June  2012, 32(6): 1997-2025. doi: 10.3934/dcds.2012.32.1997

On a phase field model for solid-liquid phase transitions

1. 

Université de Lyon, CNRS UMR 5208 & Université Lyon 1, Institut Camille Jordan, 43 bd du 11 novembre 1918, F-69622 Villeurbanne cedex, France, France

2. 

Université Blaise Pascal & CNRS UMR 6620, Laboratoire de Mathématiques, Campus des Cézeaux, B.P. 80026, F-63177 Aubière cedex, France

3. 

CEA-Grenoble (DEN/DTP/SMTH), 17, rue des martyrs, F-38054 Grenoble cedex 9, France

Received  January 2011 Revised  July 2011 Published  February 2012

A new phase field model is introduced, which can be viewed as a nontrivial generalisation of what is known as the Caginalp model. It involves in particular nonlinear diffusion terms. By formal asymptotic analysis, it is shown that in the sharp interface limit it still yields a Stefan-like model with: 1) a generalized Gibbs-Thomson relation telling how much the interface temperature differs from the equilibrium temperature when the interface is moving or/and is curved with surface tension; 2) a jump condition for the heat flux, which turns out to depend on the latent heat and on the velocity of the interface with a new, nonlinear term compared to standard models. From the PDE analysis point of view, the initial-boundary value problem is proved to be locally well-posed in time (for smooth data).
Citation: Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997
References:
[1]

D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, in "Annual Review of Fluid Mechanics," Vol. 30, Annual Reviews, Palo Alto, CA, (1998), 139-165.

[2]

E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246 (2009), 3260-3295.

[3]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[4]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9 (1998), 417-445. doi: 10.1017/S0956792598003520.

[5]

C. Cavaterra, C. G. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399. doi: 10.1016/j.na.2009.11.002.

[6]

P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81.

[7]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[9]

M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.

[10]

M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277.

[11]

A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal., 71 (2009), 2278-2290. doi: 10.1016/j.na.2009.01.061.

[12]

A. Miranville and R. Quintanilla, A Caginalp phase-field system with a nonlinear coupling, Nonlinear Anal. Real World Appl., 11 (2010), 2849-2861. doi: 10.1016/j.nonrwa.2009.10.008.

[13]

Pierre Ruyer, "Modèle de Champ de Phase pour l'Étude de l'Ébullition," Ph.D thesis, École Polytechnique, 2006.

show all references

References:
[1]

D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, in "Annual Review of Fluid Mechanics," Vol. 30, Annual Reviews, Palo Alto, CA, (1998), 139-165.

[2]

E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246 (2009), 3260-3295.

[3]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.

[4]

G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math., 9 (1998), 417-445. doi: 10.1017/S0956792598003520.

[5]

C. Cavaterra, C. G. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamic boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399. doi: 10.1016/j.na.2009.11.002.

[6]

P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81.

[7]

E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[9]

M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.

[10]

M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277.

[11]

A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal., 71 (2009), 2278-2290. doi: 10.1016/j.na.2009.01.061.

[12]

A. Miranville and R. Quintanilla, A Caginalp phase-field system with a nonlinear coupling, Nonlinear Anal. Real World Appl., 11 (2010), 2849-2861. doi: 10.1016/j.nonrwa.2009.10.008.

[13]

Pierre Ruyer, "Modèle de Champ de Phase pour l'Étude de l'Ébullition," Ph.D thesis, École Polytechnique, 2006.

[1]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631

[2]

Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523

[3]

Giada Basile, Tomasz Komorowski, Stefano Olla. Diffusion limit for a kinetic equation with a thermostatted interface. Kinetic and Related Models, 2019, 12 (5) : 1185-1196. doi: 10.3934/krm.2019045

[4]

Leonid Berlyand, Mykhailo Potomkin, Volodymyr Rybalko. Sharp interface limit in a phase field model of cell motility. Networks and Heterogeneous Media, 2017, 12 (4) : 551-590. doi: 10.3934/nhm.2017023

[5]

Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation. Communications on Pure and Applied Analysis, 2012, 11 (1) : 1-18. doi: 10.3934/cpaa.2012.11.1

[6]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[7]

Eduard Feireisl. On weak solutions to a diffuse interface model of a binary mixture of compressible fluids. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 173-183. doi: 10.3934/dcdss.2016.9.173

[8]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4963-4998. doi: 10.3934/dcdsb.2020321

[9]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039

[10]

Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49

[11]

Danielle Hilhorst, Hideki Murakawa. Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium. Networks and Heterogeneous Media, 2014, 9 (4) : 669-682. doi: 10.3934/nhm.2014.9.669

[12]

Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks and Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23

[13]

Alexander Mielke. Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 479-499. doi: 10.3934/dcdss.2013.6.479

[14]

Yansu Ji, Jianwei Shen, Xiaochen Mao. Pattern formation of Brusselator in the reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022103

[15]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325

[16]

Keng Deng. On a nonlocal reaction-diffusion population model. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65

[17]

Zhiting Xu, Yingying Zhao. A reaction-diffusion model of dengue transmission. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2993-3018. doi: 10.3934/dcdsb.2014.19.2993

[18]

Feng-Bin Wang. A periodic reaction-diffusion model with a quiescent stage. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 283-295. doi: 10.3934/dcdsb.2012.17.283

[19]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246

[20]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (2)

[Back to Top]