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liquid/solid phase change; deformation; convection; numerical simulation; finite differences
We analyse the effect of the mechanical response of the solid phase during liquid/solid phase change by numerical simulation of a benchmark test based on the well-known and debated experiment of melting of a pure gallium slab counducted by Gau & Viskanta in 1986. The adopted mathematical model includes the description of the melt flow and of the solid phase deformations. Surprisingly the conclusion reached is that, even in this case of pure material, the contribution of the solid phase to the balance of the momentum of the system influences significantly the numerical solution and is necessary in order to get a better match with the experimental observations. Here an up-to-date list of the most meaningful mathematical models and numerical simulations of this test is discussed and the need is shown of an accurate revision of the numerical simulations of melting/solidification processes of pure materials (e.g. artificial crystal growth) produced in the last thirty years and not accounting for the solid phase mechanics.
[1] Apostol, T. M.: Calculus. Vol. II: Multi-variable calculus and linear algebra, with applications to differential equations and probability. 2nd ed. Blaisdell Publishing Company Waltham (1969). MR 0248290 | Zbl 0185.11402
[2] Bailey, C., Chow, P., Cross, M., Freyer, Y., Pericleous, K.: Multiphysics modelling of the metals casting process. Proc. R. Soc. Lond. A. 452 (1996), 459-486.
[3] Baldoni, F.: Thermomechanics of Solidification. Pittsburgh University Press Pittsburgh (1997). Zbl 0945.74521
[4] Bansch, E., Smith, A.: Simulation of dendritic crystal growth in thermal convection. Interfaces and Free Boundaries 2 (2000), 95-115. DOI 10.4171/IFB/14 | MR 1759501
[5] O. Bertrand, B. Binet, H. Combeau, S. Couturier, Y. Delannoy, D. Gobin, M. Lacroix, P. Le Quere, M. Medale, J. Mercinger, H. Sadat, G. Vieira: Melting driven by natural convection. A comparison exercise: first results. Int. J. Therm. Sci. 38 (1999), 5-26. DOI 10.1016/S0035-3159(99)80013-0
[6] Brent, A. D., Volle, V. R., Reid, K. J.: Enthalpy-porosity technique for modeling convection-diffusion phase change: application to the melting of a pure metal. Numer. Heat Transfer 13 (1988), 297-318. DOI 10.1080/10407788808913615
[7] Cerimele, M. M., Mansutti, D., Pistella, F.: A front-fixing method for flows in liquid/solid phase change with a benchmark test. CD-Rom Proceedings of ECCOMAS 2000, Barcelona, September 11-14, 2000.
[8] Chalmers, B.: Principles of Solidification. J. Wiley & Sons New York (1964).
[9] Chen, P. Y. P., Timchenko, V., Leonardi, E., Davis, G. de Vahl, III, H. C. de Groh: A numerical study of directional solidification and melting in microgravity. Proceedings of the ASME, Heat Transfer Division Vol. 3 (1998), 75-83.
[10] Chiesa, F. M., Guthie, R. I. L.: Natural convection heat transfer rate during the solidification and melting of metals and alloy systems. J. Heat Transfer 99 (1977), 520-526.
[11] Costanza, G., Gauzzi, F., Montanari, R.: Structures of solid and liquid during melting and solidification of indium. Ann. New York Acad. Sci. 974 (2002), 68-78. DOI 10.1111/j.1749-6632.2002.tb05897.x
[12] Crank, J.: Free and Moving Boundary Problems. Oxford Science Publication. Clarendon Press Oxford (1984). MR 0776227
[13] Cross, M., Bailey, C., Pericleous, K., Williams, A., Bojarevics, V., Croft, N., Taylor, G.: The multiphysics modeling of solidification and melting processes. JOM-e 54 (2002).
[14] Dantzig, J.: Modelling liquid-solid phase change with melt convection. Int. J. Numer. Methods Eng. 28 (1989), 1769-1785. DOI 10.1002/nme.1620280805 | MR 1008137
[15] Fabritiis, G. De, Mancini, A., Mansutti, D., Succi, S.: Mesoscopic models of liquid/solid phase transitions. Int. J. Modern Physics C. 9 (1998), 1405-1415. DOI 10.1142/S0129183198001278
[16] III, H. C. de Groh, Lindstrom, T.: Interface shape and convection during solidification and melting of succinonitrile. NASA Technical Memorandum 106487 (1994).
[17] Derebail, R., Koster, J. N.: Numerical simulation of natural convection of gallium in a narrow gap. Int. J. Heat Mass Transfer 40 (1997), 1169-1180. DOI 10.1016/0017-9310(96)00044-0 | Zbl 0925.76643
[18] Davis, G. De Vahl, Hanjalic, K., Quere, P. Le, Bontoux, P.: Progress in Computational Heat and Mass Transfer. Proc 4th Int. Conf. Comput. Heat Mass Transfer, May 17-20, 2005, Paris. Lavoisier Paris (2005).
