The Navier–Stokes–Cahn–Hilliard model with a high-order polynomial free energy

Jaemin Shin, Junxiang Yang, Chaeyoung Lee, Junseok Kim

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


In this paper, we present a high-order polynomial free energy for the phase-field model of two-phase incompressible fluids. The model consists of the Navier–Stokes (NS) equation and the Cahn–Hilliard (CH) equation with a high-order polynomial free energy potential. In practice, a quartic polynomial has been used for the bulk free energy in the CH equation. It is well known that the CH equation does not satisfy the maximum principle and the phase-field variable takes shifted values in the bulk phases instead of taking the minimum values of the double-well potential. This phenomenon substantially changes the original volume enclosed by the isosurface of the phase-field function. Furthermore, it requires fine resolution to keep small shapes. To overcome these drawbacks, we propose high-order (higher than fourth order) polynomial free energy potentials. The proposed model is tested for an equilibrium droplet shape in a spherical symmetric configuration and a droplet deformation under a simple shear flow in a fully three-dimensional fluid flow. The computational results demonstrate the superiority of the proposed model with a high-order polynomial potential to the quartic polynomial function in volume conservation property.

Original languageEnglish
Pages (from-to)2425-2437
Number of pages13
JournalActa Mechanica
Issue number6
Publication statusPublished - 2020 Jun 1

Bibliographical note

Funding Information:
The corresponding author (J. S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003053). J. Yang is supported by China Scholarship Council (201908260060). The authors thank the reviewers for their constructive and helpful comments on the revision of this article.

Publisher Copyright:
© 2020, Springer-Verlag GmbH Austria, part of Springer Nature.

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanical Engineering


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