A free boundary problem describing an irreversible crystallization process
The author discusses the following problem: A plate of half thickness one, initial temperature \(T_ 0\), melting temperature \(T_ m\) is immersed at \(t=0\) in a melt of constant temperature \(T_ 1\), and crystallization temperature \(T_ k\), \(T_ 0<T_ m<T_ k<T_ 1\). The plate melts in contact with the crystal with interphase \(S_ 1(t)\). The crystal and the melt are separated by an interphase \(S_ 2(t)\), and a basic assumption for the paper is \(S_ 1(0)=S_ 2(0)=1\), \(0<S_ 1(t)<1<S_ 2(t)\), \(t>0\). In this two-phase Stefan problem, the interphase \(S_ 1(t)\) in the melting plate is controlled by heat flux: \(S'_ 1(t)=u_ x(S_ 1(t)^-,t)-u_ x(S_ 1(t)^+,t)\); as the crystallization is irreversible, if the temperature falls below \(T_ k\) due to specific heat, the interphase \(S_ 2(t)\) will stop. This gives, after introducing dimensionless variables, a Stefan-Signorini type condition for \(S_ 2: \quad u(S_ 2(t)^-,t)\geq 0,\quad u_ x(S_ 2(t)^-,t)+g(t)\geq 0,\quad u(S_ 2(t)^-,t)[u_ x(S_ 2(t)^- ,t)+g(t)]=0,\quad S'_ 2(t)=u_ x(S_ 2(t)^-,t)+g(t).\)There is continuity of temperature u and heat flux across the interface \(x=1\) between (melting) plate and crystal. The author proves global solvability of an approximate problem and then establishes global existence of solution for the present problem by means of a compactness argument. References include 16 items.
scientific article; zbMATH DE number 4210725