The formation and expansion of a toroidal drop moving in a viscous fluid

Abstract

The deformation of an initially spherical liquid drop moving under the action of gravity in another fluid with which it is completely miscible is investigated under conditions of small values of the drop Reynolds number. It is found experimentally that such a drop evolves into an open torus which subsequently expands, and this phenomenon is examined theoretically for two limiting drop geometries: (i) a slightly deformed spherical drop, and (ii) a highly expanded, slender open torus. Under the assumptions of zero interfacial tension and creeping flow, the theory provides a qualitative description for the initial stages of the drop evolution [case (i)], but is unable to account for the observed drop expansion during latter stages of deformation [case (ii)]. On the other hand, if small inertial effects are retained in the analysis, the theory predicts that a slender open fluid torus possessing an arbitrary cross-sectional geometry will expand without change of shape to first order in Reynolds number. Quantitative comparisons of theoretically predicted rates of expansion with experimental measurements suggest the possible existence of a small, time-dependent interfacial tension across the drop interface.