Numerical Methods and Analysis for Induced-Charge Electrokinetic Flow with Deformable Interfaces

这个项目是对生物学和微技术中出现的流体动力学基本问题的研究。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场对电解液中电池、小泡和液滴的形状和位置的控制。这些所谓的电动技术是在微型设备和生物应用中操纵颗粒和流体的最常见方法之一。例如，电场被用来诱导细胞和囊泡的形状变化，这被用来推断膜的性质。电场也被用来在膜上形成暂时性的孔，这是一种重要的技术，可以将药物输送或基因治疗的分子加载到细胞中。拟议研究的影响包括开发新的数学模型和数值方法，这将有利于研究生物和工程中的电动现象的科学家和工程师。该项目的另一个影响将是研究生的教育和参与。他们所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做有价值的准备。在电场驱动的离子液体的界面流动过程中，离子云在界面上形成屏蔽云，形成电化学双电层或德拜层。电场既作用于离子云，又产生离子云，并驱动离子云和周围的流体运动。这就是所谓的“诱导荷电动能流”，这是一种重要的应用现象。我们发展了一种快速而准确的混合或多尺度数值方法，将层动力学的渐近分析结合到界面自由边界问题的新的边界积分公式中，从而解决了在薄双层的实际重要极限中此类流动的数值计算的显著困难。当前项目的一个中心主题是开发一种用于膜周围电动流动的混合方法。该算法将把对薄膜上弹性应力和静电应力的高波数或小尺度分量的分析纳入能够处理问题固有的多时间和空间尺度的非刚性方法中。该方法将用于研究液滴、囊泡和细胞的电变形中的典型问题，以及通过电形成和电融合来检验囊泡的制造。数值研究将得到分析研究的补充，这些分析研究将被用来证明现有的电流体动力流动的简化或“集中参数”模型的合理性，并推导出新的模型。研究人员还建议开发一种混合方法来解决离子表面活性剂的问题，它结合了电动流动和可溶性表面活性剂的特点。

This project is an investigation of fundamental problems from fluid dynamics that arise in biology and microtechnology. Its focus is on the development of new mathematical models and efficient numerical methods to study the manipulation by electric fields of the shape and position of cells, vesicles, and drops in electrolytic fluids. These so-called 'electrokinetic techniques' are among the most common methods for manipulating particles and fluids in micro-scale devices and biological applications. For example, electric fields are applied to induce shape changes in cells and vesicles, and this is used to infer membrane properties. Electric fields are also used to form transient pores in membranes, which is an important technique to load cells with molecules for drug delivery or gene therapy. 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 phenomena in biology and engineering. An additional impact of this project will be 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 interfacial flow of an ionic fluid that is driven by an electric field, a screening cloud of ions develops at the interface and forms an electrochemical double layer or 'Debye layer'. The electric field both acts on the ion cloud it induces and drives both it and the surrounding fluid into motion. This is known as 'induced-charge electrokinetic flow', and it is an important phenomenon in applications. We address a significant difficulty in the numerical computation of such flows in the practically important limit of thin double layers, by developing a fast and accurate hybrid or multiscale numerical method that incorporates an asymptotic analysis of the layer's dynamics into a novel boundary integral formulation of the interfacial free boundary problem. A central theme of the current project is the development of a hybrid method for electrokinetic flow about a membrane. The algorithm 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 electrodeformation of drops, vesicles, and cells, and to examine vesicle manufacture by electroformation and coalescence by electrofusion. The numerical investigations will be complemented by analytical studies that will be used to justify existing reduced or 'lumped parameter' models for electrohydrodynamic flow, and to derive new models. The investigators also propose to develop a hybrid method for problems with ionic surfactant, which combines features of both electrokinetic flow and soluble surfactant.