  We theoretically investigate the role of solutal free convection on the
diffusion of a buoyant solute at the microfluidic scales, $\simeq
5$--$500~\mu$m. We first consider a horizontal microfluidic slit, one half of
which initially filled with a binary solution (solute and solvent), and the
other half with pure solvent. The buoyant forces generate a gravity current
that couples to the diffusion of the solute. We perform numerical resolutions
of the 2D model describing the transport of the solute in the slit. This study
allows us to highlight different regimes as a function of a single parameter,
the Rayleigh number $\text{Ra}$ which compares gravity-induced advection to
solute diffusion. We then derive asymptotic analytical solutions to quantify
the width of the mixing zone as a function of time in each regime and establish
a diagram that makes it possible to identify the range of $\text{Ra}$ and times
for which buoyancy does not impact diffusion. In a second step, we present
numerical resolutions of the same model but for a 3D microfluidic channel with
a square cross-section. We observe the same regimes as in the 2D case, and
focus on the dispersion regime at long time scales. We then derive the
expression of the 1D dispersion coefficient for a channel with a rectangular
section, and analyse the role of the transverse flow in the particular case of
a square section. Finally, we show that the impact of this transverse flow on
the solute transport can be neglected for most of the microfluidic experimental
configurations.
