Effect of Insoluble Surfactants on the Pressure-Driven Motion of a Drop in a Tube in the Limit of High Surface Coverage

Abstract

The effect of surfactants on the axisymmetric pressure-driven motion of a droplet in a tube is investigated under low Reynolds number conditions. In the presence of surface-active impurities, the motion of the drop results in constant redistribution of the adsorbed surfactant along the interface by convection and diffusion, leading to nonuniformities in interfacial tension (i.e., Marangoni stresses) which modify the viscous stress balance at the interface. These, in turn, affect the mobility of the drop and its steady or transient shape deformations. In this study, the steady-state behavior of such a drop is examined in the presence of nondilute concentrations of bulk-insoluble surfactants on the surface of the drop. The boundary integral method is used in conjunction with a finite-difference scheme to solve the unsteady surface convective-diffusion equation for surfactant transport. The Frumkin adsorption framework is used to explore the effects of monolayer saturation and nonideal surfactant interactions on the steady drop shape and speed in the limit of high surface coverage. It is found that for surfactants with strong cohesive interactions, the drop mobility increases with increasing surface coverage, attaining maximum mobility at about 50% initial coverage. Furthermore, surface-convection-dominated systems with strong cohesive interactions exhibit surface flow patterns corresponding to the formation of a stagnant cap at the trailing end of the drop. All other systems exhibit surface flow patterns corresponding to uniform retardation of interface mobility as a fully immobilized interface is approached with increasing surface convection.