# American Institute of Mathematical Sciences

June  2017, 37(6): 3243-3284. doi: 10.3934/dcds.2017138

## Two-phase incompressible flows with variable density: An energetic variational approach

 1 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, Hubei Province, China 2 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China 3 Department of Mathematics, Penn State University, University Park, PA 16802, USA

* Corresponding author

Received  May 2016 Revised  January 2017 Published  February 2017

In this paper, we study a diffuse-interface model for two-phase incompressible flows with different densities. First, we present a derivation of the model using an energetic variational approach. Our model allows large density ratio between the two phases and moreover, it is thermodynamically consistent and admits a dissipative energy law. Under suitable assumptions on the average density function, we establish the global existence of a weak solution in the 3D case as well as the global well-posedness of strong solutions in the 2D case to an initial-boundary problem for the resulting Allen-Cahn-Navier-Stokes system. Furthermore, we investigate the longtime behavior of the 2D strong solutions. In particular, we obtain existence of a maximal compact attractor and prove that the solution will converge to an equilibrium as time goes to infinity.

Citation: Jie Jiang, Yinghua Li, Chun Liu. Two-phase incompressible flows with variable density: An energetic variational approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3243-3284. doi: 10.3934/dcds.2017138
##### References:
 [1] H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506.  doi: 10.1007/s00205-008-0160-2.  Google Scholar [2] H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Commun. Math. Phys., 289 (2009), 45-73.  doi: 10.1007/s00220-009-0806-4.  Google Scholar [3] H. Abels, H. Garcke and G. Grün, Thermodynamically consistent diffuse interface models for incompressible two-phase flows with different densities, preprint, arXiv: 1011.00528 (2010). Google Scholar [4] H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities Math. Mod. Meth. Appl. Sic. 22 (2012), 1150013, 40pp. doi: 10.1142/S0218202511500138.  Google Scholar [5] H. Abels, D. Depner and H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities, J. Math. Fluid Mech., 15 (2013), 453-480.  doi: 10.1007/s00021-012-0118-x.  Google Scholar [6] H. Abels, D. Depner and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. I. H. Poincaré-AN, 30 (2013), 1175-1190.  doi: 10.1016/j.anihpc.2013.01.002.  Google Scholar [7] H. W. Alt, The entropy principle for interfaces. Fluids and solids, Adv. Math. Sci. Appl., 19 (2009), 585-663.   Google Scholar [8] D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.  Google Scholar [9] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids Elsevier, 1989. Google Scholar [10] V. I. Arnold, Mathematical Methods of Classical Mechanics Springer-Verlag, New York-Heidelberg, 1978.  Google Scholar [11] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212.   Google Scholar [12] F. Boyer, Nonhomogeneous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré -AN, 18 (2001), 225-259.  doi: 10.1016/S0294-1449(00)00063-9.  Google Scholar [13] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics Cambridge, 1995. Google Scholar [14] H. Ding, P. D. M. Spelt and C. Shu, Diffusive interface model for incompressible two-phase flows with large density ratios, J. Comp. Phys., 22 (2007), 2078-2095.   Google Scholar [15] S. Ding, Y. Li and W. Luo, Global solutions for a coupled compressible Navier-Stokes/Allen-Cahn system in 1D, J. Math. Fluid Mech., 15 (2013), 335-360.  doi: 10.1007/s00021-012-0104-3.  