Constant-fraction mushy-layer similarity solution ($h(t)=2\lambda \sqrt{\kappa t}$)

For a diffusion-controlled mushy layer of constant solid fraction $\varphi$, a sharp moving edge $h=2\lambda \sqrt{\kappa t}$ reduces the thermal heat equation to ${\theta }^{\prime \prime }+2\eta {\theta }^{\prime }=0$ and the exterior solute diffusion equation to ${\mathcal{C}}^{\prime \prime }+2Le\eta {\mathcal{C}}^{\prime }=0$, 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 $\varphi$. Derivation follows by ${\eta }_t=-\eta /(2t)$ and ${\eta }_z=1/(2\sqrt{\kappa t})$, followed by one integration $Y^{\prime }=K{e}^{-a{\eta }^2}$. 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 $0\le \varphi \le 1$.