  We consider a situation commonly encountered in microfluidics: two streams of
miscible liquids are brought at a junction to flow side by side within a
microchannel, allowing solutes to diffuse from one stream to the other and
possibly react. We focus on two model problems: (i) the transverse transport of
a single solute from a stream into the adjacent one, (ii) the transport of the
product of a diffusion-controlled chemical reaction between solutes originating
from the two streams. Our description is made general through a
non-dimensionalized formulation that incorporates both the parabolic Poiseuille
velocity profile along the channel and thermal diffusion in the transverse
direction. Numerical analysis over a wide range of the streamwise coordinate
$x$ reveal different regimes. Close to the top and the bottom walls of the
microchannel, the extent of the diffusive zone follows three distinct power law
regimes as $x$ is increased, characterized respectively by the exponents 1/2,
1/3 and 1/2. Simple analytical arguments are proposed to account for these
results.
