A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and it is represented following the model of Solomon, Wilson and Alexiades. An exact similarity solution is obtained when a restriction on data is verified, and it is analysed the relation between the problem considered here and the problem with a temperature condition at the fixed boundary. Moreover, it is proved that the solution to the problem with the convective boundary condition converges to the solution to a problem with a temperature condition when the heat transfer coefficient at the fixed boundary goes to infinity, and it is given an estimation of the difference between these two solutions. Results in this article complete and improve the ones obtained in Tarzia, Compt. Appl. Math., 9 (1990), 201-211.
Explicit solutions for the Solomon-Wilson-Alexiades's mushy zone model with convective or heat flux boundary conditions
我们完成了Solomon-Wilson-Alexiades的糊状区模型（Letters Heat Mass Transfer，9（1982），319-324）的单相Lamé-Clapeyron-Stefan问题。我们得到明确的解决方案时，对流或热通量边界条件施加在固定面的半无限材料。我们还得到了得到这些显式解的数据上的充要条件。此外，当满足这些条件时，这两个问题等价于在固定面上具有温度边界条件的同一个问题，因此也得到了表征两个自由界面之一的系数的不等式。
Explicit solution for Stefan problem with latent heat depending on the position and a convective boundary condition at the fixed face using Kummer functions
利用Kummer函数得到了半无限材料中单相Stefan问题的相似型显式解。受[D.A. Tarzia，具有对流和温度边界条件的两相Lamé-Clapeyron-Stefan问题的Neumann解之间的关系，Thermal Sci.（2016）DOI 10.2298/TSCI 140607003 T，出版中]和[Y. Zhou，L. J. Xia，精确解for Stefan problem with general power-type latent heat using Kummer function，Int. J. Heat Mass Transfer，84（2015）114-118]，我们考虑了相变问题，其中潜热被定义为具有非负实指数的位置的幂函数和固定面处的对流边界条件$x=0$。证明了解的存在唯一性。分析了该问题与周、夏等已解决的温度、流量边界条件问题之间的关系。此外，还研究了当表征固定边界处传热的系数趋于无穷大时解的极限行为。对某些算例进行了解的数值计算，并将所得结果与求解Stefan问题的一般算法所得结果进行了比较。
