Stationary dimpled drops under linear flow

Abstract

The axially symmetric deformation of a drop in a viscous fluid, under the influence of an externally imposed flow having simultaneous rotating and compressional or extensional components, is addressed. In the previous studies, two families of stationary drop shapes were constructed by simulating the dynamics of drop deformation: stable singly connected shapes with respect to axisymmetric disturbances, and unstable toroidal shapes. These two branches coexist at the same flow conditions, but were not connected. In this study, we obtain a new family of branches of unstable highly deformed stationary drops connecting with the stable flattened shapes and the toroidal ones. We use a method based on classical control theory. The controller is designed for a two-state dynamic model of the system and is employed on a high-order nonlinear dynamic model of the drop deformation. Through this feedback-control-centred approach, an extended collection of unstable stationary solutions is constructed, which spans the range from the loss of stability to the dimpled shapes almost collapsed at the centre. In the latter region, which was never obtained in previous studies, a multiplicity of solutions is identified.