We examine the effect of surfactants on a spherical gas bubble rising steadily in an infinite fluid at low and order one Reynolds number with order one and larger Peclet numbers. Our mathematical model is based on the Navier-Stokes equations coupled with a convection-diffusion equation together with appropriate interfacial conditions. The nonlinearity of the equations and boundary conditions, and the coupling between hydrodynamics and surfactant transport make the problem very challenging.
When a bubble rises in a fluid containing surface-active agents, surfactant adsorbs onto the bubble surface at the leading edge, convects to the trailing edge by the surface flow and desorbs into the bulk along the interface. This adsorption develops a surface concentration gradient on the interface that makes the surface tension at the back end relatively lower than that at the front end, and thus retards the bubble velocity. Because of surfactant impurities unavoidably present in materials, this retardation can cause a problem in materials processing in space and glass processing when bubbles are created during chemical reactions. Thus the study of how to remobilize (remove the surfactant gradient on the surface) the bubble surface becomes necessary. Many studies have been done on this retarding effects of the surfactant on a moving bubble. However, most were focused on the retarding effect due to a trace amount of surfactant, in which case the bubble velocity monotonically decreases as the bulk concentration increases. The question of how to remobilize the bubble surface remains unanswered. In this work, we will show that the bubble velocity can be controlled by remobilizing the bubble interface using the surfactant concentration. This technique not only can be used to maximize the bubble velocity, but also can be used to maximize mass transfer on purifying materials and extracting materials from mixtures.
In the first part of the work, we illustrate numerically that the bubble interface can be remobilized by increasing the bulk concentration of surfactant, for any fixed Peclet number, at low Reynolds number. For any fixed bulk concentration, the bubble velocity decreases with increasing Peclet number. The larger the Peclet number is the larger the required bulk concentration needed to bring the velocity back to the clean surface value. In the second part of the work, we will show that the remobilization still remains effective for order one Reynolds numbers. Moreover, when the rate of convection on the surface is much larger than the rate of diffusion at the back end, a stagnant cap develops near the back stagnation point that makes the bubble surface there act like a solid boundary. Wakes form at higher Reynolds numbers that drastically reduce the terminal velocity, and disappear as the bubble interface remobilizes. Finally, we consider the problem analytically for asymptotically large Peclet numbers. When the Peclet number is very large, a stagnant cap forms at the back end which makes one part of the bubble surface clean of surfactant, and the other part completely immobile. Also boundary layers develop along the bubble surface with different thicknesses on the clean part of the surface and on the stagnant cap. The asymptotic structures are obtained and the governing equations posed and partly addressed numerically and analytically.