Collaborative Research: Phase-field models, algorithms and simulations for multiphase complex fluids

两种或两种以上不混溶的粘性和/或复杂流体组分的混合物广泛用于许多科学和工程应用中，特别是在设计涉及聚合物、复合材料、凝胶、液晶等的先进材料中。预计所提出的模型和数值方法/模拟将有助于更好地理解与多相复杂流体相关的复杂物理和数学问题，并为先进材料的设计以及复杂流体的流变学和流体动力学性质提供有价值的信息。拟议的研究还将为本科生和研究生提供宝贵的机会，从事与生物和工程材料系统密切相关的跨学科研究，学习计算和应用数学的关键技能，并开发最先进的科学和工程应用数值工具。多相复杂流体混合物的流动通常涉及微观结构的耦合，界面形态和宏观流体力学。这些非线性耦合的复杂性提出了许多数学建模和算法开发，数值分析和实施的挑战。本研究旨在克服这些挑战，为耦合微结构、运动物质界面和流体动力学的非线性多相复杂流体系统设计高效准确的数值算法。为应对这些数字挑战所做的努力很少。该项目将产生满足离散能量耗散规律的数值格式，该格式允许大的时间步长和可控误差，并准确地捕获界面动力学。此外，开发的预测工具和数值模拟将扩展数学分析和数值代码对当前感兴趣的物理问题的适用性，并有助于更好地理解紧迫的科学和工程应用。

Mixtures of two or more immiscible viscous and/or complex fluid components are widely used in many science and engineering applications, in particular, in designing advanced materials involving polymers, composites, gels, liquid crystals, etc. It is expected that the proposed models and numerical methods/simulations will contribute to a better understanding of the complex physical and mathematical issues related to multiphase complex fluids, and provide valuable information for the design of advanced materials and on the rheological and hydrodynamic properties of complex fluids. The proposed research will also provide valuable opportunities for undergraduate and graduate students to engage in interdisciplinary research with strong ties to biological and engineering material systems, to learn critical skills of computational and applied mathematics, and to develop state-of-the-art numerical tools for science and engineering applications.Flows of multiphase complex fluid mixtures usually involve the coupling of microstructures, interfacial morphology and macroscopic hydrodynamics. The complexity of these nonlinear couplings presents many mathematical challenges for modeling and algorithm development, numerical analysis and implementation. The proposed research aims at overcoming these challenges to design efficient and accurate numerical algorithms for nonlinear multiphase complex fluid systems that couple the microstructure, moving material interfaces and hydrodynamics. Very few efforts have been made to address these numerical challenges. This project will result in numerical schemes which satisfy discrete energy dissipation laws, and which allow large time steps and controllable error and capture the interfacial dynamics accurately. In addition, the developed predictive tools and numerical simulations will extend the applicability of mathematical analysis and numerical codes to physical problems of current interest, and contribute to a better understanding of pressing science and engineering applications.