Free-surface flows, and in particular, their tendency to break into smaller drops, or merge to form larger drops are common aspects in several industrial contexts, like printing applications, coating flows, sintering processes, electronics and drug delivery, cell and tissue engineering, and multi-phase flows. Apart from such obvious industrial significance, they are familiar to anyone who has witnessed a dripping faucet, or raindrops falling on the windshield of an automobile. Underneath this veil of inherent familiarity are a whole slew of unexpected dynamics that characterize these processes. Coalescence and breakup, are two prominent examples of finite time singularities which occur owing to the dramatic changes in topology -- two initially disconnected masses merge and become one, in the former case; and a contiguous mass of liquid disrupts to form two or more daughter droplets, in the latter. The general scope of my dissertation study is to understand these processes -- especially, the rich nonlinear behavior in the vicinity of the singularity, at extremely small length scales that lie in the limit of the continuum approximation. For this study, powerful and well-benchmarked Arbitrary Lagrangian-Eulerian (ALE) algorithms based on the Galerkin/Finite Element Method (G/FEM) to numerically solve either the 1--D slender jet or the 3--D axisymmetric Navier-Stokes equations are developed. Some of the most significant results in this dissertation include the answer to the formation of beads--on--a--string structures in the breakup, and an estimation of the extensional viscosity, of a thinning viscoelastic filament. Another significant result is the discovery of a universal asymptotic initial regime of drop coalescence, where contrary to common knowledge, all three forces -- inertial, viscous and capillary -- are important. The true significance of this discovery is the similarity in the vicinity of the singularity, of drop breakup and coalescence.
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