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A new phase field model for material fatigue in an oscillating elastoplastic beam
June  2015, 35(6): 2497-2522. doi: 10.3934/dcds.2015.35.2497

## On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids

 1 Dipartimento di Matematica ed Informatica “U. Dini”, viale Morgagni 67/a, I-50134 Firenze 2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin 3 Dipartimento di Matematica "F.Casorati", Università di Pavia, Via Ferrata, 1, I-27100 Pavia

Received  January 2014 Revised  April 2014 Published  December 2014

We introduce a diffuse interface model describing the evolution of a mixture of two different viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the flow are taken into account. In the mathematical model, the evolution of the velocity $u$ is ruled by the Navier-Stokes system with temperature-dependent viscosity, while the order parameter $\psi$ representing the concentration of one of the components of the fluid is assumed to satisfy a convective Cahn-Hilliard equation. The effects of the temperature are prescribed by a suitable form of the heat equation. However, due to quadratic forcing terms, this equation is replaced, in the weak formulation, by an equality representing energy conservation complemented with a differential inequality describing production of entropy. The main advantage of introducing this notion of solution is that, while the thermodynamical consistency is preserved, at the same time the energy-entropy formulation is more tractable mathematically. Indeed, global-in-time existence for the initial-boundary value problem associated to the weak formulation of the model is proved by deriving suitable a priori estimates and showing weak sequential stability of families of approximating solutions.
Citation: Michela Eleuteri, Elisabetta Rocca, Giulio Schimperna. On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2497-2522. doi: 10.3934/dcds.2015.35.2497
##### References:
 [1] H. Abels, Diffuse Interface Models for Two-phase Flows of Viscous Incompressible Fluids, Lecture notes, Max Planck Institute for Mathematics in the Sciences, No. 36/2007, 2007. [2] H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Rational Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2. [3] D. M. Anderson, G. B. MacFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annual review of fluid mechanics, Palo Alto, 30 (1998), 139-165. doi: 10.1146/annurev.fluid.30.1.139. [4] S. Bartels and T. Roubíček, Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion, Math. Modelling Numer. Anal., 45 (2011), 477-504. doi: 10.1051/m2an/2010063. [5] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. [6] M. Bulícek, E. Feireisl and J. Málek, A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Analysis: Real World Applications, 10 (2009), 992-1015. doi: 10.1016/j.nonrwa.2007.11.018. [7] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. [8] J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102. [9] M. Eleuteri, J. Kopfová and P. Krejči, A thermodynamic model for material fatigue under cyclic loading, Proceedings of the 8th International Symposium on Hysteresis and Micromagnetic Modeling, Physica B: Condensed Matter, 407 (2012), 1415-1416. doi: 10.1016/j.physb.2011.10.017. [10] M. Eleuteri, J. Kopfová and P. Krejči, Non-isothermal cyclic fatigue in an oscillating elastoplastic beam, Comm. Pure Appl. Anal., 12 (2013), 2973-2996. doi: 10.3934/cpaa.2013.12.2973. [11] M. Eleuteri, J. Kopfová and P. Krejči, Fatigue accumulation in a thermo-visco-elastoplastic plate, Discrete Cont. Dynam. Syst, Ser. B, 19 (2014), 2091-2109. doi: 10.3934/dcdsb.2014.19.2091. [12] M. Eleuteri, L. Lussardi and U. Stefanelli, Thermal control of the Souza-Auricchio model for shape memory alloys, Discrete Cont. Dynam. Syst., Ser. S, 6 (2013), 369-386. [13] E. Feireisl, Mathematical theory of compressible, viscous, and heat conducting fluids, Comput. Math. Appl., 53 (2007), 461-490. doi: 10.1016/j.camwa.2006.02.042. [14] E. Feireisl, M. Frémond, E. