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January  2021, 26(1): 337-366. doi: 10.3934/dcdsb.2020141

The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures

Andrea Giorgini 1,, , Roger Temam 1, and  Xuan-Truong Vu 2,

1. 

Department of Mathematics & The Institute for Scientific Computing, and Applied Mathematics, Indiana University, Bloomington, Indiana 47405, USA
2. 

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607, USA

* Corresponding author: Andrea Giorgini

Received  January 2020 Published  January 2021 Early access  April 2020

We study the well-posedness for the mildly compressible Navier-Stokes-Cahn-Hilliard system with non-constant viscosity and Landau potential in two and three dimensional domains.

Keywords: Mildly compressible mixtures, Navier-Stokes equations, Cahn-Hilliard equation, uniqueness, strong solutions.
Mathematics Subject Classification: Primary: 35Q35, 35D30, 35D35; Secondary: 76T05.
Citation: Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141
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, Comm. Math. Phys., 289 (2009), 45-73.  doi: 10.1007/s00220-009-0806-4.

[3]

H. Abels, Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow, SIAM J. Math. Anal., 44 (2012), 316-340.  doi: 10.1137/110829246.

[4]

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.

[5]

H. Abels and E. Feireisl, On a diffuse interface model for two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698.  doi: 10.1512/iumj.2008.57.3391.

[6]

H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013, 40 pp. doi: 10.1142/S0218202511500138.

[7]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[8]

A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications., Applied Mathematical Sciences, 151. Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-12905-0.

[9]

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

[10]

F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows, Comput. Fluids, 31 (2002), 41-68.  doi: 10.1016/S0045-7930(00)00031-1.

[11]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[12]

C. S. Cao and C. G. Gal, Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234.  doi: 10.1088/0951-7715/25/11/3211.

[13]

L. Cherfils, E. Feireisl, M. Michálek, A. Miranville, M. Petcu and D. Pražák, The compressible Navier-Stokes-Cahn-Hilliard equations with dynamic boundary conditions, Math. Models Methods Appl. Sci., 29 (2019), 2557-2584.  doi: 10.1142/S0218202519500544.

[14]

F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Universitext. Springer, London, EDP Sciences, Les Ulis, 2012. doi: 10.1007/978-1-4471-2807-6.

[15]

H. Ding, P. D. M. Spelt and C. Shu, Diffuse interface model for incompressible two-phase flows with large density ratios, J. Comput. Phys., 226 (2007), 2078-2095. 

[16]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.

[17]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chinese Ann. Math. Ser. B, 31 (2010), 655-678.  doi: 10.1007/s11401-010-0603-6.

[18]

C. G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes systems with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), Art. 50, 47 pp. doi: 10.1007/s00526-016-0992-9.

[19]

C. G. Gal, M. Grasselli and H. Wu, Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities, Arch. Ration. Mech. Anal., 234 (2019), 1-56.  doi: 10.1007/s00205-019-01383-8.

[20]

M.-H. Giga, A. Kirshtein and C. Liu, Variational modeling and complex fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 73–113. doi: 10.1007/978-3-319-13344-7_2.

[21]

A. Giorgini, A. Miranville and R. Temam, Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system, SIAM J. Math. Anal., 51 (2019), 2535-2574.  doi: 10.1137/18M1223459.

[22]

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.

[23] P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19556.
[24]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

[25]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn–Hilliard fluids and topological transitions, Proc. Roy. Soc. Lond. A, 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.

[26]

V. N. Starovoĭtov, Model of the motion of a two-component liquid with allowance of capillary forces, J. Appl. Mech. Tech. Phys., 35 (1994), 891-897.  doi: 10.1007/BF02369582.

[27]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[28]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, 2001. doi: 10.1090/chel/343.

[29]

G. Tomassetti, An interpretation of Temam's stabilization term in the quasi-incompressible Navier-Stokes system, arXiv: 1909.11168.

[30]

K. E. Wardle and T. Lee, Finite element lattice Boltzmann simulations of free surface flow in a concentric cylinder, Comput. Math. Appl., 65 (2013), 230-238.  doi: 10.1016/j.camwa.2011.09.020.

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, Comm. Math. Phys., 289 (2009), 45-73.  doi: 10.1007/s00220-009-0806-4.

[3]

H. Abels, Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow, SIAM J. Math. Anal., 44 (2012), 316-340.  doi: 10.1137/110829246.

[4]

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.

[5]

H. Abels and E. Feireisl, On a diffuse interface model for two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698.  doi: 10.1512/iumj.2008.57.3391.

[6]

H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013, 40 pp. doi: 10.1142/S0218202511500138.

[7]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[8]

A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications., Applied Mathematical Sciences, 151. Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-12905-0.

[9]

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

[10]

F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows, Comput. Fluids, 31 (2002), 41-68.  doi: 10.1016/S0045-7930(00)00031-1.

[11]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183. Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[12]

C. S. Cao and C. G. Gal, Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234.  doi: 10.1088/0951-7715/25/11/3211.

[13]

L. Cherfils, E. Feireisl, M. Michálek, A. Miranville, M. Petcu and D. Pražák, The compressible Navier-Stokes-Cahn-Hilliard equations with dynamic boundary conditions, Math. Models Methods Appl. Sci., 29 (2019), 2557-2584.  doi: 10.1142/S0218202519500544.

[14]

F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Universitext. Springer, London, EDP Sciences, Les Ulis, 2012. doi: 10.1007/978-1-4471-2807-6.

[15]

H. Ding, P. D. M. Spelt and C. Shu, Diffuse interface model for incompressible two-phase flows with large density ratios, J. Comput. Phys., 226 (2007), 2078-2095. 

[16]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.

[17]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chinese Ann. Math. Ser. B, 31 (2010), 655-678.  doi: 10.1007/s11401-010-0603-6.

[18]

C. G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes systems with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), Art. 50, 47 pp. doi: 10.1007/s00526-016-0992-9.

[19]

C. G. Gal, M. Grasselli and H. Wu, Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities, Arch. Ration. Mech. Anal., 234 (2019), 1-56.  doi: 10.1007/s00205-019-01383-8.

[20]

M.-H. Giga, A. Kirshtein and C. Liu, Variational modeling and complex fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2018), 73–113. doi: 10.1007/978-3-319-13344-7_2.

[21]

A. Giorgini, A. Miranville and R. Temam, Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system, SIAM J. Math. Anal., 51 (2019), 2535-2574.  doi: 10.1137/18M1223459.

[22]

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.

[23] P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19556.
[24]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

[25]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn–Hilliard fluids and topological transitions, Proc. Roy. Soc. Lond. A, 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.

[26]

V. N. Starovoĭtov, Model of the motion of a two-component liquid with allowance of capillary forces, J. Appl. Mech. Tech. Phys., 35 (1994), 891-897.  doi: 10.1007/BF02369582.

[27]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[28]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, 2001. doi: 10.1090/chel/343.

[29]

G. Tomassetti, An interpretation of Temam's stabilization term in the quasi-incompressible Navier-Stokes system, arXiv: 1909.11168.

[30]

K. E. Wardle and T. Lee, Finite element lattice Boltzmann simulations of free surface flow in a concentric cylinder, Comput. Math. Appl., 65 (2013), 230-238.  doi: 10.1016/j.camwa.2011.09.020.

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