= Constant-fraction mushy-layer similarity solution
{title2=$h(t)=2\lambda\sqrt{\kappa t}$}
For a diffusion-controlled <mushy layer> of constant <solid fraction> $\phi$, a sharp moving edge $h=2\lambda\sqrt{\kappa t}$ reduces the thermal <heat equation> to $\theta''+2\eta\theta'=0$ and the exterior solute <diffusion equation> to $\mathcal C''+2Le\eta\mathcal C'=0$, where $Le=\kappa/D$ is the <Lewis number>. Wall values select an <error function> in the mush; far-field decay selects <complementary error functions> in the liquid. Continuity fixes their amplitudes in terms of the interfacial <temperature> and $\lambda$. The <Stefan condition>, solute conservation and <marginal equilibrium at a mush-liquid boundary> give three algebraic/transcendental equations for these two constants and $\phi$. Derivation follows by $\eta_t=-\eta/(2t)$ and $\eta_z=1/(2\sqrt{\kappa t})$, followed by one integration $Y'=Ke^{-a\eta^2}$. Dividing interface equations can discard degenerate branches, so the undivided equations must be retained when a denominator vanishes. Physical admissibility additionally requires $\lambda>0$ and $0\leq\phi\leq1$.
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