= Strain-driven steady mush
{title2=$\mathbf q=\alpha(x,-z)$}
A uniform-permeability <mushy layer> on an impermeable eutectic floor, under the quadratic roof pressure $p_0-E^2x^2/2$, has <Darcy flux> $\mathbf q=\alpha(x,-z)$, where $\alpha=\Pi E^2/\mu$. With pulling velocity $-V\mathbf e_z$, pure primary crystals, <solid fraction> $\phi$, liquid <concentration> $C$ and constant phase properties, its one-dimensional balances are
$$
\kappa T''+(V+\alpha z)T'-\frac{VL}{c_p}\phi'=0,\qquad [V(1-\phi)C]'=-\alpha zC',\qquad T=-mC.
$$
The <solidification> product at the eutectic floor retains bulk composition $C_s=(1-\phi(0))C_E$. Integration gives $C_s=C_0-(\alpha/V)\int_0^h(C-C_0)\,dz$: horizontal strain removes solute before the material freezes. A negative predicted $C_s$ is inadmissible, and diagnoses failure of the assumed steady branch rather than a negative-composition product.
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