Single Contaminated Drops Falling through Stagnant Liquid at Low Reynolds Numbers

Numerical simulations of contaminated spherical drops falling through a stagnant liquid at low Reynolds numbers are carried out using the finite difference method. The numerical results are used to describe the behavior of the surfactant concentrations and to understand the surfactant effects on the fluid motions in detail. The predicted interfacial surfactant concentration, Gamma, is almost zero for angles, theta, below a certain value (the stagnant-cap angle, theta(cap)), whereas it steeply increases and reaches a large value for theta > theta(cap) (the stagnant-cap region). The increase in the initial surfactant concentration, C-0, in the drop enhances the adsorption from the drop to the interface, which results in the increase in Gamma and the decrease in theta(cap). Peaks appear in the predicted Marangoni stresses around theta(cap), which causes similar peaks in the pressure distribution. The high-pressure spots prevent the fluid motion along the interface, which results in the formation of the stagnant-cap region and the attenuation of the tangential velocity in the continuous phase. The surfactant flux from the bulk to the interface decreases C in the vicinity of the interface for theta < theta(cap) and the weak diffusion cannot compensate for the reduction in C by adsorption, which results in C at the interface smaller than C-0. The pattern of the low C region is determined by the advection and does not smear out because of a small diffusive flux.