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

这一建议是为了研究流体力学中的基本自由边界问题，这些问题是由不同的应用驱动的，但具有相似的数学特征。第一类问题涉及表面活性剂在非混相流体的大量流动中的溶解度对液滴、气泡和射流的变形和破裂的影响。表面活性剂在整体流动中的缓慢扩散（或大的佩莱特数）导致表面活性剂的浓度在靠近非混相流体界面的薄层上发生大的梯度，这带来了一个重大的困难。为了确定界面的动力学性质，必须精确地求解界面附近的表面活性剂梯度，这对传统的数值方法提出了重大挑战。在这个项目中，层的长细将用于开发快速和准确的“混合”数值方法，将层的动力学的单独分析还原纳入界面自由边界问题的数值解中。第二类问题涉及细长的弹性纤维或长丝在流体界面受到毛细力影响时的变形。本项目将开发一种鲁棒和高效的数值方法，使用细长体理论的一维积分方程来描述纤维的动力学，并考虑流体，长丝和自由表面之间的相互作用。表面活性剂，如醇、硫酸盐和洗涤剂，广泛用于控制化学和制药工业中流体系统的乳化或混合动力学。在基本层面上，乳化过程是通过单个液滴和射流的行为发生的，因为它们在紧张流动中破裂或尖端流动。该项目旨在确定在实践中很重要的破裂模式的细节，但迄今为止难以通过传统的数值方法准确捕获。同样，细长的弹性细丝和悬臂的粘着和粘着对于特定的微机电装置的正常运行是重要的，而细丝-流体界面力正在被研究作为将细丝组装成更大结构的手段，如纳米管环和生物细丝结构。该项目旨在通过将小型结构的解析技术纳入快速、准确、可靠的数值方法，阐明在这些过程中发生的基本影响。

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.