 Solute-diffusion characteristics in a fixed-temperature mush

In a stationary mushy layer with fixed $T=T_E+Gz$, pure solid and local liquidus $C=-T/m$, solute conservation is ${\partial }_t[(1-\varphi )C]=D{\partial }_z[(1-\varphi )C_z]$. Since $C_z=-G/m$ is constant, it reduces to $(z+T_E/G){\varphi }_t=D{\varphi }_z$. The method of characteristics makes the displayed $\eta$ constant and gives $\varphi =f(\eta )$ within the region that remains mush. For negative $T_E$, the characteristics move upward where the equilibrium concentration is positive. Extrapolating a zero-solid-fraction level is not sufficient to determine a boundary when additional flux constraints create a liquid cap or a solid-fraction jump.