Analysis and numerical computations of free boundaries in fluid dynamics: surfactant solubility and elastic fibers

这个建议是为调查的基本自由边界问题，在流体力学的动机不同的应用程序，但共享相似的数学特征。 第一类问题涉及表面活性剂在不混溶流体的整体流动中的溶解度对液滴、气泡和射流的变形和破裂的影响。 一个显著的困难是由表面活性剂在本体流中的缓慢扩散（或大的Peclet数）引入的，这导致表面活性剂浓度的大梯度发生在邻近不混溶流体之间的界面的薄层上。 解决界面附近的表面活性剂的梯度必须完成高精度，以确定界面的动态，但传统的数值方法提出了一个重大的挑战。 在这个项目中，细长的层将被用来开发快速和准确的“混合”数值方法，将一个单独的分析减少层的动态到数值解的界面自由边界问题。 第二类问题涉及细长弹性纤维或细丝在流体界面处受毛细力影响时的变形。 本课题将开发一种基于细长体理论的一维积分方程，考虑流体、纤维和自由表面之间的相互作用，来描述纤维动力学的数值方法。表面活性剂，如醇类、硫酸盐和洗涤剂，在化学和制药工业中广泛用于控制流体体系的乳化或混合动力学。 在基本水平上，乳化过程通过单个流体滴和射流的行为发生，因为它们在应变流中通过破裂或尖端流动而破裂。 该项目旨在确定在实践中很重要但迄今难以用传统数值方法准确捕捉的破裂模式的细节。 类似地，细长的弹性细丝和杠杆的粘附和静摩擦在特定的微机电装置的正确操作中是重要的，并且粘性流体界面力正在被研究作为将细丝组装成更大的结构（例如纳米管环和生物粘附结构）的手段。 该项目旨在通过将用于解析小尺度结构的分析技术纳入快速、准确、稳健的数值方法中，来阐明此类过程中发生的基本效应。

This proposal is for the investigation of fundamental free boundary problems in fluid mechanics that are motivated by different applications but share similar mathematical features. The first class of problems concerns the effect that the solubility of a surfactant in the bulk flow of immiscible fluids can have on the deformation and breakup of liquid drops, bubbles, and jets. A significant difficulty is introduced by the slow diffusion (or large Peclet number) of surfactant in the bulk flow, which causes a large gradient in concentration of surfactant to occur across a thin layer adjacent to the interface between immiscible fluids. Resolving the surfactant gradient near the interface must be accomplished with great accuracy to determine the interface's dynamics but presents a significant challenge for traditional numerical methods. In this project, the slenderness of the layer will be used to develop fast and accurate 'hybrid' numerical methods that incorporate a separate analytical reduction of the layer's dynamics into numerical solution of the interfacial free boundary problem. The second class of problems concerns the deformation of a slender elastic fiber or filament when it is influenced by capillary forces at a fluid interface. This project will develop a robust and efficient numerical method that uses one-dimensional integral equations from slender body theory to describe the fiber's dynamics and takes into account the interaction between fluid, filament, and free surface.Surfactants, such as alcohols, sulfates, and detergents, are widely used to control the dynamics of emulsification or blending of fluid systems in applications in the chemical and pharmaceutical industries. At a fundamental level, the emulsification process occurs via the behavior of single fluid drops and jets as they break up by bursting or tip-streaming in a straining flow. This project seeks to determine details of the mode of breakup that are important in practice but have so far been difficult to capture accurately by traditional numerical methods. Similarly, sticking and stiction of slender elastic filaments and cantilevers is important in the proper operation of specific micro-electromechanical devices, and filament-fluid interfacial forces are being investigated as a means to assemble filaments into larger structures, such as nanotube rings and biological filamented structures. This project seeks to elucidate fundamental effects that occur in such processes by incorporating analytical techniques for the resolution of small-scale structures into fast and accurate, robust numerical methods.