= Solute-diffusion characteristics in a fixed-temperature mush
{title2=$\eta=z^2+2(T_E/G)z+2Dt$}
In a stationary <mushy layer> with fixed $T=T_E+Gz$, pure solid and local <liquidus> $C=-T/m$, solute conservation is $\partial_t[(1-\phi)C]=D\partial_z[(1-\phi)C_z]$. Since $C_z=-G/m$ is constant, it reduces to $(z+T_E/G)\phi_t=D\phi_z$. The <method of characteristics> makes the displayed $\eta$ constant and gives $\phi=f(\eta)$ 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.
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