Existence for a stationary model of binary alloy solidification

— A proof of existence is given for a stationary model of alloy solidification. The System is composed ofheat équation, soluté équation and Navier-Stokes équations. In the latter, Carman-Kozeny penalization of porous medium models the mushy zone. The problem is first regularized and a séquence of regularized solutions is built thanks to Leray-Schauder's fixed point Theorem. A solution is then extracted by compactness argument Résumé. — Une preuve d'existence pour un modèle de solidification d'alliage est donnée. Le système est composé de l'équation de la chaleur, de celle du soluté et de celles de Navier-Stokes. Dans ces dernières, la pénalisation des milieux poreux de Carman-Kozeny modélise la zone pâteuse. Le problème est tout d'abord régularisé et une suite de solutions régularisées est construite grâce au théorème de point fixe de Leray-Schauder. Une solution est ensuite extraite par un argument de compacité. In a macroscopic approach, the solidification of a binary alloy is governed by the heat and soluté équations coupled with Navier-Stokes équations. The latter ones contain a penalization term so that they apply not only to the zone where the alloy is liquid but also to the zone where both liquid and solid states coexist This is the so-called mushy zone. Since the liquid and mushy zones are a priori unknown the évolutive problem is very hard to handle mathematically, even for a simpler case [5]. We consider hère the stationary problem and we prove an existence resuit using Leray-Schauder*s homotopy Theorem (see Theorem 10.3, p. 222 in [7]). This work extends [4] and [2] to the alloy case. The outline of this paper is as follows : Section 1 is devoted to a description of the physical model. In Section 2, the assumptions on the data are stated and (*) Manuscript received March 28, 1995. (*) Ecole d'ingénieurs de FEtat de Vaud, route de Cheseaux 1, CH-1400 Yverdon (Switzerland). () Ecole Polytechnique Fédérale de Lausanne, Département de mathématiques, MAEcublens, CH-1015 Lausanne (Switzerland). This work was partially funded by the « Fonds National de la Recherche Scientifique Suisse ». M AN Modélisation mathématique et Analyse numérique 0764-583X/95/06/$ 4.00 Mathematical Modelling and Numerical Analysis © AFCET Gauthier-Villars 688 Ph. BLANC et al the définition of a weak solution is given. A regularized problem is studied in Section 3 and our main existence resuit is established in Section 4. To complete this introduction, we give some notations. We dénote by W'(Q) the space of functions in L{Q) whose derivatives up to the m-th order are If -integrable. For p ~ 2 we shall write H(Q). The space H0(Q) is the completion of C^(Q) with respect to the norm || . ||Hi. For JQ (f|g)fl =[ f .g dQ =( ^f g e (L{Q)) where p and q are conjugate exponents (p~ l + cf 1 = 1 ). Moreover, the study of Navier-Stokes équations requires the space 'V(Q) = {u e (H0(Q)) \div (u ) = 0} equipped with the norm /2 ! conciseness of writing, we note and ÏQ dQ for feL(Q), for f e ( L ' ( f l ) ) n ,