Path selection rules for droplet trains in single-lane microfluidic networks

We investigate the transport of periodic trains of droplets through microfluidic networks having one inlet, one outlet, and nodes consisting of T junctions. Variations of the dilution of the trains, i.e., the distance between drops, reveal the existence of various hydrodynamic regimes characterized by the number of preferential paths taken by the drops. As the dilution increases, this number continuously decreases until only one path remains explored. Building on a continuous approach used to treat droplet traffic through a single asymmetric loop, we determine selection rules for the paths taken by the drops and we predict the variations of the fraction of droplets taking these paths with the parameters at play including the dilution. Our results show that as dilution decreases, the paths are selected according to the ascending order of their hydrodynamic resistance in the absence of droplets. The dynamics of these systems controlled by time-delayed feedback is complex: We observe a succession of periodic regimes separated by a wealth of bifurcations as the dilution is varied. In contrast to droplet traffic in single asymmetric loops, the dynamical behavior in networks of loops is sensitive to initial conditions because of extra degrees of freedom.