-
Previous Article
Discontinuity waves as tipping points: Applications to biological & sociological systems
- DCDS-B Home
- This Issue
-
Next Article
On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics
Mathematical modeling of phase transition and separation in fluids: A unified approach
1. | Facoltà di Ingegneria, Università e-Campus, 22060 Novedrate (CO) |
2. | DICATAM, Università di Brescia, Via Valotti, 9 - 25133 Brescia |
3. | DIBRIS, Università di Genova, Via Opera Pia 13, 16145 Genova |
References:
[1] |
H. W. Alt and I. Pawlow, On the entropy principle of phase transition models with a conserved parameter,, Adv. Math. Sci. Appl., 6 (1996), 291.
|
[2] |
A. Berti and C. Giorgi, A phase-field model for liquid-vapor transitions,, J. Non-Equilibrium Thermodyn, 34 (2009), 219.
doi: 10.1515/JNETDY.2009.012. |
[3] |
A. Berti and C. Giorgi, Phase-field modeling of transition and separation phenomena in continuum thermodynamics,, AAPP Phys. Math. Nat. Sci., 91 (2013).
|
[4] |
A. Berti and C. Giorgi, A phase-field model for quasi-incompressible solid-liquid transitions,, to appear in Meccanica., ().
doi: 10.1007/s11012-014-9909-x. |
[5] |
A. Berti, C. Giorgi and E. Vuk, Free energies and pseudo-elastic transitions for shape memory alloys,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 293. Google Scholar |
[6] |
V. Berti, M. Fabrizio and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity,, Physica D, 236 (2007), 13.
doi: 10.1016/j.physd.2007.07.009. |
[7] |
E. Bonetti and M. Frémond, A phase transition model with the entropy balance,, Math. Methods Appl. Sci., 26 (2003), 539.
doi: 10.1002/mma.366. |
[8] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer New York, (1996).
doi: 10.1007/978-1-4612-4048-8. |
[9] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial energy,, J. Chem. Phys., 28 (1958), 258.
doi: 10.1063/1.1744102. |
[10] |
B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity,, Arch. Rational Mech. Anal., 13 (1963), 167.
doi: 10.1007/BF01262690. |
[11] |
M. Fabrizio, C. Giorgi and A. Morro, A Thermodynamic approach to non-isotermal phase-field evolution in continuum physics,, Phys. D, 214 (2006), 144.
doi: 10.1016/j.physd.2006.01.002. |
[12] |
M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to ferromagnetism and phase transitions,, Internat. J. Engrg. Sci., 47 (2009), 821.
doi: 10.1016/j.ijengsci.2009.05.010. |
[13] |
M. Fabrizio, C. Giorgi and A. Morro, Isotropic-nematic phase transitions in liquid crystals,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 565.
doi: 10.3934/dcdss.2011.4.565. |
[14] |
M. Fabrizio, C. Giorgi and A Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids,, Eur. J. Mech. B Fluids, 30 (2011), 281.
doi: 10.1016/j.euromechflu.2010.12.003. |
[15] |
M. Frémond, Non-Smooth Thermomechanics,, Springer New york, (2002).
doi: 10.1007/978-3-662-04800-9. |
[16] |
M. Frémond, Phase Changes in Mechanics,, Springer New york, (2012). Google Scholar |
[17] |
C. Giorgi, Continuum thermodynamics and phase-field models,, Milan J. Math., 77 (2009), 67.
doi: 10.1007/s00032-009-0101-z. |
[18] |
A. E. Green and N. Laws, On a global entropy production inequality,, Quart. J. Mech. Appl. Math., 25 (1972), 1.
doi: 10.1093/qjmam/25.1.1. |
[19] |
K. Hutter and Y. Wang, Phenomenological thermodynamics and entropy principles,, in Entropy, (2003).
|
[20] |
R. A. L. Jones, Soft condensed matter,, Eur. J. Phys., 23 (2002).
doi: 10.1088/0143-0807/23/6/703. |
[21] |
A. Karma and W. J. Rappel, Quantitative phase-field modelling of dendritic growth in two and three dimensions,, Phys. Rev. E, 57 (1998), 4323.
