Abstract
We investigate the behavior of droplets flowing through flexible rectangular channels and uncover a fascinating phenomenon: pancake-shaped confined droplets migrate transversely toward the channel center. Our findings reveal that this migration is driven by a gradient in interfacial energy, created by the deformation of the channel under the flow of the carrier phase. To explain this behavior, we introduce a simple hydrodynamic model that accurately predicts the trajectories of individual droplets and their dependence on key experimental parameters, such as droplet volume and total flow rate. In concentrated emulsions, this effect results in a striking segregation phenomenon, where all droplets converge and concentrate in the central region of the channel. These insights pave the way for deeper understanding and control of droplet dynamics in confined environments.