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Surfactant adsorption and Marangoni flow in liquid jets. 2. Modeling
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This paper is concerned with the interfacial behavior of surfactant solutions on short time scales. A gravity-driven laminar liquid jet is used to create a rapidly expanding liquid surface, which exposes the surfactant solution to highly nonequilibrium conditions. This expansion causes the surface tension to differ locally from its equilibrium value, generating a (Marangoni) shear stress that acts on the jet surface and retards the surface acceleration. A theory for the flow very near the nozzle shows that the cube-root dependence of the surface velocity on the distance traveled is altered through the adsorption of surfactant. In a boundary-layer treatment, both the surface velocity and the surface concentration increase linearly from the nozzle exit over a short distance, which we term the detachment region. The length of the detachment region is found to vary with the bulk concentration raised to the power 3/2. A numerical model of the surfactant adsorption process in the jet has been developed within the framework of the CFD code FIDAP. The numerical solution confirms the general features of the theory and shows that the maximum reduction in surface velocity occurs very close to the nozzle exit, except at high concentrations. A comparison with experiments on C16TAB at concentrations below the critical micelle concentration, which are described in part 1 of this series of papers, shows good agreement.