Kronecker delta. Rather than investigating separately heat and mass transfer, we will thus consider the scalar field $\chi_i$, which evolution is governed by the transport equation $$\label{eq:AD}
\frac{\mathrm{D}\chi_i}{\mathrm{D}t}=K_i\nabla^2\chi_i,$$ where $\mathrm{D\bullet}/\mathrm{D}t$ is the Lagrangian derivative, with boundary conditions $$\begin{aligned}
\label{eq:general_BC1}
\chi_1&=&\chi_2,\\
\label{eq:general_BC2}
k_1{\bf \nabla}\chi_1\cdot{\bf n}&=&k_2{\bf \nabla}\chi_2\cdot{\bf n},\end{aligned}$$ at the metal/silicate interface. $\chi_i$ has a diffusivity $K_i$ (*i.e.* thermal diffusivity $\kappa_i$ or mass diffusivity $\kappa_i^c$) and a conductivity $k_i$ (equal to either $\lambda_i$ or $D_{1/2}^{\delta_{i1}} \kappa_i^c$). We will also make use of the ratio $q_i=k_i/K_i$, which is equal to either $\rho_i c_{p_i}$ or $D_{1/2}^{\delta_{i1}}$, where $\rho_i$ and $c_{p_i}$ are respectively density and specific heat capacity of the phase $i$. The correspondence between the material properties of $\chi_i$ and their thermal and compositional counterparts are summarized in table \[tab:equivalence\].

  --------------- ----------------------------- -------------- ------------------------------------ -------------------------
          Thermal             $T_i$               $\kappa_i$               $\lambda_i$                   $\rho_ic_{p_i}$
    Compositional  $C_i/D_{1/2}^{\delta_{i1}}$   $\kappa_i^c$   $\kappa_i^c D_{1/2}^{\delta_{i1}}$   $D_{1/2}^{\delta_{i1}}$
          General           $\chi_i$                $K_i$                     $k_i$                           $q_i$
  --------------- ----------------------------- -------------- ------------------------------------ -------------------------

  : Summary of the correspondence between thermal and compositional cases. $\delta_{ij}$ is the Kronecker symbol. The general scalar field $\chi_i$ is associated with modified diffusivity $K_i$ and conductivity $k_i$ in each phase $i$. $q_i$ is also defined as the ratio between $k_i$ and $K_i$.

\[tab:equivalence\]

In the following 