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

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

Jie Jiang 1,, , Yinghua Li 2, and  Chun Liu 3,

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.

Keywords: Two-phase flow, incompressible Navier-Stokes, variable density, global existence, longtime behavior.
Mathematics Subject Classification: Primary: 35Q30, 35Q35, 35B40; Secondary: 76T99.
Citation: Jie Jiang, Yinghua Li, Chun Liu. Two-phase incompressible flows with variable density: An energetic variational approach. Discrete and Continuous Dynamical Systems, 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.

[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.

[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).

[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.

[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.

[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.

[7]

H. W. Alt, The entropy principle for interfaces. Fluids and solids, Adv. Math. Sci. Appl., 19 (2009), 585-663. 

[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.

[9]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids Elsevier, 1989.

[10]

V. I. Arnold, Mathematical Methods of Classical Mechanics Springer-Verlag, New York-Heidelberg, 1978.

[11]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. 

[12]

F. Boyer, Nonhomogeneous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré -AN, 18 (2001), 225-259.  doi: 10.1016/S0294-1449(00)00063-9.

[13]

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

[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. 

[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.

[16]

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

[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.

[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.

[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.

[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.

[21]

A. Haraux, Systémes Dynamiques Dissipatifs et Applications Masson, Paris, 1991.

[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.

[23]

R. Kubo, The fluctuation-dissipation theorem, Report on Progress in Physics, 1966.

[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.

[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.

[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.

[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.

[28]

L. Onsager, Reciprocal relations in irreversible processes-Ⅰ, Phys. Rev., 37 (1931), 405. 

[29]

L. Onsager, Reciprocal relations in irreversible processes-Ⅱ, Phys. Rev., 38 (1931), 2265. 

[30]

L. Rayleigh, Some general theorem relating to vibrations, Proc. Lond. Math. Soc., 4 (1873), 357-368. 

[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.

[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.

[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.

[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.

[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.

[36]

K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems Cambridge Univ. Press, Cambridge, New York, 1995. doi: 10.1017/CBO9780511662362.

[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.

[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.

[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.

[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.

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.

[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.

[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).

[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.

[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.

[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.

[7]

H. W. Alt, The entropy principle for interfaces. Fluids and solids, Adv. Math. Sci. Appl., 19 (2009), 585-663. 

[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.

[9]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids Elsevier, 1989.

[10]

V. I. Arnold, Mathematical Methods of Classical Mechanics Springer-Verlag, New York-Heidelberg, 1978.

[11]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. 

[12]

F. Boyer, Nonhomogeneous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré -AN, 18 (2001), 225-259.  doi: 10.1016/S0294-1449(00)00063-9.

[13]

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

[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. 

[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.

[16]

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

[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.

[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.

[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.

[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.

[21]

A. Haraux, Systémes Dynamiques Dissipatifs et Applications Masson, Paris, 1991.

[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.

[23]

R. Kubo, The fluctuation-dissipation theorem, Report on Progress in Physics, 1966.

[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.

[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.

[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.

[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.

[28]

L. Onsager, Reciprocal relations in irreversible processes-Ⅰ, Phys. Rev., 37 (1931), 405. 

[29]

L. Onsager, Reciprocal relations in irreversible processes-Ⅱ, Phys. Rev., 38 (1931), 2265. 

[30]

L. Rayleigh, Some general theorem relating to vibrations, Proc. Lond. Math. Soc., 4 (1873), 357-368. 

[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.

[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.

[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.

[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.

[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.

[36]

K. Taira, Analytic Semigroups and Semilinear Initial Boundary Value Problems Cambridge Univ. Press, Cambridge, New York, 1995. doi: 10.1017/CBO9780511662362.

[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.

[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.

[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.

[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.

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