Complex Dynamics of Droplet Traffic in a Bifurcating Microfluidic Channel: Periodicity, Multistability, and Selection Rules

The binary path selection of droplets reaching a T junction is regulated by time-delayed feedback and nonlinear couplings. Such mechanisms result in complex dynamics of droplet partitioning: numerous discrete bifurcations between periodic regimes are observed. We introduce a model based on an approximation that makes this problem tractable. This allows us to derive analytical formulae that predict the occurrence of the bifurcations between consecutive regimes, establish selection rules for the period of a regime, and describe the evolutions of the period and complexity of droplet pattern in a cycle with the key parameters of the system. We discuss the validity and limitations of our model which describes semiquantitatively both numerical simulations and microfluidic experiments.