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

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.