Shear-induced pair-trajectories of viscous drops in a viscoelastic medium: transition from passing to tumbling

We simulated pair-interactions between viscous drops suspended in a viscoelastic matrix. Simulations revealed a phase plot showing that at small Capillary number Ca, below a critical Weissenberg number Wi drops slide past each other separating in the stream-wise direction, while above this number they rotate around each other in a tumbling/rotating trajectory. We offer a crude physics-based analytical model for this phenomenon, predicated on the premise that the elasticity results in a large region of spiraling streamlines around a single drop which during a pair interaction traps the second drop. We treat the additional tension along the curved streamlines due to the first normal stress difference as an enhancement to the interfacial tension. An approximate force balance between the imposed planar shear flow and the perturbation flow driven by the interfacial tension and elastic tensions around the drop delineates the region of the curved streamlines. Demanding that it be large enough to trap the second drop gives us the scaling between the critical Ca and Wi values defining the phase boundary seen in the simulation. The model based on single drop dynamics is admittedly crude and entirely neglects the direct strong elastic interactions seen between the separating drops, which could play a critical role in determining the trajectories. [NSF Award 2019507, ACI-1548562 (CTS180042)]