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Rational Mech. Anal., 205 (2012), 651-672. doi: 10.1007/s00205-012-0517-4. [15] E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to some models of phase changes with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089. [16] E. Feireisl, E. Rocca, G. Schimperna and A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Comm. Math. Sci., 12 (2014), 317-343. doi: 10.4310/CMS.2014.v12.n2.a6. [17] M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [18] M. Frémond and E. Rocca, A model for shape memory alloys with the possibility of voids, Discrete Contin. Dyn. Syst., 27 (2010), 1633-1659. doi: 10.3934/dcds.2010.27.1633. [19] C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013. [20] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5. [21] 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. [22] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479. doi: 10.1103/RevModPhys.49.435. [23] A. D. Ioffe, On lower semicontinuity of integral functionals, SIAM J. Control Optimization, 15 (1977), 521-538. doi: 10.1137/0315035. [24] P. Krejčí and E. Rocca, Well-posedness of an extended model for water-ice phase transitions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 439-460. doi: 10.3934/dcdss.2013.6.439. [25] P. Krejčí, E. Rocca and J. Sprekels, Liquid-solid phase transitions in a deformable container, contribution to the book "Continuous Media with Microstructure'' on the occasion of Krzysztof Wilmanski's 70th birthday, Springer, (2010), 285-300. [26] P. Krejčí, E. Rocca and J. Sprekels, A bottle in a freezer, SIAM J. Math. Anal., 41 (2009), 1851-1873. doi: 10.1137/09075086X. [27] P. Krejčí, E. Rocca and J. Sprekels, A nonlocal phase-field model with nonconstant specific heat, Interfaces Free Bound., 9 (2007), 285-306. doi: 10.4171/IFB/165. [28] F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions, Z. Anal. Anwendungen, 21 (2002), 335-350. doi: 10.4171/ZAA/1081. [29] F. Luterotti and U. Stefanelli, Errata and addendum to: "Existence result for the one-dimensional full model of phase transitions", [Z. Anal. Anwendungen, 21 (2002), 335-350], Z. Anal. Anwendungen, 22 (2003), 239-240. doi: 10.4171/ZAA/1081. [30] F. Luterotti, G. Schimperna and U. Stefanelli, Existence results for a phase transition model based on microscopic movements, Differential equations: Inverse and direct problems, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 251 (2006), 245-263. doi: 10.1201/9781420011135.ch13. [31] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. [32] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H. [33] E. Rocca and R. Rossi, Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials, J. Differential Equations, 245 (2008), 3327-3375. doi: 10.1016/j.jde.2008.02.006. [34] E. Rocca and R. Rossi, Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials, Appl. Math., 53 (2008), 485-520. doi: 10.1007/s10492-008-0038-5. [35] E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage, Math. Models Methods Appl. Sci., 24 (2014), 1265-1341. doi: 10.1142/S021820251450002X. [36] R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis, Interfaces Free Bound., 15 (2013), 1-37. doi: 10.4171/IFB/293. [37] R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, Quaderno 05/2012 del Seminario Matematico di Brescia, (2014), 1-59. doi: 10.1051/cocv/2014015. [38] T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297. doi: 10.1137/080729992. [39] T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current, Zeit. Angew. Math. Phys., 61 (2010), 1-20. doi: 10.1007/s00033-009-0007-1. [40] V. N. Starovoitov, The dynamics of a two-component fluid in the presence of capillary forces, Math. Notes, 62 (1997), 244-254. doi: 10.1007/BF02355911. [41] P. Sun, C. Liu and J. Xu, Phase field model of thermo-induced Marangoni effects in the mixtures and its numerical simulations with mixed finite element method, Commun. Comput. Phys., 6 (2009), 1095-1117. [42] J. B. Zelďovich and Y. P. Raizer, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, Academic Press, New York, 1967. doi: 10.1115/1.3607836. [43] L. Zhao, H. Wu and H. Huang, Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids, Commun. Math. Sci., 7 (2009), 939-962. doi: 10.4310/CMS.2009.v7.n4.a7.

