Adsorption of surfactants onto the surface of a spherical rising bubble and its effect on the terminal velocity of the bubble

Adsorption of surfactants onto the surface of a single rising bubble was estimated by numerically solving the Navier–Stokes equation and the convection-diffusion equations of the surfactants in the bulk and on the surface at steady state. Then, the effects of Reynolds number (Re) and bulk concentration of the surfactants on adsorption and on the terminal velocity of the bubble were also numerically studied. The results revealed that (a) adsorption of surfactants at the front of the bubble with respect to the rise direction is lower than that at the back, and the difference increases as Re increases and∕or the bulk concentration of the surfactants decreases, (b) the critical Re $(Rec)$ at which a bubble behaves as a solid particle can be estimated, and (c) the averaged adsorption of surfactants coincides with the equilibrium adsorption corresponding to a zero net adsorption-desorption flux for $Re<Rec$⁠, and inversely, the averaged adsorption exceeds the equilibrium adsorption for $Re>Rec$⁠.