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Surfactants are widely used in the manipulation of drop motion in microchannels, which is commonly involved in many applications, e.g., surfactant assisted oil recovery and droplet microfluidics. This study is dedicated to a crucial fundamental problem, i.e., the effects of a soluble surfactant on drop motion and their underlying mechanisms, which is an extension of our previous work of an insoluble-surfactant-covered droplet in a square microchannel [Z. Y. Luo, X. L. Shang, and B. F. Bai, “Marangoni effect on the motion of a droplet covered with insoluble surfactant in a square microchannel,” Phys. Fluids 30, 077101 (2018)]. We make essential improvements to our own three-dimensional front-tracking finite-difference model, i.e., by further integrating the equation governing surfactant transport in the bulk fluid and surfactant mass exchange between the drop surface and bulk fluid. We find that the soluble surfactant generally enlarges the droplet-induced extra pressure loss compared to the clean droplet, and enhancing surfactant adsorption tends to intensify such an effect. We focus specifically on the influences of four soluble-surfactant-relevant dimensionless parameters, including the Biot number, the dimensionless adsorption depth, the Damkohler number, and the bulk Peclet number. Most importantly, we discuss the mechanisms underlying the soluble surfactant effect, which consists of two aspects similar to the insoluble case, i.e., the reduced surface tension to decrease droplet-induced extra pressure loss and the enlarged Marangoni stress playing the opposite role. Surprisingly, we find that the enlarged Marangoni stress always makes the predominant contribution over the reduced surface tension in the effects of above-mentioned four soluble-surfactant-relevant dimensionless parameters on drop motion. This finding explains why the droplet-induced extra pressure loss increases with the film thickness, which is opposite to that observed for clean droplets.
Surfactants are widely used in the manipulation of drop motion in microchannels, which is commonly involved in many applications, e.g., surfactant assisted oil recovery and droplet microfluidics. This study is dedicated to a crucial fundamental problem, i.e., the effects of a soluble surfactant on drop motion and their underlying mechanisms, which is an extension of our previous work of an insoluble-surfactant-covered droplet in a square microchannel [Z. Y. Luo, X. L. Shang, and B. F. Bai, “Marangoni effect on the motion of a droplet covered with insoluble surfactant in a square microchannel,” Phys. Fluids 30, 077101 (2018)]. We make essential improvements to our own three-dimensional front-tracking finite-difference model, i.e., by further integrating the equation governing surfactant transport in the bulk fluid and surfactant mass exchange between the drop surface and bulk fluid. We find that the soluble surfactant generally enlarges the droplet-induced extra pressure loss compared to the clean droplet, and enhancing surfactant adsorption tends to intensify such an effect. We focus specifically on the influences of four soluble-surfactant-relevant dimensionless parameters, including the Biot number, the dimensionless adsorption depth, the Damkohler number, and the bulk Peclet number. Most importantly, we discuss the mechanisms underlying the soluble surfactant effect, which consists of two aspects similar to the insoluble case, i.e., the reduced surface tension to decrease droplet-induced extra pressure loss and the enlarged Marangoni stress playing the opposite role. Surprisingly, we find that the enlarged Marangoni stress always makes the predominant contribution over the reduced surface tension in the effects of above-mentioned four soluble-surfactant-relevant dimensionless parameters on drop motion. This finding explains why the droplet-induced extra pressure loss increases with the film thickness, which is opposite to that observed for clean droplets.
In this study, we present numerical simulations examining the impact of soluble surfactant on the interface dynamics of a rising droplet. To achieve this, the droplet interface is tracked using an arbitrary Lagrangian–Eulerian approach, and the bulk and interfacial surfactant concentration evolution equations fully coupled with the incompressible Navier–Stokes equations are solved. We systematically evaluate the boundary of interfacial dynamics evolution by varying certain dimensionless parameters. Specifically, we study the effects of changes in parameters such as the Langmuir number, the Biot number, the Damkohler number, the bulk Peclet number, and the elastic number on interfacial tangential velocity, interfacial concentration and its gradient, interfacial viscous shear stress, and droplet rising velocity. Our findings confirm the validity of the stagnant-cap model for describing the interfacial fluidity of a surfactant-laden rising droplet. Increasing the Langmuir number and decreasing the Damkohler number can inhibit interface fluidity, but there is a threshold for the Damkohler number. Additionally, the overall increase in interface tension may mask the hindering effect of the locally increased concentration gradient on the interfacial fluidity. The Biot number has no impact on the steady state of the interface, but a slow adsorption rate may result in a bimodal retardation before the interface reaches a steady state. A clear threshold exists for the Peclet number to hinder the interface velocity, and a too high Peclet number leads to strong nonlinearity in the interface physical quantities. Variations in the elastic number significantly affect the evolution of the interface, causing the interface velocity to pass through several states, ranging from almost no retardation, uniform retardation, stagnant-cap retardation to complete retardation.
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