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##### References:
 [1] H. Abels, Diffuse Interface Models for Two-phase Flows of Viscous Incompressible Fluids, Lecture notes, Max Planck Institute for Mathematics in the Sciences, No. 36/2007, 2007. [2] H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Rational Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2. [3] D. M. Anderson, G. B. MacFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annual review of fluid mechanics, Palo Alto, 30 (1998), 139-165. doi: 10.1146/annurev.fluid.30.1.139. [4] S. Bartels and T. Roubíček, Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion, Math. Modelling Numer. Anal., 45 (2011), 477-504. doi: 10.1051/m2an/2010063. [5] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. [6] M. Bulícek, E. Feireisl and J. Málek, A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Analysis: Real World Applications, 10 (2009), 992-1015. doi: 10.1016/j.nonrwa.2007.11.018. [7] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. [8] J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102. [9] M. Eleuteri, J. Kopfová and P. Krejči, A thermodynamic model for material fatigue under cyclic loading, Proceedings of the 8th International Symposium on Hysteresis and Micromagnetic Modeling, Physica B: Condensed Matter, 407 (2012), 1415-1416. doi: 10.1016/j.physb.2011.10.017. [10] M. Eleuteri, J. Kopfová and P. Krejči, Non-isothermal cyclic fatigue in an oscillating elastoplastic beam, Comm. Pure Appl. Anal., 12 (2013), 2973-2996. doi: 10.3934/cpaa.2013.12.2973. [11] M. Eleuteri, J. Kopfová and P. Krejči, Fatigue accumulation in a thermo-visco-elastoplastic plate, Discrete Cont. Dynam. Syst, Ser. B, 19 (2014), 2091-2109. doi: 10.3934/dcdsb.2014.19.2091. [12] M. Eleuteri, L. Lussardi and U. Stefanelli, Thermal control of the Souza-Auricchio model for shape memory alloys, Discrete Cont. Dynam. Syst., Ser. S, 6 (2013), 369-386. [13] E. Feireisl, Mathematical theory of compressible, viscous, and heat conducting fluids, Comput. Math. Appl., 53 (2007), 461-490. doi: 10.1016/j.camwa.2006.02.042. [14] E. Feireisl, M. Frémond, E. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Rational Mech. Anal., 205 (2012), 651-672. doi: 10.1007/s00205-012-0517-4. [15] E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to some models of phase changes with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369. doi: 10.1002/mma.1089. [16] E. Feireisl, E. Rocca, G. Schimperna and A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Comm. Math. Sci., 12 (2014), 317-343. doi: 10.4310/CMS.2014.v12.n2.a6. [17] M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [18] M. Frémond and E. Rocca, A model for shape memory alloys with the possibility of voids, Discrete Contin. Dyn. Syst., 27 (2010), 1633-1659. doi: 10.3934/dcds.2010.27.1633. [19] C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013. [20] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5. [21] 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. [22] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479. doi: 10.1103/RevModPhys.49.435. [23] A. D. Ioffe, On lower semicontinuity of integral functionals, SIAM J. Control Optimization, 15 (1977), 521-538. doi: 10.1137/0315035. [24] P. Krejčí and E. Rocca, Well-posedness of an extended model for water-ice phase transitions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 439-460. doi: 10.3934/dcdss.2013.6.439. [25] P. Krejčí, E. Rocca and J. Sprekels, Liquid-solid phase transitions in a deformable container, contribution to the book "Continuous Media with Microstructure'' on the occasion of Krzysztof Wilmanski's 70th birthday, Springer, (2010), 285-300. [26] P. Krejčí, E. Rocca and J. Sprekels, A bottle in a freezer, SIAM J. Math. Anal., 41 (2009), 1851-1873. doi: 10.1137/09075086X. [27] P. Krejčí, E. Rocca and J. Sprekels, A nonlocal phase-field model with nonconstant specific heat, Interfaces Free Bound., 9 (2007), 285-306. doi: 10.4171/IFB/165. [28] F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions, Z. Anal. Anwendungen, 21 (2002), 335-350. doi: 10.4171/ZAA/1081. [29] F. Luterotti and U. Stefanelli, Errata and addendum to: "Existence result for the one-dimensional full model of phase transitions", [Z. Anal. Anwendungen, 21 (2002), 335-350], Z. Anal. Anwendungen, 22 (2003), 239-240. doi: 10.4171/ZAA/1081. [30] F. Luterotti, G. Schimperna and U. Stefanelli, Existence results for a phase transition model based on microscopic movements, Differential equations: Inverse and direct problems, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 251 (2006), 245-263. doi: 10.1201/9781420011135.ch13. [31] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. [32] O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Phys. D, 43 (1990), 44-62. doi: 10.1016/0167-2789(90)90015-H. [33] E. Rocca and R. Rossi, Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials, J. Differential Equations, 245 (2008), 3327-3375. doi: 10.1016/j.jde.2008.02.006. [34] E. Rocca and R. Rossi, Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials, Appl. Math., 53 (2008), 485-520. doi: 10.1007/s10492-008-0038-5. [35] E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage, Math. Models Methods Appl. Sci., 24 (2014), 1265-1341. doi: 10.1142/S021820251450002X. [36] R. Rossi and T. Roubíček, Adhesive contact delaminating at mixed mode, its thermodynamics and analysis, Interfaces Free Bound., 15 (2013), 1-37. doi: 10.4171/IFB/293. [37] R. Rossi and M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, Quaderno 05/2012 del Seminario Matematico di Brescia, (2014), 1-59. doi: 10.1051/cocv/2014015. [38] T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297. doi: 10.1137/080729992. [39] T. Roubíček and G. Tomassetti, Thermodynamics of shape-memory alloys under electric current, Zeit. Angew. Math. Phys., 61 (2010), 1-20. doi: 10.1007/s00033-009-0007-1. [40] V. N. Starovoitov, The dynamics of a two-component fluid in the presence of capillary forces, Math. Notes, 62 (1997), 244-254. doi: 10.1007/BF02355911. [41] P. Sun, C. Liu and J. Xu, Phase field model of thermo-induced Marangoni effects in the mixtures and its numerical simulations with mixed finite element method, Commun. Comput. Phys., 6 (2009), 1095-1117. [42] J. B. Zelďovich and Y. P. Raizer, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, Academic Press, New York, 1967. doi: 10.1115/1.3607836. [43] L. Zhao, H. Wu and H. Huang, Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids, Commun. Math. Sci., 7 (2009), 939-962. doi: 10.4310/CMS.2009.v7.n4.a7.
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