Droplet sorting in a loop of flat microfluidic channels

Motivated by recent experiments, we numerically study the droplet traffic in microfluidic channels forming an asymmetric loop with a long and a short arm. The loop is connected to an inlet and an outlet channel by two right angled T-junctions. Assuming flat channels, we employ the boundary element method (BEM) to numerically solve the two-dimensional Darcy equation that governs two phase flow in the Hele-Shaw limit. The occurrence of different sorting regimes is summarized in sorting diagrams in terms of droplet size, distance between consecutive droplets in the inlet channel, and loop asymmetry for mobility ratios of the liquid phases larger and smaller than one. For large droplet distances, the traffic is regulated by the ratio of the total hydraulic resistances of the long and short arms. At high droplet densities and below a critical droplet size, droplet-droplet collisions are observed for both mobility ratios.