effect on $t_{1/2}/t_g$ of increasing Re$_{1}$ while keeping Bo and Pe$_{1}$ constants at $10^{4}$. At $\mathrm{Bo}=10^{4}$ the deformation regime changes from spherical to backward-facing bag shape, forward-facing bag shape, and finally stirring regime as Re$_{1}$ is increased. The equilibration time decreases significantly with Re$_{1}$ (as $t_{1/2}/t_g \sim \mathrm{Re_1}^{-0.59}$) while $\mathrm{Re_1 \lesssim 10^3}$, but $t_{1/2}/t_g$ seems to reach a plateau at Re$_{1}\gtrsim 10^{3}$ when entering the stirring regime. While higher Re$_{1}$ simulations would be needed to confirm this point, this suggests that the equilibration time may become independent of Re$_{1}$ (and hence on viscosity) in the stirring regime at high Re$_{1}$. Figure \[fig:drop\_t12\]c shows $t_{1/2}/t_g$ as a function of the Bond number, for Pe$_{1}$ fixed at $10^{4}$ and Re$_{1}$ fixed at either 10 or $10^{4}$. We find that $t_{1/2}/t_g$ decreases as Bo increases, irrespectively of Re$_1$, before reaching a plateau. This is consistent with the fact that increasing Bo allows the metal phase to be more deformed, which helps equilibration. This is particularly drastic in the stirring regime, in which the vigorous stretching and folding of the metal phase leads to very fast equilibration.

![Equilibration time $t_{1/2}$ normalized by $t_{g}=\sqrt{(\rho_1/\Delta\rho)s_0/g}$ as a function of the Péclet number in the inner phase 