= Diffusive steadily pulled equilibrium mush
{title2=$h_0=(-mC_0-T_E)/G-D/V$}
For a linear imposed <temperature> $T=T_E+Gz$, local <liquidus> equilibrium gives $C=-(T_E+Gz)/m$ in the <mushy layer>. With equal densities, pure solid, pulling velocity $-V$, and diffusion only in liquid volume fraction $q=1-\phi$, steady solute flux is $-VqC-DqC_z=-VC_0$. The exterior concentration is $C_0+(C_i-C_0)e^{-V(z-h_0)/D}$. <Marginal equilibrium at a mush-liquid boundary> gives $C_i-C_0=GD/(mV)$ and hence the displayed thickness. In the mush, $q[C+(D/V)C_z]=C_0$, so $\phi=(h_0-z)/(mC_0/G+h_0-z)$. A positive thickness requires the thermal interval to exceed the diffusive length $D/V$.
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