its gradient. The influence of surfactant characteristics is studied by varying arbitrarily some physical-chemical parameters. In all cases, starting from the flow around a clean bubble, the results describe the temporal evolution of the relevant scalar and dynamic interfacial quantities as well as the changes in the flow structure and the increase of the drag coefficient. Since surface diffusion is extremely weak compared to advection, part of the bubble (and in certain cases all the interface) tends to become stagnant. This results in a dramatic increase of the drag which in several cases reaches the value corresponding to a rigid sphere. The present results confirm the validity of the well-known stagnant-cap model for describing the flow around a bubble contaminated by slightly soluble surfactants. They also show that a simple relation between the cap angle and the bulk concentration cannot generally be obtained because diffusion from the bulk plays a significant role.

Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble

“…The approach developed in this paper may be coupled with a realistic model of surfactant transport 12,44 to assess the changes induced in the wake dynamics by the adsorption/desorption and surface transport mechanisms that drive the contamination of bubbles in many real fluids, especially water. Nevertheless the next step of this stream of research performed in our group will still consider clean bubbles.…”

Surface viscosity and Marangoni stresses at surfactant laden interfaces

“…We assume here that k s is constant; however, the flux onto the interface will generally depend on the concentration of surfactant in the bulk, c, directly below the interface (Cuenot, Magnaudet & Spennato 1997). Neglecting the dependence on the bulk concentration avoids the introduction of a second advection-diffusion equation for c.…”

Evaporation-induced saline Rayleigh convection inside a colloidal droplet

“…3, however, the Marangoni-induced flow on the droplet surface can be retarded by diverse conditions of the droplet surface. Regardless of the strength of Marangoni flow, it has been reported that the Marangoni-induced flow can be retarded even though the solute is pure water; some explanations given are the presence of a surfactant in the droplet [37][38][39] and contaminants in the air. 40,41 In the case of the surfactant, the surface concentration of a droplet exposed to the surrounding air becomes gradually dense and ultimately saturated, so it is not likely to have a local movement on the droplet surface.…”

“…Such a layer retards the rising speed of a bubble since a part of the bubble surface becomes stagnant and the bubble behaves as a rigid sphere, imposing a no-slip condition on the external flow. 38,43 Hu and Larson 5 showed theoretically that a tiny concentration gradient of the contaminants can greatly suppress the surface velocity produced by the thermal Marangoni stress. In the present case, similar to the case of the rising bubble, the contaminant adsorbed at the airwater interface will be swept towards the contact line of a droplet and change the boundary condition to the no-slip condition.…”