mass diffusivity of phase 1, $\dot{\epsilon}$ the stretching rate, $\rho_1$ the density of phase 1, $\Delta\rho$ the density difference, and $\nu_1$ the kinematic viscosity of phase 1.

Based on these timescales and the remaining parameters of the set of equations, we build two sets of dimensionless numbers depending on the geometry of the flow we consider. In the case of a stretched sheet or ligament in which we ignore the effect of gravity, using an advection timescale equal to the stretching timescale allows to build the following set of independent dimensionless parameters: $$\mathrm{Re}_1=\frac{s_0^2\dot{\epsilon}}{\nu_1},\
\mathrm{Pe}_1=\frac{s_0^2\dot{\epsilon}}{K_1},\ 
\mathrm{Oh}_{\kappa_1}=\frac{K_1\sqrt{\rho_1}}{\sqrt{\gamma s_0}},\
\frac{K_1}{K_2}, \ 
\frac{k_1}{k_2}, \ 
\frac{\rho_1}{\rho_2},\
\frac{\nu_1}{\nu_2},
\label{eq:numbers-stretching}$$ where Re$_1$ is the Reynolds number, Pe$_1$ the Péclet number, and Oh$_{\kappa_1}$ the thermal/compositional Ohnesorge number. We will also make use of the Weber number defined as $$\mathrm{We}_1=\frac{\rho_1 s_0^3\dot{\epsilon}^2}{\gamma},
\label{eq:weber-def}$$ which is equal to $\left( \mathrm{Oh}_{\kappa_1} \mathrm{Pe}_{1}\right)^{2}$. In the case of a free-falling mass of metal, we use an advection timescale equal to the free-fall time and the problem is now described by the following set of independent dimensionless parameters: $$\mathrm{Re}_1=\sqrt{\frac{\Delta\rho}{\rho_1}\frac{g s_0^3}{\nu_1^2}},\
\mathrm{Pe}_1=\sqrt{\frac{\Delta\rho}{\rho_1}\frac{g s_0^3}{K_1^2}},\ 
\mathrm{Bo}=\frac{\Delta\rho g s_0^2}{\gamma},\
\frac{K_1}{K_2}, \ 
\frac{k_1}{k_2}, \ 
\frac{\rho_1}{\rho_2},\
\frac{\nu_1}{\nu_2}.
\label{eq:numbers-free-fall}$$ The Ohnesorge number can be obtained from Bo and Pe$_1$ as $\mathrm{Oh}_{\kappa_{1}}=\mathrm{Bo}^{1/2}/\mathrm{Pe}_{1}$.

The 