Drop deformation and breakup in a vortex at finite inertia

Abstract

Deformation and breakup of a viscous drop in a potential vortex are numerically simulated. Capillary number, Reynolds number, and viscosity and density ratios are varied to investigate their effects on the drop dynamics. The vortex locally gives rise to an extensional flow near the drop with the axis of extension rotating at a constant rate, as the drop revolves around the vortex centre. The rotation of the axis plays a critical role in the competing dynamics between the flow-induced stretching and the interfacial tension. The relation between the rotating extensional flow and a shear flow is explored. For low capillary numbers, a periodic state is reached, where the drop deforms into an ellipsoidal shape and undergoes steady rotation with a distinct phase lag behind the imposed flow. For density-and-viscosity-matched drops, increased interfacial tension results in decreased deformation and reduced phase lag. Increased inertia promotes deformation. In the presence of inertia, decreasing capillary number leads to a negative phase lag. The rotation of the extension axis inhibits deformation at low values of the Reynolds number. But at high Reynolds numbers, rotation-induced centrifugal forces promote deformation. At low and high viscosity ratios, an increase in viscosity ratio leads to enhancement and reduction in deformation, respectively. At density ratios larger than unity, the drop deformation displays resonance in that it varies non-monotonically with a distinct peak with variation of interfacial tension and density ratio. The peak corresponds to the natural frequency of the drop deformation matching with the frequency of rotation due to the vortex. A simple physical model is used to explain various observations including asymptotic scalings. We also explore different mechanisms for drop breakup at different Reynolds number, and provide critical capillary numbers as functions of other parameters. In particular, vortex-induced resonance offers an alternative mechanism for size-selective drop breakup. Details of flow fields and transients are also presented and discussed.