Dynamics of aqueous ferrofluid droplets at coflowing liquid-liquid interface under a non-uniform magnetic field

We report the dynamical behaviour of aqueous ferrofluid droplets (FFDs) of different sizes and concentrations at the interface of coflowing immiscible oils in a microchannel in the presence of the magnetic field. We show that the migration and sorting phenomena are governed by the interplay between magnetic force $Fm$⁠, non-inertial lift force $Fl$⁠, and interfacial tension force $Fi$⁠. Depending on the force ratios, namely, the lift force ratio $[KL=Fl/Fi$] and magnetic force ratio [$KM=Fm/Fi$], three different regimes are identified. The results also reveal that the FFD trajectory and migration length (⁠$Lmig*$⁠) depend on the net force ratio $[FR, net=Fm+Fl/Fi]$ and magnet location $xm*$⁠. Finally, size-based sorting of FFD is demonstrated, which shows a remarkable improvement in terms of the size range and resolution compared to when the magnetic field is not present.