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Rasheed, A.; Belmiloudi, A.

An analysis of a phase-field model for isothermal binary alloy solidification with convection under the influence of magnetic field. (English) Zbl 1387.80004

J. Math. Anal. Appl. 390, No. 1, 244-273 (2012).
Summary: This paper is devoted to the study of a new phase-field model with convection under the influence of magnetic field for the isothermal solidification of binary mixtures in two-dimensional geometry. A brief description of the system governing the model is given. The existence of weak solutions as well as the regularity and uniqueness results are established under certain conditions on the nonlinearities. A maximum principle under additional assumptions are also derived. A numerical calculation is performed.

Cited in 1 Review
Cited in 2 Documents

MSC:

80A22 Stefan problems, phase changes, etc.
35B50 Maximum principles in context of PDEs
35Q35 PDEs in connection with fluid mechanics

Keywords:

phase-field equations; existence; uniqueness; maximum principle; isothermal solidification; binary alloys; magnetohydrodynamics; numerical simulation; nickel-copper mixture

Software:

DASSL; VLUGR2
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Full Text: DOI

References:

[1] Al-Rawahi, N.; Tryggvason, G., Numerical simulation of dendritic solidification with convection: Two-dimensional geometry, J. Comput. Phys., 180, 471-496 (2002) · Zbl 1143.76529
[2] Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., A phase-field model of solidification with convection, Phys. D, 135, 175-194 (2000) · Zbl 0951.35112
[3] Belmiloudi, A., Stabilization, Optimal and Robust Control: Theory and Applications in Biological and Physical Sciences (2008), Springer-Verlag: Springer-Verlag London, Berlin · Zbl 1169.93010
[4] Belmiloudi, A., Robust and optimal control problems to a phase-field model for the solidification of a binary alloy with a constant temperature, J. Dyn. Control Syst., 10, 4, 453-499 (2004) · Zbl 1065.49003
[5] Belmiloudi, A., Robin-type boundary control problems for the nonlinear Boussinesq type equations, J. Math. Anal. Appl., 273, 428-456 (2002) · Zbl 1032.76017
[6] Belmiloudi, A.; Yvon, J. P., Robust control of a non-isothermal solidification model, WSEAS Trans. Syst., 4, 12, 2291-2300 (2005) · Zbl 1142.49313
[7] Brezis, H., Analyse Fonctionnelle (1983), Masson: Masson Paris · Zbl 0511.46001
[8] Brown, P. N.; Hindmarsh, A. C.; Petzold, L. R., Using Krylov methods in the solution of large-scale differential-algebraic systems, SIAM J. Sci. Comput., 15, 6, 1467-1488 (1994) · Zbl 0812.65060
[9] Caginalp, G., An analysis of a phase field model for a free boundary, Arch. Ration. Mech. Anal., 92, 205-245 (1986) · Zbl 0608.35080
[10] Eck, C., Analysis of a two-scale phase field model for liquid-solid phase transitions with equiaxed dendritic microstructures, Multiscale Model. Simul., 3, 1, 28-49 (2004) · Zbl 1074.35013
[11] C. Eck, M. Redeker, An adaptive two-scale model for phase transitions with evolution of equiaxed dendritic microstructures, SimTech preprint 2010-79, 2010.; C. Eck, M. Redeker, An adaptive two-scale model for phase transitions with evolution of equiaxed dendritic microstructures, SimTech preprint 2010-79, 2010.
[12] Glicksman, M. E.; Huang, S. C., Convective heat transfer during dendritic growth, (Convective Transport and Instability Phenomena (1982), Karlsruhe), 557
[13] Grugicic, M.; Cao, G.; Millar, R. S., Computer modeling of the evolution of dendrite microstructure in binary alloys during non-isothermal solidification, J. Mater. Synthesis Processing, 10, 4, 191-203 (2002)
[14] Hadid, H. B.; Henry, D.; Kaddeche, S., Numerical study of convection in the horizontal Bridgman configuration under the action of a constant magnetic field. Part 1. Two-dimensional flow, J. Fluid Mech., 333, 23-56 (1997) · Zbl 0897.76033
[15] Hoffman, K. H.; Jiang, L., Optimal control of a phase field model for solidification, Numer. Funct. Anal. Optim., 13, 11-27 (1992) · Zbl 0724.49003
[16] Kaouil, B.; Noureddine, M.; Nassif, R.; Boughaleb, Y., Phase-field modelling of dendritic growth behaviour towards the cooling/heating of pure nickel, M. J. Condens. Matter, 6, 1, 109-112 (2005)
