Linear and Energy-Stable Method with Enhanced Consistency for the Incompressible Cahn–Hilliard–Navier–Stokes Two-Phase Flow Model

The Cahn–Hilliard–Navier–Stokes model is extensively used for simulating two-phase incompressible fluid flows. With the absence of exterior force, this satisfies energy dissipation law. present work focuses on developing a linear, decoupled, and dissipation-preserving time-marching scheme hydrodynamics coupled Cahn–Hilliard model. An efficient time-dependent auxiliary variable approach first introduced to design equivalent equations. Based forms, BDF2-type linear constructed. In each time step, unique solvability law can be analytically estimated. To enhance stability consistency, we correct modified by practical relaxation technique. Using finite difference method in space, fully discrete described, numerical solutions separately implemented. Numerical results indicate that proposed has desired accuracy, stability. Moreover, flow-coupled phase separation, falling droplet, dripping droplet are well simulated.