Google Scholar [16] D. A. Edwards, H. Brenner and D. T. Wasan, Interfacial Transport Process and Rheology Butterworths/Heinemann, London, 1991. Google Scholar [17] E. FEreisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Different. Equations, 12 (2000), 647-673.  doi: 10.1023/A:1026467729263.  Google Scholar [18] C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. I. H. Poincaré-AN, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.  Google Scholar [19] C. G. Gal and M. Grasselli, Longtime behavior of a model for homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst, 28 (2010), 1-39.  doi: 10.3934/dcds.2010.28.1.  Google Scholar [20] M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831.  doi: 10.1142/S0218202596000341.  Google Scholar [21] A. Haraux, Systémes Dynamiques Dissipatifs et Applications Masson, Paris, 1991.  Google Scholar [22] J. U. Kim, Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal., 18 (1987), 89-96.  doi: 10.1137/0518007.  Google Scholar [23] R. Kubo, The fluctuation-dissipation theorem, Report on Progress in Physics, 1966. Google Scholar [24] J. L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations: Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Equations, vol. 30 of Mathematics Studies, New York, 1977, North-Holland, 285–346. Google Scholar [25] C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.  Google Scholar [26] C. Liu, J. Shen and X. F. Yang, Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density, J. Sci. Comput., 62 (2015), 601-622.  doi: 10.1007/s10915-014-9867-4.  Google Scholar [27] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, P. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.  Google Scholar [28] L. Onsager, Reciprocal relations in irreversible processes-Ⅰ, Phys. Rev., 37 (1931), 405.   Google Scholar [29] L. Onsager, Reciprocal relations in irreversible processes-Ⅱ, Phys. Rev., 38 (1931), 2265.   Google Scholar [30] L. Rayleigh, Some general theorem relating to vibrations, Proc. Lond. Math. Soc., 4 (1873), 357-368.   Google Scholar [31] J. Shen and X. F. Yang, A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput., 32 (2010), 1159-1179.  doi: 10.1137/09075860X.  Google Scholar [32] J. Shen, X. Yang and Q. Wang, On mass conservation in phase field models for binary fluids, Commun. Comput. Phys., 13 (2013), 1045-1065.  doi: 10.4208/cicp.300711.160212a.  Google Scholar [33] W. Shen and S. Zheng, Maximal attractor for the coupled Cahn-Hilliard equations, Nonlinear Anal., 49 (2002), 21-34.  doi: 10.1016/S0362-546X(00)00246-7.  Google Scholar [34] J. Simon, Compact sets in the space $L^p(0, T; B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [35] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.  Google Scholar [36] K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems Cambridge Univ. Press, Cambridge, New York, 1995. doi: 10.1017/CBO9780511662362.  Google Scholar [37] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Applied Mathematics and Sciences, Vol. 68 Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [38] H. Wu and X. Xu, Analysis of a diffuse-interface model for the binary viscous incompressible fluids with thermo-induced Marangoni effects, Commun. Math. Sci., 11 (2013), 603-633.  doi: 10.4310/CMS.2013.v11.n2.a15.  Google Scholar [39] X. Xu, L. Zhao and C. Liu, Axisymmetric solutions to coupled Navier-Stokes/Allen-Cahn equations, SIAM J.Math.Anal., 41 (2010), 2246-2282.  doi: 10.1137/090754698.  Google Scholar [40] S. Zheng, Nonlinear Evolution Equations Pitman series Monographs and Survey in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Florida, 2004. doi: 10.1201/9780203492222.  Google Scholar

