Topics in Mathematical Theories of Elastic Complex Fluids

复杂流体是既具有固体性质又具有粘性液体性质的粘弹性材料。这些材料在我们的日常生活中非常丰富，包括各种各样的混合物、聚合物溶液、胶体分散体、生物流体、电流变流体、液晶和液晶聚合物。与固体和简单液体不同，复杂流体的模型方程随着新的实验证据的出现而不断发展。我们提出研究的体系由聚合物组成，由于内部弹性形态和动力学而诱导微观结构。这些材料的数学描述和自洽理论的多尺度、多物理性质需要结合非线性偏微分方程、变分法、渐近展开、动力学理论和统计物理等工具。我们建议发展数学理论来理解这些迷人的材料。重点是了解微观结构在动力学输运和诱导弹性应力之间的特殊耦合中的作用。复杂流体表现出许多复杂的流变学和流体动力学特征，这些特征对生物和工业过程非常重要。应用包括注射表面活性剂治疗气道闭合性疾病；喷墨打印机喷口、喷油器、灭火器的聚合物添加剂；结构振动的磁流变阻尼等。需要新的数学描述和自洽理论来解决诸如分子间和扭曲弹性相互作用，它们与流体力学和外加电场或磁场的耦合等问题。本提案致力于发展新的数学理论，用于数值算法的建模、分析和设计，以了解这些重要的材料。

Complex fluids are viscoelastic materials that posses both solids and viscous liquids properties. These materials are abundant in our daily life, including wide varieties of mixtures, polymeric solutions, colloidal dispersions, biofluids, electro-rheological fluids, liquid crystals and liquid crystal polymers. Unlike solids and simple liquids, the model equations for complex fluids continue to evolve as new experimental evidence become available. The systems that we propose to study consist of polymers with induced microstructure due to internal elastic morphology and dynamics.The multiscale, multi-physics nature of the mathematical descriptions and self-consistent theories of these materials requires the combination of tools from nonlinear PDE, calculus of variations, asymptotic expansion, kinetic theory and statistical physics. We propose to develop the mathematical theories to understand these fascinating materials. The main focus is on understanding the role of microstructures in the special coupling between the kinetic transport and the induced elastic stresses.Complex fluids exhibit many intricate rheological and hydrodynamic features that are very important to biological and industrial processes. Applications include the treatment of airway closure disease by surfactant injection; polymer additive to jets in inkjet printers, fuel injection, fire extinguishers; magneto-rheological damping of structural vibrations etc. New mathematical descriptions and self-consistent theories are needed to resolve problems such as the intermolecular and distortional elastic interactions, their coupling to hydrodynamics and applied electric or magnetic fields. This proposal is devoted to develop new mathematical theories to use in the modeling, analysis and the designing of numerical algorithms, in order to understand these important materials.