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A linear mass and energy conserving numerical scheme for two-phase flows with thermocapillary effects

Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China

E-mail Address: shaolihua@ustb.edu.cn

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Zhenlin Guo

Mechanics Division, Beijing Computational Science Research Center, Building 9, East Zone ZPark II, No. 10, East Xibeiwang Road, Haidian District, Beijing 100193, P. R. China

E-mail Address: zguo@csrc.ac.cn

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Yanxiao Sun

Mechanics Division, Beijing Computational Science Research Center, Building 9, East Zone ZPark II, No. 10, East Xibeiwang Road, Haidian District, Beijing 100193, P. R. China

E-mail Address: sunyanxiao@csrc.ac.cn

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https://doi.org/10.1142/S0129183124500359Cited by:0 (Source: Crossref)

Abstract

In this paper, a thermodynamically consistent phase-field model is employed to simulate the thermocapillary migration of a droplet. The model equations consist of a general Navier–Stokes equation for the two-phase flows, a Cahn–Hilliard equation for the diffuse interface, and a heat equation, and meanwhile satisfy the balance laws of mass, energy and entropy. In particular, the total energy of the system includes kinetic energy, potential energy and internal energy, which leads to a highly coupled and nonlinear equation system. We therefore develop a linear mass and energy conserving, semi-decoupled numerical method for the numerical simulations. As the model contains a heat (energy) equation, a simple error term introduced by the temporal discretization of the momentum equation can be absorbed into the heat equation, such that the numerical solutions satisfy the conservation laws of mass and energy exactly at the temporal discrete level. Several numerical tests are carried out to validate our numerical method.

Keywords:

PACS: 47.11.-j, 02.60.Cb

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References

1. R. S. Subramanian and R. Balasubramaniam , The Motion of Bubbles and Drops in Reduced Gravity ( Cambridge University Press, 2001). Google Scholar
2. A. A. Darhuber and S. M. Troian , Annu. Rev. Fluid Mech. 37, 425 (2005). Crossref, Web of Science, ADS, Google Scholar
3. S. Sun, J. Li, J. Zhao and Q. Wang , J. Sci. Comput. 83, 1 (2020). Crossref, Web of Science, Google Scholar
4. L. Yue, Z. Chai, H. Wang and B. Shi , Phys. Rev. E 105, 015314 (2022). Crossref, Web of Science, ADS, Google Scholar
5. H. Liu, A. J. Valocchi, Y. Zhang and Q. Kang , J. Comput. Phys. 256, 334 (2014). Crossref, Web of Science, ADS, Google Scholar
6. Q. Yang, Y. Liu, X. Jia, T. Zhang, H. Tian, J. Fan, Q. Xu and F. Song , Phys. Rev. E 106, 015111 (2022). Crossref, Web of Science, ADS, Google Scholar
7. J.-Q. Li, T.-H. Fan, T. Taniguchi and B. Zhang , Int. J. Heat Mass Transfer 117, 412 (2018). Crossref, Web of Science, Google Scholar
8. Y. Xiao, Z. Zeng, L. Zhang, J. Wang, Y. Wang, H. Liu and C. Huang , Phys. Fluids 34, 022114 (2022). Crossref, Web of Science, Google Scholar
9. Y. Hu, D. Li, X. Niu and S. Shu , Int. J. Heat Mass Transfer 138, 809 (2019). Crossref, Web of Science, Google Scholar
10. T. Mitchell, M. Majidi, M. Rahimian and C. Leonardi , Phys. Fluids 33, 032108 (2021). Crossref, Web of Science, Google Scholar
11. D. DasGupta, P. K. Mondal and S. Chakraborty , Phys. Rev. E 90, 023011 (2014). Crossref, Web of Science, ADS, Google Scholar
12. H. Liu, L. Wu, Y. Ba and G. Xi , Int. J. Heat Mass Transfer 104, 337 (2017). Crossref, Web of Science, Google Scholar
13. H. Liu, Y. Zhang and A. J. Valocchi , J. Comput. Phys. 231, 4433 (2012). Crossref, Web of Science, ADS, Google Scholar
14. H. Liu, A. J. Valocchi, Y. Zhang and Q. Kang , Phys. Rev. E 87, 013010 (2013). Crossref, Web of Science, ADS, Google Scholar
15. J. Lowengrub and L. Truskinovsky , Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 454, 2617 (1998). Crossref, Web of Science, Google Scholar
16. Z. Guo and P. Lin , J. Fluid Mech. 766, 226 (2015). Crossref, Web of Science, ADS, Google Scholar
17. J. Shen, J. Xu and J. Yang , SIAM Rev. 61, 474 (2019). Crossref, Web of Science, Google Scholar
18. G. Zhu, H. Chen, J. Yao and S. Sun , Appl. Math. Model. 70, 82 (2019). Crossref, Web of Science, Google Scholar
19. M. Wang, Q. Huang and C. Wang , J. Sci. Comput. 88, 33 (2021). Crossref, Web of Science, Google Scholar
20. X. Yang , J. Comput. Phys. 327, 294 (2016). Crossref, Web of Science, ADS, Google Scholar
21. X. Yang and L. Ju , Comput. Methods Appl. Mech. Eng. 315, 691 (2017). Crossref, Web of Science, ADS, Google Scholar
22. X. Yang , Comput. Methods Appl. Mech. Eng. 400, 115479 (2022). Crossref, Web of Science, ADS, Google Scholar
23. N. Young, J. Goldstein and M. Block , J. Fluid Mech. 6, 350 (1959). Crossref, Web of Science, ADS, Google Scholar
24. J. W. Cahn and J. E. Hilliard , J. Chem. Phys. 28, 258 (1958). Crossref, Web of Science, ADS, Google Scholar
25. W. Feng, Z. Guo, J. S. Lowengrub and S. M. Wise , J. Comput. Phys. 352, 463 (2018). Crossref, Web of Science, ADS, Google Scholar