transfer problems {#sec:equations-equivalence}
--------------------------------------------------------------------------------

We study here the evolution of either temperature $T_i$ or concentration $C_i$ in a two-phase flow, where $i \in \lbrace 1,2 \rbrace$ refers to the metallic ($i=1$) or the silicate phase ($i=2$). In the Boussinesq approximation and in the absence of volumetric heat sources (radioactive heating or latent heat), the evolution of temperature and composition within both phases is governed by two transport equations of identical mathematical form. Mathematically speaking, the only difference between heat and mass transfer problems lies in the boundary conditions at the metal/silicates interface, which can be written $$\begin{aligned}
\label{eq:boundaryT1}
T_1&=&T_2,\\
\label{eq:boundaryT2}
\lambda_1{\bf \nabla}T_1\cdot{\bf n}&=&\lambda_2{\bf \nabla}T_2\cdot{\bf n},\end{aligned}$$ for temperature, and $$\begin{aligned}
\label{eq:boundaryC1}
D_{1/2}&=&\frac{C_1}{C_2},\\
\label{eq:boundaryC2}
\kappa_1^c{\bf \nabla}C_1\cdot{\bf n}&=&\kappa_2^c{\bf \nabla}C_2\cdot{\bf n},\end{aligned}$$ for composition, where $\lambda_i$ is the thermal conductivity, $\kappa_i^c$ the mass diffusivity, $D_{1/2}$ the partition coefficient and ${\bf n}$ the unit vector normal to the interface pointing toward phase 2. Equations \[eq:boundaryT2\] and \[eq:boundaryC2\] express the continuity of heat and mass flux at the interface, respectively.

Temperature is continuous across the interface, but concentration is not. Yet the two problems can be made mathematically equivalent by introducing a new variable $\chi_i$ defined as either $$\chi_i=T_i \quad\text{or}\quad \chi_i = \frac{C_i}{D_{1/2}^{\delta_{i1}}},$$ where $\delta_{ij}$ is the 