Flows of Free Liquid Fibers and Sheets

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This project is concerned with the formation and evolution of free liquid filaments, jets and sheets of viscous, viscoelastic, and viscoplastic fluids under the influence of internal and external forces. The formulation of these fluid flows involves an unknown free surface that describes the region in an ambient medium occupied by the fluid. Understanding how these free surfaces evolve, what flow scenarios and material parameters are prone to trigger flow instabilities, and how the transition to flow catastrophes such as capillary breakup proceeds is a fundamental challenge in mathematics and the sciences. Capillary, inertial and viscoelastic forces or balances thereof are known to stabilize or destabilize fluid filaments. In the case of highly viscous fluids, liquid filaments thin uniformly before breakup occurs and drops are formed, while in contrast viscoelasticity is expected to induce a beads-on-string morphology of the liquid filaments. The inherently nonlinear nature and the high degree of complexity of the cascade of events leading to thinning, necking, breakup or other transient patterns are topics that this project addresses. It encompasses a rigorous derivation and justification of averaged lower-dimensional equations for thin filaments and sheets indicating boundaries of validity of these models, a discussion of capillary thinning and breakup of viscoplastic fluid jets, and stability studies in the context of fiber spinning and film drawing for viscous and viscoelastic liquids. Analyzing and justifying mathematical models of fiber and film flows and explaining observed flow phenomena arising in capillary thinning and engineering applications requires the development of new mathematical tools, combining machinery from partial differential equations, stability theory, asymptotic analysis and multiscale analysis. A variety of high-tech applications, including food processing, ink-jet printing, atomization of paints and aerosols, jet stabilization, fiber spinning, and electrospinning of nanofibers, are all characterized by similar objectives: to either induce breakup of thin fluid filaments (or sheets) in a controlled manner, to suppress it or to keep flow instabilities from occurring. To reach these objectives, a deep mathematical understanding of the physical mechanisms and rheological parameters governing the onset of instabilities and the transition to flow catastrophes is necessary. The investigator develops the mathematical framework to derive, analyze, and justify flow models describing the evolution of free liquid filaments and sheets. Under examination are the occurrence of flow instabilities, the transition to breakup, and novel rheological phenomena in several exemplary flows. This research has the potential to significantly enhance our knowledge of the physical principles and limitations governing flows in these vital areas of engineering and science. The project also serves the purpose to train both graduate and undergraduate students in mathematical fluid mechanics and applied analysis in general and fiber/sheet forming flows in particular.