Surfactants are chemicals that,when dissolved in a system of multiple immiscible fluids, tend to form layers at the fluid-fluid interfaces and thus reduce the surface tension.
In this paper, a diffuse interface model for surfactants in multi-phase flow with three or more fluids is derived. A system of Cahn-Hilliard equations is coupled with a Navier-Stokes system and an advection-diffusion equation for the surfactant ensuring the thermodynamic consistency. The governing equations are presented in Section 2.7.
Specifically, the case of local chemical equilibrium with respect to the surfactant is considered.
The authors derive a general moving boundary problem for multi-phase flow of immiscible, incompressible fluids with surfactant attached to the equations presented in the Section 2.7. The surfactant is subject to advection-diffusion equations in the bulk and on the interfaces and impacts on the flow via the capillary term and the Marangoni force. A general phase field model is then derived and summarized in Section 3.4 following the same procedures. A detailed asymptotic analysis has been performed, which links the two models in the sense that the sharp interface limit of the phase field model is the moving boundary problem. Some numerical simulations of surfactant diffusion through a stationary triple junction and of a Marangoni effects showcase the capability of the model and support the results of the asymptotic analysis.
The authors plan to address the non-instantaneous case in forthcoming work using some techniques already considered in the Section 5.