Stagnant mushy-layer model ($E=c_{p}T+L\varphi ,B=\varphi C$)

A stagnant mushy-layer model has no bulk advection, equal phase mass densities, pure solid, and local equilibrium on the liquidus. With liquid fraction $\varphi$, specific enthalpy $E=c_{p}T+L\varphi$ and bulk solute concentration $B=\varphi C$, neglecting solute diffusion inside the mushy layer gives $T_t+(L/c_p){\varphi }_t=\kappa T_{zz}$ and $(\varphi C{)}_t=0$. The latter fixes composition at each stationary material level; it does not automatically set every level to the initial liquid concentration when solute can diffuse ahead of the growing mush.

For $\mathcal{C}=C_0/\Delta C\gg 1$ and inverse Stefan number $\mathcal{S}=O(\mathcal{C})$, a stagnant mushy-layer model has nearly constant effective heat capacity, giving ${\Omega }^2{\theta }_t=\kappa {\theta }_{zz}$ with ${\Omega }^2=1+\mathcal{S}/\mathcal{C}$. With negligible liquid solutal diffusivity, roof value $\theta =-1$, positive liquid superheat ${\theta }_{\infty }$, and edge $h=2\lambda \sqrt{\kappa t}$, the mush profile is $\theta =-1+\mathrm{erf}(\Omega \eta )/\mathrm{erf}(\Omega \lambda )$. Matching heat flux requires $\Omega e^{{\lambda }^2}\mathrm{erfc}\lambda ={\theta }_{\infty }{e}^{{\Omega }^2{\lambda }^2}\mathrm{erf}(\Omega \lambda )$. The square on $\Omega$ in the exponential is essential. The porosity $\mathcal{C}/(\mathcal{C}-\theta )$ records the leading composition balance; the linear thermal profile is asymptotic, not exact at finite $\mathcal{C}$.

At a sharp mush–liquid edge with linear liquidus $T_L=T_m-mC$, marginal equilibrium makes the liquid temperature gradient tangent to the local liquidus: $T_z=-m{C}_z$. Together with $T=T_L$ at the edge and $T\ge T_L$ in the exterior liquid, it supplies the usual absence-of-constitutional supercooling closure. When solute diffusion is neglected inside the mushy layer but retained outside it, interfacial solute conservation can require a small solid fraction jump; imposing unit porosity as an additional finite-diffusivity condition generally overdetermines this sharp-edge reduction.