No-flux constraint in a fixed-temperature mush ($J_C=-D(1-\varphi )C_z$)

For nonzero imposed thermal gradient and linear liquidus equilibrium, $C_z=-G/m\ne 0$ wherever a mixture of crystals and pore liquid persists. A sealed boundary requires zero solute flux, so a local-equilibrium mush trace there would require $1-\varphi =0$, not a zero-solid-fraction roof. In particular a pure-liquid trace with the equilibrium concentration gradient has nonzero diffusive flux. A sealed column must develop a nonequilibrium-in-the-liquid temperature/composition relation, a separate liquid cap, or an appropriate phase-fraction jump; simply transporting the initial zero-fraction level violates the boundary condition.

Assume a column is sealed to salt at both ends, diffusion occurs only in liquid, and its imposed temperature increases strictly upward. A finite-width equilibrium mushy layer cannot be stationary: its solute flux is nonzero unless its liquid fraction vanishes. The final state is pure ice below a height $h_f$ and uniform-composition brine above it. Their interface obeys $C_f=-(T_E+G{h}_f)/m$. The displayed mass conservation equation fixes $h_f$ from the initial salt content. For a diffusive steadily pulled equilibrium mush, let $H=-T_E/G$, $c=m{C}_0/G$, $d=D/V$ and $h_0=H-c-d$. The initial salt content is $N=C_0[h_0+d\log (1+h_0/c)]$. Thus the liquid-cap depth ${\ell }_f=h_0-h_f$ obeys ${\ell }_f({\ell }_f+c+d)=c[h_0+d\log (1+h_0/c)]$, which has one positive root. This conclusion assumes no subsequent exchange with an underlying eutectic reservoir.