On the energy analysis of two-phase flows simulated with the diffuse interface method

The phase-field method (PFM) is employed to simulate two-phase flows with the fully coupled Cahn-Hilliard-Navier-Stokes equations governing the temporal evolution. The methodology minimizes the total energy functional, accounting for diffusive and viscous dissipations. A new perspective is presented by analyzing the interplay between kinetic energy, mixing energy, and viscous dissipation using the temporal evolution of the total energy functional. The classical surface energy is approximated with mixing energy under specific conditions, and the accuracy of this substitution is rigorously evaluated. The energy-based surface tension formulation derived from the Korteweg stress tensor demonstrates exceptional accuracy in capturing variations in the mixing energy. These concepts are demonstrated by considering two benchmark problems: droplet oscillation and capillary thread breakup. Key findings include validating mixing-energy theory for highly deformed interfaces, as well as the discovery of distinct energy dissipation patterns during thread breakup and droplet oscillations. The results highlight the robustness of the free energy-based PFM in accurately capturing complex interfacial dynamics, while maintaining energy conservation.