and diffusivities, both for analytical (sections \[sec:basic-sheet\] and \[sec:basic-ligament-analytical\] for sheets and ligaments, respectively) and numerical calculations (supplementary information and section \[sec:basic-ligament-tension\] for sheet and ligaments, respectively). In numerical simulations, ligaments are free to fragment. We also consider the case of a free-falling, initially round metal mass. Numerical calculations on deformed drops assume identical viscosities and diffusivities between the two phases (sections \[sec:path\], \[sec:basic-droplet\] and \[sec:stirring\]). Analytical calculations on non-deformed drops do take into account the conductivity and diffusivity contrasts (section \[sec:basic-droplet\]).

![Setup for the 2D sheet (a), ligament (b) and drop (c) calculations. Dashed arrows represent the imposed velocity field and the red lines indicate the numerical box boundary. (a): $L_0$ is the box size and $s_0$ the initial thickness of the sheet. (b): $L_0/2$ is the box size and $s_0$ the initial diameter of the ligament. The metal phase corresponding to the inner cylinder (black lines) is surrounded by the silicate phase, the shaded radial plan corresponds to the 2D numerical simulation area. (c): $L_0$ is the box size and $s_0$ the initial diameter of the drop. The imposed velocity field compensates the mean falling velocity $\langle u_y \rangle$.[]{data-label="fig:geometries"}](geometries.pdf){width="1\linewidth"}

Governing equations, and equivalence between the heat and mass 