Simultaneous determination of two unknown thermal coefficients through a mushy zone model with an overspecified convective boundary condition

Abstract

We have considered the simultaneous determination of two unknown thermal coefficients for a semi-infinite material which was under a phase-change process with a mushy zone according to the model of Solomon, Wilson and Alexiades. It was assumed that the material was initially liquid at its melting temperature and that the solidification process began when a heat flux was imposed at the fixed face. The associated free boundary value problem was overspecified with a convective boundary condition aiming at the simultaneous determination of the temperature in the solid region, one of the two free boundaries of the mushy zone and two thermal coefficients. These were chosen among the latent heat by unit mass, the thermal conductivity, the mass density, the specific heat and the two coefficients that characterize the mushy zone. It was assumed that the other free boundary of the mushy zone, the bulk temperature, the heat flux and heat transfer coefficients at the fixed face were known. Depending on the choice of the unknown thermal coefficients, fifteen phase-change problems arose. In this paper, we present those fifteen problems and obtain necessary and sufficient conditions on data for each of them in order to obtain their solutions. We show that there are twelve cases in which it is possible to find a unique solution and that there are infinite solutions for the remaining three cases. Moreover, for each problem, we give explicit formulae for the temperature of the material, the unknown free boundary and the two unknown thermal coefficients.