show all references

##### References:
 [1] H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506.  doi: 10.1007/s00205-008-0160-2.  Google Scholar [2] H. Abels, Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities, Commun. Math. Phys., 289 (2009), 45-73.  doi: 10.1007/s00220-009-0806-4.  Google Scholar [3] H. Abels, H. Garcke and G. Grün, Thermodynamically consistent diffuse interface models for incompressible two-phase flows with different densities, preprint, arXiv: 1011.00528 (2010). Google Scholar [4] H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities Math. Mod. Meth. Appl. Sic. 22 (2012), 1150013, 40pp. doi: 10.1142/S0218202511500138.  Google Scholar [5] H. Abels, D. Depner and H. Garcke, Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities, J. Math. Fluid Mech., 15 (2013), 453-480.  doi: 10.1007/s00021-012-0118-x.  Google Scholar [6] H. Abels, D. Depner and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. I. H. Poincaré-AN, 30 (2013), 1175-1190.  doi: 10.1016/j.anihpc.2013.01.002.  Google Scholar [7] H. W. Alt, The entropy principle for interfaces. Fluids and solids, Adv. Math. Sci. Appl., 19 (2009), 585-663.   Google Scholar [8] D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.  Google Scholar [9] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids Elsevier, 1989. Google Scholar [10] V. I. Arnold, Mathematical Methods of Classical Mechanics Springer-Verlag, New York-Heidelberg, 1978.  Google Scholar [11] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212.   Google Scholar [12] F. Boyer, Nonhomogeneous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré -AN, 18 (2001), 225-259.  doi: 10.1016/S0294-1449(00)00063-9.  Google Scholar [13] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics Cambridge, 1995. Google Scholar [14] H. Ding, P. D. M. Spelt and C. Shu, Diffusive interface model for incompressible two-phase flows with large density ratios, J. Comp. Phys., 22 (2007), 2078-2095.   Google Scholar [15] S. Ding, Y. Li and W. Luo, Global solutions for a coupled compressible Navier-Stokes/Allen-Cahn system in 1D, J. Math. Fluid Mech., 15 (2013), 335-360.  doi: 10.1007/s00021-012-0104-3.  Google Scholar [16] D. A. Edwards, H. Brenner and D. T. Wasan, Interfacial Transport Process and Rheology Butterworths/Heinemann, London, 1991. Google Scholar [17] E. FEreisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions, J. Dynam. Different. Equations, 12 (2000), 647-673.  doi: 10.1023/A:1026467729263.  Google Scholar [18] C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. I. H. Poincaré-AN, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.  Google Scholar [19] C. G. Gal and M. Grasselli, Longtime behavior of a model for homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst, 28 (2010), 1-39.  doi: 10.3934/dcds.2010.28.1.  Google Scholar [20] M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831.  doi: 10.1142/S0218202596000341.  Google Scholar [21] A. Haraux, Systémes Dynamiques Dissipatifs et Applications Masson, Paris, 1991.  Google Scholar [22] J. U. Kim, Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal., 18 (1987), 89-96.  doi: 10.1137/0518007.  Google Scholar [23] R. Kubo, The fluctuation-dissipation theorem, Report on Progress in Physics, 1966. Google Scholar [24] J. L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations: Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Equations, vol. 30 of Mathematics Studies, New York, 1977, North-Holland, 285–346. Google Scholar [25] C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.  Google Scholar [26] C. Liu, J. Shen and X. F. Yang, Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density, J. Sci. Comput., 62 (2015), 601-622.  doi: 10.1007/s10915-014-9867-4.  Google Scholar [27] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, P. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.  Google Scholar [28] L. Onsager, Reciprocal relations in irreversible processes-Ⅰ, Phys. Rev., 37 (1931), 405.   Google Scholar [29] L. Onsager, Reciprocal relations in irreversible processes-Ⅱ, Phys. Rev., 38 (1931), 2265.   Google Scholar [30] L. Rayleigh, Some general theorem relating to vibrations, Proc. Lond. Math. Soc., 4 (1873), 357-368.   Google Scholar [31] J. Shen and X. F. Yang, A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput., 32 (2010), 1159-1179.  doi: 10.1137/09075860X.  Google Scholar [32] J. Shen, X. Yang and Q. Wang, On mass conservation in phase field models for binary fluids, Commun. Comput. Phys., 13 (2013), 1045-1065.  doi: 10.4208/cicp.300711.160212a.  Google Scholar [33] W. Shen and S. Zheng, Maximal attractor for the coupled Cahn-Hilliard equations, Nonlinear Anal., 49 (2002), 21-34.  doi: 10.1016/S0362-546X(00)00246-7.  Google Scholar [34] J. Simon, Compact sets in the space $L^p(0, T; B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [35] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.  Google Scholar [36] K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems Cambridge Univ. Press, Cambridge, New York, 1995. doi: 10.1017/CBO9780511662362.  Google Scholar [37] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Applied Mathematics and Sciences, Vol. 68 Springer, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [38] H. Wu and X. Xu, Analysis of a diffuse-interface model for the binary viscous incompressible fluids with thermo-induced Marangoni effects, Commun. Math. Sci., 11 (2013), 603-633.  doi: 10.4310/CMS.2013.v11.n2.a15.  Google Scholar [39] X. Xu, L. Zhao and C. Liu, Axisymmetric solutions to coupled Navier-Stokes/Allen-Cahn equations, SIAM J.Math.Anal., 41 (2010), 2246-2282.  doi: 10.1137/090754698.  Google Scholar [40] S. Zheng, Nonlinear Evolution Equations Pitman series Monographs and Survey in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Florida, 2004. doi: 10.1201/9780203492222.  Google Scholar
 [1] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [2] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [3] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [4] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [5] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [6] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [7] Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020350 [8] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 [9] Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164 [10] Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082 [11] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [12] Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073 [13] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [14] Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020049 [15] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [16] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [17] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [18] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [19] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267 [20] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

2019 Impact Factor: 1.338