Flows of Free Liquid Fibers and Sheets

这个项目涉及在内力和外力的影响下，粘性、粘弹性和粘塑性流体的自由液丝、射流和薄片的形成和演化。这些流体流动的公式涉及一个未知的自由面，该自由面描述了被流体占据的环境介质中的区域。了解这些自由表面是如何演变的，什么样的流动场景和材料参数容易引发流动不稳定性，以及如何过渡到流动灾变(如毛细管破裂)，这是数学和科学中的一个根本挑战。已知毛细管力、惯性力和粘弹力或其平衡力可以稳定或破坏流体细丝的稳定。在高粘度流体的情况下，液体细丝在发生解体和形成液滴之前是均匀变细的，而相反地，粘弹性预计会导致液体细丝的串珠形态。导致稀疏、缩颈、分手或其他瞬时模式的一连串事件的内在非线性性质和高度复杂性是本项目要解决的主题。它包括对细丝和薄片的平均低维方程的严格推导和证明，这些方程表明了这些模型的有效性边界，讨论了毛细管变薄和粘塑性流体射流的破裂，以及粘性和粘弹性液体的纤维纺丝和薄膜拉伸的稳定性研究。分析和证明纤维和薄膜流动的数学模型，并解释在毛细管变薄和工程应用中出现的观察到的流动现象，需要开发新的数学工具，结合偏微分方程组、稳定性理论、渐近分析和多尺度分析。各种高科技应用，包括食品加工、喷墨打印、涂料和气雾剂的雾化、喷射稳定化、纤维纺丝和纳米纤维的静电纺丝，都具有相似的目标：要么以受控的方式诱导细流体细丝(或薄片)的破裂，要么抑制它，或者防止流动不稳定的发生。为了达到这些目标，有必要对控制不稳定性开始和向流动灾变过渡的物理机制和流变参数有一个深刻的数学理解。研究人员开发了数学框架，以推导、分析和证明描述自由液体细丝和薄片演变的流动模型。正在研究的是流动不稳定性的发生，向破碎的转变，以及在几个典型流动中的新的流变现象。这项研究有可能极大地提高我们对管理这些重要工程和科学领域的流动的物理原理和限制的知识。该项目还旨在为研究生和本科生提供数学流体力学和一般应用分析方面的培训，特别是纤维/板材成型流程方面的培训。

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.