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