[17] Laurencot, Ph., Weak solutions to a phase-field model with non-constant thermal conductivity, Quart. Appl. Math., 4, 739-760 (1997) · Zbl 0893.35060
[18] Li, M.; Takuya, T.; Omura, N.; Miwa, K., Effects of magnetic field and electric current on the solidification of AZ91D magnesium alloys using an electromagnetic vibration technique, J. Alloys Compounds, 487, 187-193 (2009)
[19] Petzold, Linda R., A description of DASSL: A differential/algebraic system solver, (Scientific Computing. Scientific Computing, IMACS Trans. Sci. Comput. (1983)), 65-68
[20] Monteiro, E.; Almeida, R.; Rouboa, A., Finite volume method analysis of heat transfer in multiblock grid during solidification in heat transfer, (Belmiloudi, A., Mathematical Modelling, Numerical Methods, and Information Technology (2011), InTech), 129-150
[21] Ramizer, J. C.; Beckermann, C.; Kerma, A.; Diepers, H. J., Phase-field modeling of binary alloy solidification with couple heat and solute diffusion, Phys. Rev. E, 69, 051607-1-051607-16 (2004)
[22] Ramizer, J. C.; Beckermann, C., Examination of binary alloy free dendritic growth theories with a phase-field model, Acta Materialia, 53, 1721-1736 (2005)
[23] Rappaz, J.; Scheid, J. F., Existence of solutions to a phase-field model for the isothermal solidification process of a binary alloy, Math. Methods Appl. Sci., 23, 491-513 (2000) · Zbl 0964.35026
[24] Rappaz, M.; Rettenmayr, M., Simulation of solidification, Curr. Opin. Solid State Mater. Sci., 3, 275-282 (1998)
[25] A. Rasheed, A. Belmiloudi, Modeling and simulations of dendrite growth using phase-field method with magnetic field effect, submitted for publication.; A. Rasheed, A. Belmiloudi, Modeling and simulations of dendrite growth using phase-field method with magnetic field effect, submitted for publication. · Zbl 1373.80005
[26] A. Rasheed, A. Belmiloudi, Phase-field method for computationally efficient modeling of the solidification of binary alloy with magnetic field effect, in: V. Dougalis, E. Gallopoulos, A. Hadjidimos, et al. (Eds.), Proceedings of the NumAn2010 Conference in Numerical Analysis; Recent Approaches to Numerical Analysis, Chania, Greece, 2010, pp. 222-230.; A. Rasheed, A. Belmiloudi, Phase-field method for computationally efficient modeling of the solidification of binary alloy with magnetic field effect, in: V. Dougalis, E. Gallopoulos, A. Hadjidimos, et al. (Eds.), Proceedings of the NumAn2010 Conference in Numerical Analysis; Recent Approaches to Numerical Analysis, Chania, Greece, 2010, pp. 222-230.
[27] Rasheed, A.; Belmiloudi, A.; Mahé, F., Dynamics of dendrite growth in a binary alloy with magnetic field effect, Discrete Contin. Dyn. Syst., 2011, Special Issue, 1224-1233 (2011) · Zbl 1306.82018
[28] Ruud, C. O.; Chandra, D.; Fernandez, J. M.; Hepworth, M. T., Copper and copper alloy viscosity, Metall. Mater. Trans. B, 7B, 3, 497-498 (1976)
[29] R. Sampath, The adjoint method for the design of directional binary alloy solidification processes in the presence of a strong magnetic field, Thesis, Cornell University, USA, 2001.; R. Sampath, The adjoint method for the design of directional binary alloy solidification processes in the presence of a strong magnetic field, Thesis, Cornell University, USA, 2001.
[30] Takaki, T.; Fukuoka, T.; Tomita, Y., Phase-field simulations during directional solidification of a binary alloy using adaptive finite element method, J. Crystal Growth, 283, 1-2, 263-278 (2005)
[31] Temam, R., Navier-Stokes Equations (1977), North-Holland: North-Holland Amsterdam · Zbl 0335.35077
[32] Tong, X.; Beckermann, C.; Kerma, A.; Li, Q., Phase-field simulations of dendritic crystal growth in a forced flow, Phys. Rev. E, 63, 061601-1-061601-16 (2001)
[33] Wang, S. L., Thermodynamically-consistent phase-field models for solidification, Phys. D, 69, 189-200 (1993) · Zbl 0791.35159
[34] Warren, J. A.; Boettinger, W. J., Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method, Acta Metall. Mater., 43, 2, 689-703 (1995)
[35] Wheeler, A. A.; Boettinger, W. J.; McFadden, G. B., Phase-field model for isothermal phase transitions in binary alloys, Phys. Rev. A, 45, 10, 7424-7439 (1992)
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