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In this paper, we consider numerical approximations of a binaryfluid-surfactant phase-field model coupled with the fluid flow, in which thesystem is highly nonlinear that couples the incompressible Navier-Stokesequations and two Cahn-Hilliard type equations. We develop two, linear andsecond order time marching schemes for solving this system, by combining the"Invariant Energy Quadratization" approach for the nonlinear potentials, theprojection method for the Navier-Stokes equation, and a subtleimplicit-explicit treatment for the stress and convective terms. We prove thewell-posedness of the linear system and its unconditional energy stabilityrigorously. Various 2D and 3D numerical experiments are performed to validatethe accuracy and energy stability of the proposed schemes.

Numerical approximations for the binary Fluid-Surfactant Phase Field Model with fluid flow: Second-order, Linear, Energy stable schemes