Abstract
Two classes of thermodynamically consistent hydrodynamic phase field models have been developed for binary fluid mixtures of incompressible viscous fluids of possibly different densities and viscosities. One is quasi-incompressible, while the other is incompressible. For the same binary fluid mixture of two incompressible viscous fluid components, which one is more appropriate? To answer this question, we conduct a comparative study in this paper. First, we visit their derivation, conservation and energy dissipation properties and show that the quasi-incompressible model conserves both mass and linear momentum, while the incompressible one does not. We then show that the quasi-incompressible model is sensitive to the density deviation of the fluid components, while the incompressible model is not in a linear stability analysis. Second, we conduct a numerical investigation on coarsening or coalescent dynamics of protuberances using the two models. We find that they can predict quite different transient dynamics depending on the initial conditions and the density difference although they predict essentially the same quasi-steady results in some cases. This study thus cast a doubt on the applicability of the incompressible model to describe dynamics of binary mixtures of two incompressible viscous fluids especially when the two fluid components have a large density deviation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abels, H.: Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Commun. Math. Phys. 289(1), 45–73 (2009)
Abels, H.: Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow. SIAM J. Math. Anal. 44(1), 316–340 (2012)
Abels, H., Garcke, H., Grün, G.: Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22(03), 1150013 (2012)
Aki, G.L., Dreyer, W., Giesselmann, J., Kraus, C.: A quasi-incompressible diffuse interface model with phase transition. Math. Models Methods Appl. Sci. 24(05), 827–861 (2014)
Aland, S., Voigt, A.: Benchmark computations of diffuse interface models for two-dimensional bubble dynamics. Int. J. Numer. Methods Fluids 69(3), 747–761 (2012)
Badalassi, V.E., Ceniceros, H.D., Banerjee, S.: Computation of multiphase systems with phase field models. J. Comput. Phys. 190(2), 371–397 (2003)
Beris, A.N., Edwards, B.: Thermodynamics of Flowing Systems. Oxford University Press, Oxford (1994)
Borcia, R., Bestehorn, M.: Phase-field model for Marangoni convection in liquid-gas systems with a deformable interface. Phys. Rev. E 67(6), 066307 (2003)
Boyer, F.: A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31(1), 41–68 (2002)
Chen, L.Q.: Phase-field models for microstructure evolution. Ann. Rev. Mater. Res. 32(1), 113–140 (2002)
Ding, H., Spelt, P.D.M., Shu, C.: Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226(2), 2078–2095 (2007)
Doi, M.: Introduction to Polymer Physics. Clarendon Press, Oxford (1996)
Du, Q., Liu, C., Wang, X.: A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198, 450–468 (2004)
Du, Q., Liu, C., Wang, X.: Retrieving topological information for phase field models. SIAM J. Appl. Math. 65(6), 1913–1932 (2005)
Garcke, H., Hinze, M., Kahle, C.: A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow. Appl. Numer. Math. 99, 151–171 (2016)
Gong, Y., Zhao, J., Yang, X., Wang, Q.: Fully discrete second-order linear schemes for hydrodynamic phase field models of viscous fluid flows with variable densities. Siam J. Sci. Comput. 40(2), B528–B553 (2018)
Grün, G.: On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities. SIAM J. Numer. Anal. 51(6), 3036–3061 (2013)
Grün, G., Klingbeil, F.: Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model. J. Comput. Phys. 257, 708–725 (2014)
Guo, Z., Lin, P., Lowengrub, J.: A numerical method for the quasi-incompressible Cahn–Hilliard–Navier–Stokes equations for variable density flows with a discrete energy law. J. Comput. Phys. 276, 486–507 (2014)
Guo, Z., Lin, P., Wang, Y.: Continuous finite element schemes for a phase field model in two-layer fluid Benard–Marangoni convection computations. Comput. Phys. Commun. 185(1), 63–78 (2014)
Hua, J., Lin, P., Liu, C., Wang, Q.: Energy law preserving \(\text{ C }^0\) finite element schemes for phase field models in two-phase flow computations. J. Comput. Phys. 230(19), 7115–7131 (2011)
Jacqmin, D.: Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 155(1), 96–127 (1999)
Kim, J., Lowengrub, J.: Phase field modeling and simulation of three-phase flows. Interfaces Free Bound. 7(4), 435–466 (2005)
Li, J., Wang, Q.: A class of conservative phase field models for multiphase fluid flows. J. Appl. Mech. 81(2), 021004 (2014)
Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 179(3), 211–228 (2003)
Lowengrub, J.S., Ratz, A., Voigt, A.: Phase field modeling of the dynamics of multicomponent vesicles spinodal decomposition coarsening budding and fission. Phys. Rev. E 79(3), 031926 (2009)
Lowengrub, J.S., Truskinovsky, L.: Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. A 454, 2617–2654 (1998)
Shen, J., Yang, X.: A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities. SIAM J. Sci. Comput. 32(3), 1159–1179 (2010)
Shen, Jie, Yang, X., Wang, Q.: Mass and volume conservation in phase field models for binary fluids. Commun. Comput. Phys. 13, 1045–1065 (2013)
Teigen, K.E., Song, P., Lowengrub, J., Voigt, A.: A diffuse-interface method for two-phase flows with soluble surfactants. J. Comput. Phys. 230(2), 375–393 (2011)
Verschueren, M., Van de Vosse, F.N., Meijer, H.E.H.: Diffuse-interface modelling of thermocapillary flow instabilities in a Hele–Shaw cell. J. Fluid Mech. 434, 153–166 (2001)
Wang, Q., Forest, M.G., Zhou, R.: A hydrodynamic theory for solutions of nonhomogeneous nematic liquid crystalline polymers with density variations. J. Fluid Eng. 126, 180–188 (2004)
Wise, S.: Three dimensional multispecies nonlinear tumor growth—I: model and numerical method. J. Theor. Biol. 253(3), 524–543 (2008)
Yue, P., Feng, J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317 (2004)
Zhao, J., Shen, Y., Happasalo, M., Wang, Z.J., Wang, Q.: A 3d numerical study of antimicrobial persistence in heterogeneous multi-species biofilms. J. Theor. Biol. 392, 83–98 (2016)
Zhao, J., Wang, Q.: Modeling cytokinesis of eukaryotic cells driven by the actomyosin contractile ring. Int. J. Numer. Methods Biomed. Eng. 32(12), e2774 (2016)
Zhao, J., Wang, Q., Yang, X.: Numerical approximations to a new phase field model for two phase flows of complex fluids. Comput. Methods Appl. Mech. Eng. 310, 77–97 (2016)
Acknowledgements
Xiaogang Yang’s work is supported by the Scientific Research Fund of Wuhan Institute of Technology through Grants K201741; Yuezheng Gong’s work is partially supported by China Postdoctoral Science Foundation through Grants 2016M591054 and the foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems (201703); Jun Li’s work is supported by the National Natural Science Foundation of China (Grant No.11301287), Tianjin Normal University Foundation for the introduction of talent (5RL154); Jia Zhao’s work is partially supported by a Research Catalyst Grant from the Office of Research and Graduate Studies at Utah State University and Qi Wang’s work is partially supported by DMS-1517347, NSFC awards #11571032, #91630207, and NSAF-U1530401.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. Balachandar.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Yang, X., Gong, Y., Li, J. et al. On hydrodynamic phase field models for binary fluid mixtures. Theor. Comput. Fluid Dyn. 32, 537–560 (2018). https://doi.org/10.1007/s00162-018-0463-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00162-018-0463-3