  Microfluidics proved to be a key technology in various applications, allowing
to reproduce large-scale laboratory settings at a more sustainable small-scale.
The current effort is focused on enhancing the mixing process of different
passive species at the micro-scale, where a laminar flow regime damps
turbulence effects. Chaotic advection is often used to improve mixing effects
also at very low Reynolds numbers. In particular, we focus on passive
micromixers, where chaotic advection is mainly achieved by properly selecting
the geometry of microchannels. In such a context, reduced order modeling can
play a role, especially in the design of new geometries. In this chapter, we
verify the reliability and the computational benefits lead by a Hierarchical
Model (HiMod) reduction when modeling the transport of a passive scalar in an
S-shaped microchannel. Such a geometric configuration provides an ideal setting
where to apply a HiMod approximation, which exploits the presence of a leading
dynamics to commute the original three-dimensional model into a system of
one-dimensional coupled problems. It can be proved that HiMod reduction
guarantees a very good accuracy when compared with a high-fidelity model,
despite a drastic reduction in terms of number of unknowns.
