calculations, the velocity field will either be imposed, or obtained by solving the Navier-Stokes equation under the assumption of incompressibility, $$\begin{aligned}
\label{eq:NS}
\rho_i\left(\frac{\partial{\bf u}}{\partial t}+{\bf u}\cdot\nabla{\bf u}\right)&=&-{\bf \nabla}P+\eta_i\nabla^2{\bf u}+\rho_i{\bf g},\\
\label{eq:imcomp}
\nabla\cdot{\bf u}&=&0,\end{aligned}$$ where ${\bf u}$ is the flow velocity, $P$ is pressure, $\eta_i$ is viscosity, and $g$ is the acceleration of gravity. The non-linear term on the left-hand side of equation \[eq:NS\] is the source of turbulence. It is included in our numerical simulations, but since our simulations are 2D, we will not be able to reach fully developed turbulent regimes. At the metal/silicates interface, the velocity field and tangential stress are continuous, while the normal stress is discontinuous due to interfacial tension, $$\left[{\bf \sigma_1}-{\bf \sigma_2}\right]\cdot{\bf n}=\gamma\, ({\bf\nabla}\cdot{\bf n})\,{\bf n},
\label{eq:tension}$$ where ${\bf \sigma_{i}}$ is the stress tensor, ${\bf\nabla}\cdot{\bf n}$ is the local curvature, and $\gamma$ is interfacial tension.

Dimensionless numbers {#sec:equations-dimensionless}
---------------------

From these equations, we define a diffusion time $t_\kappa=s_0^2/K_{1}$, an advection time related to stretching $t_{\dot{\epsilon}}=1/\dot{\epsilon}$, a free-fall time related to gravity $t_g=\sqrt{(\rho_{1} /\Delta\rho) s_0/g}$, a viscous time $t_\nu=s_0^2/\nu_{1}$ and a capillary time $t_\gamma=\sqrt{\rho_{1} s_0^3/\gamma}$, with $s_0$ the typical length (initial thickness of the sheet, or diameter of the ligament or drop), $K_{1}$ the thermal or 