Numerical Methods and Analysis for Interfacial Flow with Ionic Fluids and Surfactants

这个项目是对流体动力学、复杂流体和可变形物质的基本问题的研究，这些问题出现在生物和微技术的应用中。它的重点是发展新的数学模型和有效的数值方法来研究电解流体中的电场和表面活性剂的流场对液滴、细胞和囊泡的形态和控制。电动技术是在微尺度和生物应用中操纵软颗粒和离子流体的最常用方法之一，可以诱导细胞和囊泡的形状变化，从中可以推断膜的性质。它们还可以诱导膜通道和孔形成，用于高级细胞治疗和治疗。表面活性剂用于增强或控制从石油开采到农业、食品和制药加工等行业的各种复杂流体流动。它们在微流体应用中也很重要。所提出的研究的影响包括发展新的数学模型和数值方法，这将有利于科学家和工程师研究生物和工程中的电动力学和表面活性剂现象。这个项目的另一个影响是研究生的教育和参与。他们接受的跨学科培训将为他们在数学和科学领域的一系列职业生涯提供宝贵的准备。在表面活性剂作用下的流体-流体界面变形过程中，界面与体之间的表面活性剂交换发生在界面附近的薄层中，该薄层的动力学控制着界面表面张力和形状。在离子流体在电场驱动下的界面流动过程中，在界面处形成一层薄薄的离子云，形成电化学双层或“德拜层”。电场诱导离子云、界面和周围流体的运动。表面活性剂对界面和流动动力学的影响以及离子流体中“诱导电荷电动力学流动”的发生都是具有广泛应用的重要现象。本项目通过开发快速准确的“混合”或多尺度数值方法，将层动力学的渐近分析纳入边界积分或类似的界面自由边界问题的公式，解决了在薄表面活性剂交换层和薄电双层的实际重要极限中这种流动的数值计算困难。目前项目的中心主题是发展一种关于可变形膜的电动流动的混合方法，以及研究表面活性剂和离子表面活性剂对界面流动影响的快速和精确的数值方法。电动流的算法将把膜上弹性和静电应力的高波数或小尺度分量的分析纳入一种非刚性方法，能够处理问题中固有的多个时间和空间尺度。该方法将用于研究液滴、囊泡和细胞变形中的典型问题，并探索膜和离子流体性质之间的相互作用。在含有表面活性剂的流体中，混合数值方法将被开发出来，以精确处理具有紧密界面相互作用的多个连接域，例如发生在多个液滴中，以及内部流体中存在表面活性剂，这两种情况都与大量数值挑战相关。该项目还涉及快速或加速算法的开发，包括可溶表面活性剂的3D流滴，以及对边界积分方法的稳定性和收敛性的基本分析，这是这项工作的基石。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is an investigation of fundamental problems in the dynamics of fluids, complex fluids, and deformable matter that occur in applications to biology and microtechnology. Its focus is on the development of new mathematical models and efficient numerical methods to study the morphology and control of drops, cells, and vesicles by electric fields in electrolytic fluids and by flow fields with surfactant. Electrokinetic techniques are among the most common methods for manipulating soft particles and ionic fluids in micro-scale and biological applications and can induce, for example, shape changes in cells and vesicles, from which membrane properties can be inferred. They can also induce membrane channel and pore formation for advanced cell treatment and therapy. Surfactants are used to enhance or control a wide range of complex fluid flows that occur in industries ranging from oil extraction to agriculture, food, and pharmaceutical processing. They are also important in microfluidic applications. Impacts of the proposed research include the development of new mathematical models and numerical methods that will be of benefit to scientists and engineers studying electrokinetic and surfactant phenomena in biology and engineering. An additional impact of this project is the education and involvement of graduate students. The interdisciplinary training they receive will be valuable preparation for a range of careers in mathematics and science.During the deformation of a fluid-fluid interface with surfactant, surfactant exchange between the interface and bulk occur in a thin layer adjacent to the interface, and the layer's dynamics control interfacial surface tension and shape. During the interfacial flow of an ionic fluid that is driven by an electric field, a thin screening cloud of ions develops at the interface to form an electrochemical double layer or 'Debye layer'. The electric field induces motion in the ion cloud, the interface, and the surrounding fluid. The influence of surfactants on interface and flow dynamics and the occurrence of 'induced-charge electrokinetic flow' in an ionic fluid are both important phenomena in a wide range of applications. This project addresses a difficulty in the numerical computation of such flows in the practically important limit of thin surfactant exchange layers and thin electrical double layers, by developing fast and accurate 'hybrid' or multiscale numerical methods that incorporate an asymptotic analysis of the layer dynamics into a boundary integral or similar formulation of the interfacial free boundary problem. Central themes of the current project are the development of a hybrid method for electrokinetic flow about a deformable membrane and the investigation of fast and accurate numerical methods for studying the influence of surfactant and ionic surfactant on interfacial flow. The algorithm for electrokinetic flow will incorporate an analysis of the high-wavenumber or small-scale component of the elastic and electrostatic stresses on a membrane into a nonstiff method that is capable of handling the multiple time and space scales inherent in the problem. The method will be used to study canonical problems in the deformation of drops, vesicles, and cells, and to explore the interaction between membrane and ionic fluid properties. In the context of surfactant-laden flows, the hybrid numerical method will be developed to accurately handle multiply connected domains with close interaction of interfaces, such as occurs with multiple drops, and the presence of surfactant in the interior fluid, both of which are associated with substantial numerical challenges. This project also involves the development of fast or accelerated algorithms, including for drops in 3D flow with soluble surfactant, as well a fundamental analysis of the stability and convergence of the boundary integral methods that are a cornerstone of this work.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.