zbMATH — the first resource for mathematics

Existence for a stationary model of binary alloy solidification. (English) Zbl 0837.35119
Summary: A proof of existence is given for a stationary model of alloy solidification. The system is composed of heat equation, solute equation and Navier-Stokes equations. In the latter, Carman-Kozeny penalization of porous medium models the mushy zone. The problem is first regularized and a sequence of regularized solutions is built thanks to Leray-Schauder’s fixed point theorem. A solution is then extracted by compactness argument.

35Q35 PDEs in connection with fluid mechanics
35D05 Existence of generalized solutions of PDE (MSC2000)
76D05 Navier-Stokes equations for incompressible viscous fluids
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI EuDML
[1] N. AHMAD, 1995, Numerical Simulation of Transport Processes in Multicomponent Systems Related to Solidification Problems, Thesis EPFL.
[2] G. AMIEZ, P.-A. GREMAUD, M. PICASSO, 1990, On a Penalty Method for the Stockes Problem in Regions With Moving Boundaries, Report DMA-EPFL N. 14.90.
[3] Ph. BLANC, L. GASSER, 1993, Existence of a Stationary Solution of a Binary Alloy Problem, Report DMA-EPFL N. 09.93. · Zbl 0837.35119
[4] J.R. CANNON, E. DIBENEDETTO, G. H. NIGHTLY, 1980, The Steady State Stefan Problem with Convection, Archive for Rational Mechanics and Analysis, 73, pp. 79-97. Zbl0436.76056 MR555585 · Zbl 0436.76056 · doi:10.1007/BF00283258
[5] J. R. CANNON, E. DIBENEDETTO, G. H. KNIGHTLY, 1983, The Bidimensional Stefan Problem with Convection : the Time Dependent Case, Comm. in Partial Dijferential Equations, 14,pp. 1549-1604. Zbl0547.35117 MR728873 · Zbl 0547.35117 · doi:10.1080/03605308308820315
[6] R. DAUTRAY, J.-L. LIONS, 1987, Analyse mathématique et calcul numérique pour les sciences et les techniques, tome 2, Masson. Zbl0749.35005 MR902801 · Zbl 0749.35005
[7] D. GlLBARG, N. S. TRUDINGER, 1977, Elliptic Partial Differential Equations of Second Order, Springer. Zbl0361.35003 MR473443 · Zbl 0361.35003
[8] O. A. LADYZHENSKAYA, N. N. URAL’TSEVA, 1968, Linear and Quasilinear Elliptic Equations, Academic Press. Zbl0164.13002 MR244627 · Zbl 0164.13002
[9] R. TEMAM, 1984, Navier-Stokes Equations, North-Holland. Zbl0568.35002 MR769654 · Zbl 0568.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.