 Binary alloy Stefan similarity solution

Consider a planar binary alloy with solid at $x<a(t)$ and liquid at $x>a(t)$, equal constant phase mass densities, no bulk flow, constant thermal properties and constant solute diffusion coefficients $D_s,D_l>0$. A similarity solution places the interface at $a(t)=2\lambda \sqrt{D_{l}t}$. In the liquid, far-field concentration $C_{\infty }$ and constant interfacial concentration $C_a$ giveThis solves the diffusion equation, has $C_l(a,t)=C_a$ and tends to $C_{\infty }$ far ahead. The solid has its own diffusion profile, with coordinate $x/(2\sqrt{D_{s}t})$. Similarly, thermal profiles use each phase's thermal diffusivity, not automatically $D_l$.

Local liquidus equilibrium gives $T_a=T_m-m{C}_a$, with slope magnitude $m>0$, and the solid-liquid segregation coefficient $k_D$ gives $C_s(a,t)=k_D{C}_a$. For the normal directed toward the liquid, the heat and solute jump conditions areThe first is the Stefan condition; the second conserves solute rejected by the moving boundary. Together with the far-field thermal data, these determine $\lambda$ and $C_a$. For $k_D=0$ and negligible solid solute flux, the second reduces to salt rejection. Neglecting solid diffusion is an additional model limit: finite diffusion on both sides can matter near a nearly stationary interface $\lambda =0$.