Numerical methods and analysis for interfacial fluid flow with soluble surfactant

该项目研究了由表面活性剂介导流动的应用所引发的基本问题。从数学的角度来看，理论方法有可能适用于更广泛的实际情况。利用这种方法，项目研究人员最近开始开发一种快速、准确的混合数值方法，以研究表面活性剂的溶解度对非混相流体两相流动的影响，这种影响在实际中很重要，但在理论上具有挑战性的限制是大Peclet数或缓慢扩散。表面活性剂通过改变非混相流体之间界面的表面张力来影响流体混合物的动力学，并且在能量上有利于留在界面上。然而，在许多情况下，表面活性剂的缓慢扩散迁移率会对界面动力学产生重要影响，因为表面活性剂可溶于靠近界面而不在界面上的整体流动。本研究的大体积Peclet数限制引入了空间尺度的分离，这对传统的数值方法提出了实质性的挑战。本文所采用的方法的概念基础是将小扩散极限下的解析奇异摄动技术与两相界面流动的快速精确数值方法相结合。这种方法的一个重要优点是，高精度的基于表面的方法，如边界积分法或边界元法，可以适用于表面活性剂溶解度的研究。如果没有研究人员正在开发的治疗方法，这些方法就不适用。该项目预计将开发创新的理论模型和数值方法，用于分析和模拟表面活性剂介导的液滴破碎和可溶性表面活性剂的尖端流动。它将开发新的，快速，高效和准确的数值方法，预计将有助于科学家和工程师研究乳化液的形成和稳定性，以及新兴的微流体应用，从化学处理技术到先进的医疗应用。该项目的另一个重要影响是研究生和博士后的教育和培训。他们在这个项目中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。

This project investigates fundamental problems that are motivated by applications to surfactant-mediated flow. The theoretical approach has the potential to be adapted to a wider variety of practical situations that are similar from a mathematical point of view. Using this approach, the project investigators have recently begun the development of a fast and accurate hybrid numerical method to study the effects of solubility of surfactant on the two-phase flow of immiscible fluids in the practically important but theoretically challenging limit of large Peclet number or slow diffusion. Surfactants influence the dynamics of fluid mixtures by altering the surface tension at interfaces between immiscible fluids and are energetically favored to remain on an interface. However, in many examples the slow diffusional mobility of a surfactant that is soluble in the bulk flow near to but not on an interface can exert an important influence on the interface dynamics. The large bulk Peclet number limit of this investigation introduces a separation of spatial scales that presents a substantial challenge for traditional numerical methods. The conceptual underpinning of the approach taken here combines analytical, singular perturbation techniques in the small diffusion limit with fast and accurate numerical methods for two-phase interfacial flow. An important benefit of this approach is that highly accurate surface-based methods, such as the boundary integral or boundary element method, can be adapted to the study of surfactant solubility. Without the treatment that is under development by the investigators these methods do not apply.The project is expected to develop innovative theoretical models and numerical methods for the analysis and simulation of surfactant-mediated drop breakup and tip-streaming with soluble surfactant. It will develop new, fast, efficient and accurate numerical methods that are expected to be useful to scientists and engineers studying emulsion formation and stability as well as emerging microfluidic applications that range from chemical processing techniques to advanced medical applications. An additional and important impact of the project is the education and training of graduate students and postdoctoral fellows. The interdisciplinary training they receive on this project will be valuable preparation for a range of careers in mathematics and science.