Effects of viscosity ratio and three dimensional positioning on hydrodynamic interactions between two viscous drops in a shear flow at finite inertia

Abstract

Drops driven toward each other by shear at finite inertia follow two distinct types of trajectories. Type I trajectory is similar to the one in Stokes flow where drops slide past each other. However, at finite inertia, drops display a new type II trajectory, where they reverse their paths. Increasing viscosity ratio results in a transition from type II to type I trajectory. The transition is caused by decreased drop deformation and increased alignment with the flow at higher drop viscosity; both decrease the zone of reversed streamlines that accompanies a drop at finite inertia. The transition is delineated in a phase diagram of Reynolds number and viscosity ratio for different capillary numbers. The critical viscosity ratio, where a type II transitions into type I, increases with Reynolds number except at higher capillary numbers, where the critical viscosity ratio shows a slight nonmonotonic variation with Reynolds number. Also, it is nonmonotonic with capillary numbers in that for a fixed Reynolds number, the critical viscosity ratio first increases with increasing capillary number and then decreases. Similar to the Stokes regime, increased viscosity ratio leads to a decreased postcollision cross-stream separation effectively decreasing the shear induced diffusion. Higher viscosity ratio results in an increased separation between drops during encounter, which results in a smaller interaction time. With drops placed initially at different shear planes, drops come under the influence of the reversed flow zone around a single drop that broadens off the central shear plane. Consequently, the trajectory changes from type I to type II as the offset in the vorticity direction increases. The change depends on the initial offset in the shear direction as well. The final displacement in the shear direction varies linearly with the initial offset. The net relative displacement in the shear direction shows a gradual decrease with increasing offset. The net relative displacement in the vorticity direction with increasing offset first increases from a zero value when drops are placed at the same shear plane to a maximum and then decreases. For certain cases, it reaches a negative value.