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

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H. AbelsD. 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. AbelsD. 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. AndersonG. 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

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P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics Cambridge, 1995. Google Scholar

[14]

H. DingP. 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. DingY. 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

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D. A. Edwards, H. Brenner and D. T. Wasan, Interfacial Transport Process and Rheology Butterworths/Heinemann, London, 1991. Google Scholar

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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

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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. GurtinD. 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

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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. LiuJ. 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. ShenX. 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. XuL. 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. AbelsD. 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. AbelsD. 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. AndersonG. 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. DingP. 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. DingY. 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. GurtinD. 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. LiuJ. 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. ShenX. 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. XuL. 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

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