Past exam of the topics in fluid mechanics course of the University of Oxford / 2010 / 3 / iii / Solution

Introduce three positive functions of the unknown front parameter:Differentiating the error function profiles gives ${\theta }^{s\prime }=2P{\theta }_{\lambda }$, ${\theta }^{l\prime }=-2Q({\theta }_{\lambda }-1+{\mathcal{R}}_{\theta })$ and ${\mathcal{C}}^{l\prime }=-2R({\theta }_{\lambda }-1)$ at the interface. The three flux conditions are therefore exactlyThese undivided equations are the safest description of the constant-fraction mushy-layer similarity solution, including degenerate parameter choices. In the nondegenerate case the third equation first givesThe second then gives the solid fractionUse the third equation to replace the $Q$ term in the first, and eliminate $\varphi$. The front parameter is a positive root of the single transcendental equationThen substitute back for ${\theta }_{\lambda },\varphi$ and $h(t)=2\lambda \sqrt{\kappa t}$. Since the physical parameters have not been assigned numbers, an implicit scalar equation is the appropriate determination of $\lambda$.

The divided formulas require $R\ne Q$ and ${\theta }_{\lambda }-1+{\mathcal{R}}_C\ne 0$. If either vanishes, return to the boxed three-equation system. For example $Le=1$ gives $R=Q$, so the last equation demands ${\mathcal{R}}_{\theta }=0$; otherwise this proposed similarity solution cannot exist. A physical branch must also have $\lambda >0$ and $0\le \varphi \le 1$. The algebra alone does not establish existence or uniqueness of such a branch for arbitrary parameters.