Reynolds number defined either as $\mathrm{Re}_{1}=t_\nu /t_{\dot{\epsilon}}$ or $\mathrm{Re}_{1}=t_\nu /t_g$ compares inertia to viscous forces. The Péclet number $\mathrm{Pe}_{1}=t_\kappa /t_{\dot{\epsilon}}$ or $\mathrm{Pe}_{1}=t_\kappa /t_g$ accounts for the relative importance of advective transport compared to diffusive transport. The thermal/compositional Ohnesorge number $\mathrm{Oh}_{\kappa}=t_\gamma /t_\kappa$ compares the influence of surface tension to thermal or mass diffusion. The number $\mathrm{Oh}_{\kappa_{1}}$ thus defined has the same form as the classical Ohnesorge number ($\nu_1\sqrt{\rho_1}/\sqrt{\gamma s_{0}}$), except here the $t_{\gamma}$ is compared to the scalar diffusion time rather than to the momentum diffusion time $t_{\nu}$. The Weber number $\mathrm{We}=(t_\gamma /t_{\dot{\epsilon}})^2$ accounts for the relative importance of fluid’s inertia compared to its interfacial tension, and the Bond number $\mathrm{Bo}=(t_\gamma /t_g)^2$ for the relative importance of gravitational forces compared to tension forces.

Numerical simulations with *Basilisk* {#sec:equations-basilisk}
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The equations introduced in section \[sec:equations-equivalence\] are solved in 2D with direct numerical simulations using the free software *Basilisk* (see basilisk.fr and @popinet_2009). This partial differential equations solver uses an adaptive Cartesian mesh, making simulations possible up to Reynolds number of $10^4$. Mesh refinement is performed when the discretization error of a selected field (*e.g.* temperature, concentration, velocity) is larger than an arbitrary value. It allows a good spatial resolution when and 