[19] Drazin, P. G., Reid, W. H.: Hydrodynamic Stability. Cambridge University Press Cambridge (1985).
[20] Epstein, M., Cheung, F. B.: Complex freezing melting interfaces in fluid flow. Ann. Rev. Fluid Mech. 15 (1983), 293-319. DOI 10.1146/annurev.fl.15.010183.001453
[21] Gadkari, D. B., Shashidharan, P., Lal, K. B., Arora, B. M.: Influence of crystal-melt interface shape on self-seeding and single crystalline quality. Bull. Mater. Sci. 24 (2001), 475-482. DOI 10.1007/BF02706718
[22] Gau, C., Viskanta, R.: Melting and solidification of a metal system in a rectangular cavity. Int. J. Heat Mass Transfer 27 (1984), 113-123. DOI 10.1016/0017-9310(84)90243-6
[23] Gau, C., Viskanta, R.: Melting and solidification of a pure metal on a vertical wall. Transaction of the ASME 108 (1986), 174-181. DOI 10.1115/1.3246884
[24] Golub, G., Loan, C. van: Matrix Computations. The Johns Hopkins University Press Baltimore (1989). MR 1002570
[25] Gondi, P., Montanari, R., Evangelista, E., Buroni, G.: X-ray study of structures of liquid metals with controlled convective motions. Microgravity Quarterly 7 (1997), 155-173.
[26] Hannoun, N., Alexiades, V., Mai, T. Z.: Resolving the controversy over tin and gallium melting in a rectangular cavity heated from the side. Numerical Heat Transfer, Part B 44 (2003), 253-276.
[27] Hannoun, N., Alexiades, V., Mai, T. Z.: A reference solution for phase change with convection. Int. J. Numer. Methods Fluids 48 (2005), 1283-1308. DOI 10.1002/fld.979 | MR 2153612 | Zbl 1112.76402
[28] Hills, R. N., Roberts, P. H.: A macroscopic model of phase coarsening. Int. J. Non-Linear Mech. 25 (1990), 319-329. DOI 10.1016/0020-7462(90)90022-2 | Zbl 0711.76092
[29] Hirasaki, G. J., Hellums, J. D.: Boundary conditions on the vector and scalar potentials in viscous three-dimensional hydrodynamics. Q. Appl. Math. 28 (1970), 293-296. Zbl 0229.76031
[30] Hoger, A., Johnson, B. E.: Linear elasticity for constrained materials: Incompressibility. J. Elasticity 38 (1995), 69-93. DOI 10.1007/BF00121464 | MR 1323555 | Zbl 0824.73007
[31] Hunter, S. C.: Mechanics of Continuous Media. Ellis Horwood Limited Chichester (1976). MR 0445984 | Zbl 0385.73002
[32] Hurle, D. T. J.: Convective transport in melt growth systems. J. Crystal Growth 65 (1983), 124-132. DOI 10.1016/0022-0248(83)90045-3
[33] Huppert, H. E.: The fluid mechanics of solidification. J Fluid Mech. 212 (1990), 209-240. DOI 10.1017/S0022112090001938 | MR 1051332
[34] Kang, K., Ryou, H.: Computation of solidification and melting using the PISO algorithm. Numer. Heat Transfer, Part B 46 (2004), 179-194. DOI 10.1080/10407790490438563
[35] Kim, S., Anghaie, S., Chen, G.: Numerical prediction of multicellular melt flow during natural convection-dominated melting. J. Thermophysics and Heat Transfer 17 (2003), 62-68. DOI 10.2514/2.6734
[36] Kumar, V., Durst, F., Ray, S.: Modeling moving-boundary problems of solidification and melting adopting an arbitrary Lagragian-Eulerian approach. Numer. Heat Transfer, Part B 49 (2006), 299-331. DOI 10.1080/10407790500379981
[37] Lamazouade, A., Ganaoui, M. El, Morvan, D., Bontoux, P.: Numerical simulation of thermo-solutal convection during Bridgman crystal growth. Revue Generale de Thermique 38 (1999), 674-683. DOI 10.1016/S0035-3159(99)80085-3
[38] Lee, Y., Korpela, S. A.: Multicellular natural convection in a vertical slot. J. Fluid Mech. 126 (1983), 91-121. DOI 10.1017/S0022112083000063 | Zbl 0533.76088
[39] Quere, P. Le, Gobin, D.: A note on possible flow instabilities in melting from the side. Int. J. Thermal Sci. 38 (1999), 595-600. DOI 10.1016/S0035-3159(99)80039-7
[40] Mansutti, D., Graziani, G., Piva, R.: A discrete vector potential model for unsteady incompressible viscous flows. J. Comput. Phys. 92 (1991), 161-184. DOI 10.1016/0021-9991(91)90296-W | Zbl 0712.76038