doi: 10.1103/PhysRevE.57.4323. |
[22] |
A. G. Lamorgese, D. Molin and R. Mauri, Phase field approach to multiphase flow modeling,, Milan J. Math., 79 (2011), 597.
doi: 10.1007/s00032-011-0171-6. |
[23] |
G. A. Maugin, The Thermomechanics of Nonlinear Irreversible Behaviours. An Introduction,, World Scientific Singapore, (1999).
doi: 10.1142/3700. |
[24] |
G. A. Maugin and W. Muschik, Thermodynamics with internal variables. Part I. General concepts,, J. Non-Equilibrium Thermodyn, 19 (1994), 217.
doi: 10.1515/jnet.1994.19.3.217. |
[25] |
A. Morro, A phase-field approach to non-isothermal transitions,, Math. Comput. Modelling, 48 (2008), 621.
doi: 10.1016/j.mcm.2007.11.001. |
[26] |
I. Müller, Thermodynamics,, Pitman Boston, (1985). Google Scholar |
[27] |
O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions,, Phys. D, 43 (1990), 44.
doi: 10.1016/0167-2789(90)90015-H. |
[28] |
O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent" phase-field model,, Phys. D, 69 (1993), 107.
doi: 10.1016/0167-2789(93)90183-2. |
[29] |
I. Singer-Loginova and H. M. Singer, The phase-field technique for modeling multiphase materials,, Rep. Prog. Phys., 71 (2008), 106501.
doi: 10.1088/0034-4885/71/10/106501. |
[30] |
P. Ván, Weakly nonlocal irreversible thermodynamics,, Ann. Phys. (8), 12 (2003), 146.
doi: 10.1002/andp.200310002. |
show all references
References:
[1] |
H. W. Alt and I. Pawlow, On the entropy principle of phase transition models with a conserved parameter,, Adv. Math. Sci. Appl., 6 (1996), 291.
|
[2] |
A. Berti and C. Giorgi, A phase-field model for liquid-vapor transitions,, J. Non-Equilibrium Thermodyn, 34 (2009), 219.
doi: 10.1515/JNETDY.2009.012. |
[3] |
A. Berti and C. Giorgi, Phase-field modeling of transition and separation phenomena in continuum thermodynamics,, AAPP Phys. Math. Nat. Sci., 91 (2013).
|
[4] |
A. Berti and C. Giorgi, A phase-field model for quasi-incompressible solid-liquid transitions,, to appear in Meccanica., ().
doi: 10.1007/s11012-014-9909-x. |
[5] |
A. Berti, C. Giorgi and E. Vuk, Free energies and pseudo-elastic transitions for shape memory alloys,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 293. Google Scholar |
[6] |
V. Berti, M. Fabrizio and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity,, Physica D, 236 (2007), 13.
doi: 10.1016/j.physd.2007.07.009. |
[7] |
E. Bonetti and M. Frémond, A phase transition model with the entropy balance,, Math. Methods Appl. Sci., 26 (2003), 539.
doi: 10.1002/mma.366. |
[8] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer New York, (1996).
doi: 10.1007/978-1-4612-4048-8. |
[9] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial energy,, J. Chem. Phys., 28 (1958), 258.
doi: 10.1063/1.1744102. |
[10] |
B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity,, Arch. Rational Mech. Anal., 13 (1963), 167.
doi: 10.1007/BF01262690. |
[11] |
M. Fabrizio, C. Giorgi and A. Morro, A Thermodynamic approach to non-isotermal phase-field evolution in continuum physics,, Phys. D, 214 (2006), 144.
doi: 10.1016/j.physd.2006.01.002. |
[12] |
M. Fabrizio, C. Giorgi and A. Morro, A thermodynamic approach to ferromagnetism and phase transitions,, Internat. J. Engrg. Sci., 47 (2009), 821.
doi: 10.1016/j.ijengsci.2009.05.010. |
[13] |
M. Fabrizio, C. Giorgi and A. Morro, Isotropic-nematic phase transitions in liquid crystals,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 565.
