$\mathrm{Pe_1}$ (a), the Reynolds number in the inner phase $\mathrm{Re_1}$ (b), and the Bond number Bo (c). $t_{1/2}$ is the time at which $\langle \chi_1 \rangle$ is half its initial value. The lines correspond to calculated scaling law.[]{data-label="fig:drop_t12"}](drop_t12.pdf){width="1\linewidth"}

Equilibration Processes in the “Building Blocks”: Sheets, Ligaments and Droplets {#sec:basic}
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The regime diagram obtained in section \[sec:path\], and experiments [@deguen_2014; @landeau_2014; @wacheul_2014; @wacheul_2018], suggest that the most relevant dynamical regime at the beginning of the fragmentation sequence is the stirring regime, in which the metal phase is stretched into thin sheets (in our 2D simulations), and sheets and ligaments in 3D. The metal phase will then eventually fragment into drops. We therefore study here heat and mass transfer in what we consider to be the three “building blocks” of the fragmentation and equilibration sequence: stretched sheets, ligaments, and drops.

A stretched isolated sheet {#sec:basic-sheet}
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We consider here the homogenization of a scalar field $\chi(x,y,t)$ in and around a stretched isolated 2D sheet. Following @ranz_1979, the evolution of $\chi$ is described in a local Lagrangian frame $(x,y)$ which moves and rotates with the fluid such that the $y-$direction is always parallel to the stretching direction, and the $x-$direction perpendicular to 