[41] Mansutti, D., Baldoni, F., Rajagopal, K. R.: On the influence of the deformation of the forming solid in the solidification of a semi-infinte water-layer of fluid. Math. Models Methods Appl. Sci. 11 (2001), 367-386. DOI 10.1142/S0218202501000891 | MR 1820678
[42] Mansutti, D., Raffo, R., Santi, R.: Liquid/Solid phase change with convection and deformations: 2D case. Progress in Industrial Mathematics at ECMI Mathematics in Industry Vol. 8, 2004 A. Di Bucchianico, R. M. M. Mattheij, M. A. Peletier Springer Berlin (2006), 268-272. MR 2228611 | Zbl 1309.80003
[43] Miller, W., Succi, S., Mansutti, D.: A lattice Boltzmann model for anisotropic liquid/solid phase transition. Phys. Rev. Lett. 86 (2001), 3578-3581. DOI 10.1103/PhysRevLett.86.3578
[44] Rady, M. A., Mohanty, A. K.: Natural convection during melting and solidification of pure metals in a cavity. Numer. Heat Transfer, Part A 29 (1996), 49-63. DOI 10.1080/10407789608913778
[45] Sampath, R., Zabaras, N.: An object oriented implementation of a front tracking finite element method for directional solidification processes. Int. J. Numer. Methods Eng. 44 (1999), 1227-1265. DOI 10.1002/(SICI)1097-0207(19990330)44:9<1227::AID-NME471>3.0.CO;2-R | Zbl 0943.76052
[46] Slattery, J. C.: Momentum, Energy and Mass Transfer in Continua. McGraw-Hill New York (1972).
[47] Song, R., Dhatt, G., Cheikh, A. Ben: Thermo-mechanical finite element model of casting systems. Int. J. Numer. Methods Eng. 30 (1990), 579-599. DOI 10.1002/nme.1620300403
[48] Stella, F., Giangi, M.: Melting of a pure metal on a vertical wall: numerical simulation. Numer. Heat Transfer, Part A 38 (2000), 193-208. DOI 10.1080/10407780050135405
[49] Stefan, J.: Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Sitzungsberichte der "Osterreichischen Akademie der Wissenschaften Mathematisch-Naturwissen-schaftliche Klasse, Abteilung 2, Mathematik, Astronomie, Physik, Meteorologie und Technik 98 (1988), 965-983 German.
[50] Szekely, J., Chambra, P. S.: The effect of natural convection on the shape and movement of the melt-solid interface in the controlled solidification. Met. Trans. B1 (1970), 1195-1203.
[51] Tenchev, R. T., Mackenzie, J. A., Scanlon, T. J., Stickland, M. T.: Finite element moving mesh analysis of phase change problems with natural convection. Int. J. Heat Fluid Flow 26 (2005), 597-612. DOI 10.1016/j.ijheatfluidflow.2005.03.003
[52] Teskeredzic, A., Demirdzic, I., Muzaferija, S.: Numerical method for heat transfer, fluid flow and stress analysis in phase-change problems. Numer. Heat Transfer, Part B 42 (2002), 437-459. DOI 10.1080/10407790190054021
[53] Truesdell, C., Rajagopal, K. R.: An Introduction to the Mechanics of Fluids. Birkhäuser Boston (2000). MR 1731441 | Zbl 0942.76001
[54] Vorst, H. Van der: Bi-CGSTAB: A fast and smoothly converging variant of the Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 13 (1992), 631-644. DOI 10.1137/0913035 | MR 1149111
[55] Viswanath, R., Jaluria, Y.: A comparison of different solution methodologies for melting and solidification problems in enclosures. Numer. Heat Transfer, Part B. 24 (1993), 77-105. DOI 10.1080/10407799308955883
[56] Voller, V. R., Cross, M., Markatos, N.: An enthalpy method for convection/diffusion phase change. Int. J. Numer. Methods Eng. 24 (1987), 271-284. DOI 10.1002/nme.1620240119 | Zbl 0609.76104
[57] Voller, V. R.: An overview of numerical methods for solving phase change problems: a review. Adv. Numer. Heat Transfer W. J. Minkowycz, E. M. Sparrow Taylor & Francis Philadelphia (1997).
[58] Yeoh, G. H., Davis, G. de Vahl, Leonardi, E., III, H. C. de Groh, Yao, M.: A numerical and experimental study of natural convection and interface shape in crystal growth. J. Crystal Growth 173 (1997), 492-502. DOI 10.1016/S0022-0248(96)00851-2
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