Pair interactions between viscous drops in a viscoelastic matrix in free shear: Transition from passing to tumbling trajectories

Abstract

Shear-induced pair interactions between viscous drops suspended in a viscoelastic matrix are numerically investigated examining the effects of elasticity and drop deformability on their post-collision trajectory. Two different trajectory types are identified depending on the Weissenberg number Wi and capillary number Ca. Drops suspended in a Newtonian matrix (Wi = 0.0) show a passing trajectory where drops slide past each other and separate in the stream-wise direction. However, when increasing the Weissenberg number above a critical value, a tumbling/doublet trajectory is observed where two drops rotate around the midpoint of the line joining their centers, as was also seen previously for rigid particles. The tumbling trajectory is explained by investigating the flow around a single drop in shear. Elasticity generates a larger region of spiraling streamlines around a drop, which, during a pair interaction, traps the second drop giving rise to the tumbling pair. Decreasing deformability (lower Ca) and increasing viscoelasticity (higher Wi) favor a tumbling trajectory. With simulations sweeping the parameter space, we obtain a phase plot of the two different trajectories as functions of Ca and Wi. Treating the tension along the curved streamlines due to the non-zero first normal stress difference in the viscoelastic medium as an enhancement to the interfacial tension, we have developed an approximate force balance model for the zone of spiraling streamlines. It qualitatively captures the observed scaling of the critical Ca and Wi values at the phase boundary. The effects of unequal size, initial configuration, and non-unity viscosity ratio are briefly investigated.