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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 |
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.
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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