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This thesis is concerned with time-dependent free-boundary problems at low Reynolds numbers. The primary objective is to use a combined experimental and numerical investigation to examine the deformation and breakup characteristics of a single phase Newtonian liquid drop suspended in a second immiscible Newtonian fluid undergoing a prescribed linear flow. Two related studies grew naturally out of this work and are also discussed here: (1) an analytic and numerical examination of the behaviour of concentric double emulsion droplets in linear flows, and (2) an introduction to the effect surfactants have on drop deformation in extensional flows.

Specifically, the breakup of a Newtonian liquid drop is studied under well defined flow conditions. Experiments are performed in a computer-controlled four-roll mill and a detailed numerical investigation using the boundary integral method is presented. Particular attention is given to the dynamics of drop breakup and many experimental and numerical examples are shown of the actual fragmentation process. Transient effects associated both with nonspherical initial shapes and time-dependent velocity gradients are studied. Although the time-dependent velocity fields are limited to step changes in flow conditions, these investigations provide valuable insight into the breakup phenomenon and are a necessary first step toward understanding more complicated flow situations. The effects of viscosity ratio, flow-type and capillary number are discussed thoroughly. Overall, these studies of drop breakup provide a nice illustration of the influence of an interfacial-tension-driven flow that arises because of curvature variations along the fluid-fluid interface (the interfacial tension is constant). The interaction of this interfacial-tension-driven flow with the prescribed time-dependent velocity field produces very interesting breakup processes, often without large scale stretching of the drop. Also, for highly elongated drops, finite-amplitude capillary waves are observed experimentally and numerically.

The study of drop deformation and breakup leads naturally to consideration of other systems involving deformable miscrostructures. Double emulsion droplets, which arise in applications involving liquid membranes, are frequently treated by drawing analogies with the known behaviour of single phase droplets. We present a fundamental investigation of concentric double emulsion drops in extensional flows. The analytic results allow calculation of the first effects of flow-induced deformation and the effective viscosity of a dilute emulsion of these particles. The analysis suggests interesting deformation and interaction of the two drop surfaces so a numerical investigation of the finite deformation of these particles is described. The critical capillary number for breakup is determined, the dependence on physical and flow parameters is outlined and possible mechanisms for breakup are discussed that differ from the single phase droplet results. Finally, the effect of different flow-types, i.e., uniaxial or biaxial extensional flows, is shown in some instances to suggest breakup of the inner droplet even though the outer droplet maintains a steady shape.

The thesis closes with an introduction to the effect surface-active agents have on drop deformation. Because the distribution of surfactant along the fluid-fluid interface produces interfacial tension variations, the calculation of the drop shape as a function of time is intimately coupled with the convection and diffusion of surfactant along the drop surface. We present an approximate numerical procedure to study finite deformations and surfactant transport. The results are not extensive but suggest several interesting aspects of the deformation of drops due to the presence of surfactant.

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