Determination of Two Unknown Thermal Coefficients Through a Mushy Zone with a Convective Overspecified Boundary Condition

We consider a semi-infinite material that is initially assumed to be liquid at its melting temperature which, without loss of generality, we assume at 0 C. At time t 0  , a heat flux (characterized by the constant 0 q 0  ) is imposed at x 0  and then solidification ensues, where three distinct regions can be distinguished (for a complete description of this mushy model see [1]): H1) Liquid, at temperature 0 C, occupying the region x r(t)  ; H2) Solid, at temperature T(x, t) 0  , occupying the region 0 x s(t)   , with s(t) r(t)  ; H3) Mushy zone, at temperature T 0  , occupying the region s(t) x r(t)   , with two assumptions on this structure which depends on two parameters 0, and 0 1      . We also consider an overspecified condition on the fixed face x 0  given by a convective boundary condition characterized by the constant 0 h 0  [3]. We deal with six unknown thermal coefficients: k 0  (thermal conductivity), c 0  (specific heat), 0   (mass density), 0  (latent heat by unit of mass), and 0, and 0 1      the two parameters of the mushy region. The goal of this paper is the simultaneous determination of two thermal coefficients among   k,c, , , ,    when the constants 0 q 0  and 0 h 0  , and the boundary s(t) are determined experimentally. We obtain explicit formulas in 15 different cases, and we also give necessary and sufficient conditions on data for the existence of a solution. These results complement [2].