doi: 10.3934/dcdss.2011.4.565. |
[14] |
M. Fabrizio, C. Giorgi and A Morro, Phase separation in quasi-incompressible Cahn-Hilliard fluids,, Eur. J. Mech. B Fluids, 30 (2011), 281.
doi: 10.1016/j.euromechflu.2010.12.003. |
[15] |
M. Frémond, Non-Smooth Thermomechanics,, Springer New york, (2002).
doi: 10.1007/978-3-662-04800-9. |
[16] |
M. Frémond, Phase Changes in Mechanics,, Springer New york, (2012). Google Scholar |
[17] |
C. Giorgi, Continuum thermodynamics and phase-field models,, Milan J. Math., 77 (2009), 67.
doi: 10.1007/s00032-009-0101-z. |
[18] |
A. E. Green and N. Laws, On a global entropy production inequality,, Quart. J. Mech. Appl. Math., 25 (1972), 1.
doi: 10.1093/qjmam/25.1.1. |
[19] |
K. Hutter and Y. Wang, Phenomenological thermodynamics and entropy principles,, in Entropy, (2003).
|
[20] |
R. A. L. Jones, Soft condensed matter,, Eur. J. Phys., 23 (2002).
doi: 10.1088/0143-0807/23/6/703. |
[21] |
A. Karma and W. J. Rappel, Quantitative phase-field modelling of dendritic growth in two and three dimensions,, Phys. Rev. E, 57 (1998), 4323.
doi: 10.1103/PhysRevE.57.4323. |
[22] |
A. G. Lamorgese, D. Molin and R. Mauri, Phase field approach to multiphase flow modeling,, Milan J. Math., 79 (2011), 597.
doi: 10.1007/s00032-011-0171-6. |
[23] |
G. A. Maugin, The Thermomechanics of Nonlinear Irreversible Behaviours. An Introduction,, World Scientific Singapore, (1999).
doi: 10.1142/3700. |
[24] |
G. A. Maugin and W. Muschik, Thermodynamics with internal variables. Part I. General concepts,, J. Non-Equilibrium Thermodyn, 19 (1994), 217.
doi: 10.1515/jnet.1994.19.3.217. |
[25] |
A. Morro, A phase-field approach to non-isothermal transitions,, Math. Comput. Modelling, 48 (2008), 621.
doi: 10.1016/j.mcm.2007.11.001. |
[26] |
I. Müller, Thermodynamics,, Pitman Boston, (1985). Google Scholar |
[27] |
O. Penrose and P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions,, Phys. D, 43 (1990), 44.
doi: 10.1016/0167-2789(90)90015-H. |
[28] |
O. Penrose and P. C. Fife, On the relation between the standard phase-field model and a "thermodynamically consistent" phase-field model,, Phys. D, 69 (1993), 107.
doi: 10.1016/0167-2789(93)90183-2. |
[29] |
I. Singer-Loginova and H. M. Singer, The phase-field technique for modeling multiphase materials,, Rep. Prog. Phys., 71 (2008), 106501.
doi: 10.1088/0034-4885/71/10/106501. |
[30] |
P. Ván, Weakly nonlocal irreversible thermodynamics,, Ann. Phys. (8), 12 (2003), 146.
doi: 10.1002/andp.200310002. |
[1] |
Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303 |
[2] |
Hussein Fakih, Ragheb Mghames, Noura Nasreddine. On the Cahn-Hilliard equation with mass source for biological applications. Communications on Pure & Applied Analysis, 2021, 20 (2) : 495-510. doi: 10.3934/cpaa.2020277 |
[3] |
Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350 |
[4] |
Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 |
[5] |
Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 763-782. doi: 10.3934/cpaa.2020289 |
[6] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[7] |
Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020467 |
[8] |
Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021020 |
[9] |
Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227 |
[10] |
Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020370 |
[11] |
Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 |
[12] |
Mia Jukić, Hermen Jan Hupkes. Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020402 |
[13] |
Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 |
[14] |
Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076 |
[15] |
Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 |
[16] |
Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi. Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020051 |
[17] |
Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141 |
[18] |
Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021024 |
[19] |
Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303 |
[20] |
Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]