Surface tension (eq. \[eq:tension\]) is finally implemented by computing the curvature of the interface.

The Path Toward Equilibration: Regime Diagram and Timescales {#sec:path}
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Dynamical shape regimes {#sec:path_regime}
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We start here by building a regime diagram (figure \[fig:drop\_regime\]) for the deformation of an initially round metal volume falling from rest into molten silicates, from 2D numerical simulations. We choose to construct a regime diagram as a function of the Reynolds number in the inner phase Re$_{1}$ (eq. \[eq:numbers-free-fall\], build with the inertial terminal velocity scaling) and of the Bond number Bo (eq. \[eq:numbers-free-fall\]), which measures the relative importance of buoyancy and surface tension. Since we use $\nu_1=\nu_2$ in these simulations, $\mathrm{Re_2}=\sqrt{\rho_1/\rho_2}~\mathrm{Re_1}$ and the regime diagram is thus a function of $\mathrm{Re_2}$ divided by 1.4. Besides, viscosities in the metal and the magma ocean also vary, with a potential influence on the regime diagram [@wacheul_2018]. Since in the case of a free-falling drop buoyancy is balanced by the largest of inertia and viscous forces, a small value of Bo implies that surface tension always dominates over the other forces, irrespectively of Re$_{1}$. Conversely, a large value of Bo implies that surface tension is small compared to either viscous forces (low 