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Existence and regularity of solutions of a phase field model for solidification with convection of pure materials in two dimensions. (English) Zbl 1034.76057
Summary: We study existence and regularity of weak solutions of a phase field type model for pure material solidification in the presence of natural convection. We assume that a non-stationary solidification occurs in a two-dimensional bounded domain. The governing equations are the phase field equation coupled with a nonlinear heat equation and modified Navier-Stokes equations. 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 hold only in the non-solid regions, which are not known a priori, we have a free boundary problem.

MSC:
76T99 Multiphase and multicomponent flows
80A22 Stefan problems, phase changes, etc.
76R10 Free convection
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30 Navier-Stokes equations
35R35 Free boundary problems for PDEs
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