Similarity solution of two-phase cylindrical Stefan solidification problem

Abstract

The mathematical model of determining temperature fields in cylindrical domain with solidification process is represented. The solidification process of cylinder due to cooling is constructed by two-phase cylindrical Stefan problem for liquid and solid zones with freezing interface. Respect to strength of the heat sink at the center of cylindrical material boundary condition  is an important to determine temperature in solid domain. The analytical solution of the problem is introduced with method of similarity principle which enables us to reduce free boundary problem to ordinary differential equations. Temperature solutions of solid and liquid zones are represented by special function which called exponential integral equation. The free boundary at freezing interface and temperatures at two phases are determined. Lemmas about exponential integral functions are introduced and used to prove that obtained operator function is contraction operator. Upper boundness of the exponential integral function is checked graphically. It is shown that existence of uniqueness of solution exists.