Weak-renormalized solutions for a system that models non-isothermal solidification

The aim of this paper is to prove the existence of weak-renormalized solutions to a system of the Navier-Stokes-Boussinesq kind. This system may be regarded as a modified version of the non-isothermal solidification problem with melt convection. This task will be accomplished satisfactorily in the two-dimensional case. Some nontrivial and deep difficulties will be found, however, in three dimensions in space.Ministerio de Ciencia e Innovació

Existence and regularity of solutions of a phase field model for solidification with convection of pure materials in two dimensions

CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOWe study the existence and regularity of weak solutions of a phase field type model for pure material solidification in presence of natural convection. We assume that the non-stationary solidification process occurs in a two dimensional bounded domain. The governing equations of the model are the phase field equation coupled with a nonlinear heat equation and a modified Navier-Stokes equation. These equations include buoyancy forces modelled by Boussinesq approximation and a Carman-Koseny term to model the flow in mushy regions. Since these modified Navier-Stokes equations only hold in the non-solid regions, which are not known a priori, we have a free boundary-value problem.We study the existence and regularity of weak solutions of a phase field type model for pure material solidification in presence of natural convection. We assume that the non-stationary solidification process occurs in a two dimensional bounded domain. The governing equations of the model are the phase field equation coupled with a nonlinear heat equation and a modified Navier-Stokes equation. These equations include buoyancy forces modelled by Boussinesq approximation and a Carman-Koseny term to model the flow in mushy regions. Since these modified Navier-Stokes equations only hold in the non-solid regions, which are not known a priori, we have a free boundary-value problem2003109125CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO300513/87-9Anderson, D.M., McFadden, G.B., Wheeler, A.A., Diffusive-interface methods in fluid mechanics (1998) Annual Review of Fluid Mechanics, 30, pp. 139-165. , Palo Alto, CABeckermann, C., Diepers, H.-J., Steinbach, I., Karma, A., Tong, X., Modeling melt convection in phase-field simulations of solidification (1999) Journal of Computational Physics, 154, pp. 468-496Blanc, Ph., Gasser, L., Rappaz, J., Existence for a stationary model of binary alloy solidification (1995) Math. Modelling. Num. Anal., 29 (6), pp. 687-699Caginalp, G., Jones, J., A derivation and analysis of phase field models of thermal alloys (1995) Annal. Phys., 237, pp. 66-107Caginalp, G., Phase field computations of single-needle crystals, crystal growth and motion by mean curvature (1994) SIAM J. Sci. Comput., 15 (1), pp. 106-126Caginalp, G., Xie, W., Phase-field and sharp-interface alloy models (1993) Phys. Rev. E, 48 (3), pp. 1897-1999Caginalp, G., Stefan and Hele-Shaw type models as asymption limits of the phase-field equations (1989) Phys. Rev. A, 39 (11), pp. 5887-5896Caginalp, G., An Analysis of phase field model of a free boundary (1986) Arch. Rat. Mech. Anal., 92, pp. 205-245Cannon, J.R., DiBenedetto, E., Knigthly, G.H., The bidimensional Stefan problem with convection time dependent case (1983) Comm. Partial. Diff. Eqs., 8 (14), pp. 1549-1604Cannon, J.R., DiBenedetto, E., Knightly, G.H., The steady state Stefan problem with convection (1980) Arch. Rat. Mech. Anal., 73, pp. 79-97Coriell, S.R., Cordes, M.R., Boettinger, W.J., Sekerka, R.F., Effects of gravity on coupled convective and interfacial instabilities during directional solidification (1980) J. Cryst. Growth, 49 (13)Davis, S.H., Hydrodynamic interations in directional solidification (1990) J. Fluid Mech., 212 (241)Deimling, K., (1984) Nonlinear Functional Analysis, , Springer-VerlagDiBenedetto, E., O'Leray, M., Three-dimensional conduction-convection problems with change of phase (1993) Arch. Rat. Mech. Anal., 123, pp. 99-116DiBenedetto, E., Friedman, A., Conduction-convection problems with change of phase (1986) J. Diff. Eqs., 62, pp. 129-185Diepers, H.-J., Beckermann, C., Steinbach, I., Simulation of convection and ripening in a binary alloy mush using the phase-field method (1999) Acta. Mater., 47 (13), pp. 3663-3678Fix, G.J., Phase field methods for free boundary problems (1983) Free Boundary Problems: Theory and Applications, pp. 580-589. , A. Fasano, M. Primicerio (Eds), Pitman, BostonHoffman, K.H., Jiang, L., Optimal control a phase field model for solidification (1992) Numer. Funct. Anal. Optimiz., 13 (1-2), pp. 11-27Kobayshi, R., Modeling and numerical simulation of dentritic crystal growth (1993) Phys. D, 63, pp. 410-479O'Leray, M., Analysis of the mushy region in conduction-convection problems with change of phase (1997) Elect. Journal. Diff. Eqs., 1997 (4), pp. 1-14Ladyzenskaja, O.A., Solonnikov, V.A., Ural'ceva, N.N., (1968) Linear and Quasilinear Equations of Parabolic Type, , American Mathematical Society, ProvidenceLions, J.L., (1969) Quelques Méthodes de Resolution de Problèmes aux Limites Non Linéaires, , Dunod-Gauthier-Villars, ParisMorosanu, C., Motreanu, D., A generalized phase-field system (1999) Journal of Math. Anal. Appl., 237, pp. 515-540Pericleous, K.A., Cross, M., Moran, G., Chow, P., Chan, K.S., Free surface Navier-Stokes flows with simultaneous heat tranfer and solidification/melting (1996) Adv. Comput. Math., 6, pp. 295-308Penrose, O., Fife, P.C., On the relation between the standard phase-field model and a thermodynamically consistent phase-field model (1993) Phys. D, 69, pp. 107-113Penrose, O., Fife, P.C., Thermodynamically consistent models of phase-field type for the kinetic phase transitions (1990) Phys D, 43, pp. 44-62Simon, J., Compact sets in the space Lp(0, TB) (1987) Annali di Matematica Pura ed Applicata, 146, pp. 65-96. , Serie QuartaTemam, R., (1977) Navier-Stokes Equations, Theory and Numerical Analysis, , North-Holland Publishing CompanyVaz, C.L.D., (2000) Análise de um Modelo Matemático de Condução-Conveçcão do Tipo Campo de Fases para Solidificação, , Ph.D. Thesis, Universidade Estadual de Campinas, BrazilVoller, V.R., Prakash, C., A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems (1987) Int. J. Mass. Tranfer, 30 (8), pp. 1709-1719Voller, V.R., Cross, M., Markatos, N.C., An enthalpy method for convection/diffusion phase field models of solidification (1987) Int. J. Num. Methods. Eng., 24 (1), pp. 271-28

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