A Dual-Reciprocity Boundary Element Method for Evaluating Bulk Convective Transport of Surfactant in Free-Surface Flows

We present a dual-reciprocity boundary element method (DRBEM) to investigate bulk surfactant transport dynamics in a free-surface flow system under steady-state conditions. This free-surface flow system consists of semi-infinite bubble progression in a rigid axisymmetric capillary tube. Once adsorbed to the air–liquid interface with a surface concentration Γ, surfactant alters the interfacial surface tension γ. As the interfacial stress balance, which governs the fluid mechanics, is a function of γ, a strong coupling exists between surfactant transport dynamics and the fluid mechanics (physicochemical hydrodynamics). To model this problem over a range of bulk concentrations, C, the bulk convective/diffusive transport of surfactant to the interface must be calculated. In this paper, DRBEM is used to simulate the bulk convection–diffusion relationship while the boundary element method (BEM) is used to solve Stokes flow, and a finite-difference method is used to solve the surface transport equation under steady-state conditions. A nonlinear Langmuir adsorption model is used to determine the surfactant equation of state γ=f(Γ). The validity of the DRBEM is first demonstrated by comparing computational and analytical solutions for a test problem. Next, the computational algorithm is used to calculate the bulk concentration field surrounding the bubble as a function of the far-downstream quantity of surfactant, Co, and its influence on interfacial dynamics. These profiles clearly demonstrate the importance of accurately calculating the bulk concentration field under moderate Co conditions. In addition, the variation of mechanical properties of this system as a function of Co indicates that the interfacial pressure jump can be significantly larger when the bulk transport of surfactant to the interface is limited.