= Strain-dominated mush temperature profile
{title2=$T=T_E+(T_0-T_E)\operatorname{erf}(z/\delta)/\operatorname{erf}(h/\delta)$}
Set $\delta=\sqrt{2\kappa/\alpha}$, $R=\alpha h/V$ and $S=L/[c_p(T_0-T_E)]$. In the regular limit $R\gg S\gg1$, at fixed finite thermal strain number $\alpha h^2/\kappa$ and bounded phase fraction, the <heat equation> reduces to $\kappa T''+\alpha zT'=0$. Integration gives the displayed <error function> profile. Matching the roof <temperature gradient> $G$ yields
$$
G=\frac{2(T_0-T_E)}{\sqrt{\pi}\delta}\frac{e^{-(h/\delta)^2}}{\operatorname{erf}(h/\delta)}.
$$
The product composition is $C_s=C_0-\alpha(T_0-T_E)\delta[1-e^{-(h/\delta)^2}]/[mV\sqrt\pi\operatorname{erf}(h/\delta)]$. 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.
Back to article page
