A Tridimensional Phase-Field Model with Convection for Change Phase of an Alloy

We consider a tridimensional phase-field model for a solidification/melting non-stationary process, which incorporatesthe physics of binary alloys, thermal properties and fluid motion of non-solidified material. The model is a free-boundary valueproblem consisting of a non-linear parabolic system including a phase-field equation, a heat equation, a concentration equationand 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.