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A tridimensional phase-field model with convection for phase change of an alloy
We consider a tridimensional phase-field model for a solidification/melting
non-stationary process, which incorporates the physics of binary alloys,
thermal properties and fluid motion of non-solidified material. The model is a
free-boundary value problem consisting of a highly non-linear parabolic system
including a phase-field equation, a heat equation, a concentration equation and
a variant of the Navier-Stokes equations modified by a penalization term of
Carman-Kozeny type to model the flow in mushy regions and a Boussinesq type
term to take into account the effects of the differences in temperature and
concentration in the flow. A proof of existence of generalized solutions for
the system is given. For this, the problem is firstly approximated and a
sequence of approximate solutions is obtained by Leray-Schauder's fixed point
theorem. A solution of the original problem is then found by using compactness
arguments.