Transverse transport of solutes between co-flowing pressure-driven streams for microfluidic studies of diffusion/reaction processes

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.