Dynamically Regularized Lagrange Multiplier Method for the Hydrodynamics-Coupled Binary Fluid-Surfactant Phase-Field Model

In this paper, we develop dynamically regularized Lagrange multiplier (DRLM) schemes for a hydrodynamics-coupled binary fluid-surfactant phase-field model, which comprises the incompressible Navier-Stokes equations and two Cahn-Hilliard type equations. To ensure the energy dissipation property, we introduce two time-dependent Lagrange multipliers corresponding to the kinetic and phase-field components, thereby reformulating the original system into an equivalent form that intrinsically incorporates the energy-evolution process. The reformulated system is temporally discretized using the backward differentiation formula. A key advantage of the proposed schemes lies in the introduction of two dynamic terms with regularization parameters, which ensure the unique determination of the multipliers and alleviate the time-step restrictions associated with the explicit treatment of nonlinear terms. The proposed schemes are fully decoupled and are theoretically proven to be mass conservative and energy dissipative with respect to the original energy functional. Each time step requires solving two sixth- and fourth-order linear equations involving the tri-Laplacian and bi-Laplacian operators, two generalized Stokes systems, and two nonlinear algebraic equations for the introduced multipliers. Numerical experiments are conducted to verify the temporal accuracy, mass conservation, and energy-dissipation properties of the proposed schemes. Furthermore, simulations of phase separation and droplet coalescence are performed to demonstrate their effectiveness.