Strain-dominated mush temperature profile ($T=T_E+(T_0-T_E)\mathrm{erf}(z/\delta )/\mathrm{erf}(h/\delta )$)

Set $\delta =\sqrt{2\kappa /\alpha }$, $R=\alpha h/V$ and $S=L/[c_p(T_0-T_E)]$. In the regular limit $R\gg S\gg 1$, at fixed finite thermal strain number $\alpha h^2/\kappa$ and bounded phase fraction, the heat equation reduces to $\kappa T^{\prime \prime }+\alpha z{T}^{\prime }=0$. Integration gives the displayed error function profile. Matching the roof temperature gradient $G$ yieldsThe product composition is $C_s=C_0-\alpha (T_0-T_E)\delta [1-e^{-(h/\delta {)}^2}]/[mV\sqrt{\pi }\mathrm{erf}(h/\delta )]$. The asymptotic reduction requires admissible phase fractions. If the thermal strain number grows singularly with $R$, the two ordering inequalities alone do not control boundary-layer errors; the full strain-driven steady mush equations retain the latent-heat term.