 Solution

Integrating $Y^{\prime \prime }+2a\eta Y^{\prime }=0$ once gives $Y^{\prime }=K{e}^{-a{\eta }^2}$. A second integration expresses the solution using the error function when the value at zero is prescribed, and the complementary error function when decay at infinity is prescribed. Write ${\theta }_{\lambda }={\theta }^s(\lambda )$ and take $\lambda >0$, $Le>0$. The wall condition and the interface value giveFor the exterior liquid, continuity at the interface fixes ${\theta }^l(\lambda )={\theta }_{\lambda }-1+{\mathcal{R}}_{\theta }$ and ${\mathcal{C}}^l(\lambda )={\theta }_{\lambda }-1$. Thus the two decaying similarity solutions areThe factor $\sqrt{Le}$ is required because solute and heat diffuse on different length scales. Both denominators are positive for a finite positive $\lambda$. These formulas use the differential equations, wall/far-field data and continuity; the remaining three flux conditions determine their unknown constants.