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In this work, we develop an efficient energy stable scheme for the hydrodynamics coupled phase-field surfactant model with variable densities. The thermodynamically consistent model consists of two Cahn–Hilliard–type equations and incompressible Navier–Stokes equation. We use two scalar auxiliary variables to transform nonlinear parts in the free energy functional into quadratic forms, and then they can be treated efficiently and semi-implicitly. A splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. By some subtle explicit-implicit treatments to nonlinear convection and stress terms, we construct a first-order energy stable scheme for the two-phase system with soluble surfactants. The developed scheme is efficient and easy-to-implement. At each time step, computations of phase-field variables, the velocity and pressure are decoupled. We rigorously prove that the proposed scheme is unconditionally energy stable. Numerical results confirm that our scheme is accurate and energy stable.

Decoupled and Energy Stable Time-Marching Scheme for the Interfacial Flow with Soluble Surfactants