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

Author

Listed:
  • Qiming Huang

    (CCS Guangzhou Plan Approval Center, Guangzhou 510235, China)

  • Junxiang Yang

    (School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou 510006, China)

Abstract

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

Suggested Citation

  • Qiming Huang & Junxiang Yang, 2022. "Linear and Energy-Stable Method with Enhanced Consistency for the Incompressible Cahn–Hilliard–Navier–Stokes Two-Phase Flow Model," Mathematics, MDPI, vol. 10(24), pages 1-16, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:24:p:4711-:d:1000509
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    References listed on IDEAS

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    1. Choi, Jeong-Whan & Lee, Hyun Geun & Jeong, Darae & Kim, Junseok, 2009. "An unconditionally gradient stable numerical method for solving the Allen–Cahn equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(9), pages 1791-1803.
    2. Lee, Chaeyoung & Jeong, Darae & Shin, Jaemin & Li, Yibao & Kim, Junseok, 2014. "A fourth-order spatial accurate and practically stable compact scheme for the Cahn–Hilliard equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 409(C), pages 17-28.
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    Cited by:

    1. Junseok Kim, 2024. "Modified Wave-Front Propagation and Dynamics Coming from Higher-Order Double-Well Potentials in the Allen–Cahn Equations," Mathematics, MDPI, vol. 12(23), pages 1-16, November.

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