A conservative Allen-Cahn model for a hydrodynamics coupled phase-field surfactant system

In this study, we develop a model for a binary fluid-surfactant system utilizing a coupling of two kinds conservative Allen-Cahn type equations and the Navier-Stokes equations. To ensure mass conservation, we incorporate hybrid Lagrange multipliers into the two Allen- Cahn type equations. Specifically, for the concentration variable, a global correction using a time-dependent Lagrange multiplier is utilized, while for the binary fluid variable, a space- time dependent Lagrange multiplier is applied to minimize the impact of dynamics of motion by mean curvature. We propose a linear second order scheme for practical solution of the model. Computational tests demonstrate that the proposed model is effective for the binary fluid-surfactant system and is capable of preserving the small features of interfaces.