Collaborative Research on Mathematical Constructs for Multiphase Complex Fluids

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本研究计划将发展一套多相复杂流体流体动力学的数学建构。复杂流体与粘性流体（如水、油）的区别在于，即使在最简单的实验中，复杂流体也需要微观结构的解析来解析其行为。对于单相复杂流体，Navier-Stokes粘性流体模型没有捕捉到剪切变薄（粘度随剪切速率的增加而下降）和剪切中产生的法向应力（相反平移的平行板沿其相互法线受到一个力）的特征现象。然而，这些特征已经被单相聚合物液体的动力学理论成功地预测了。当结合形成多相混合物时，无论是天然的（生物膜或肺气道粘膜层）还是合成的（分散在聚合物基质中的纳米棒或纳米血小板），不同的流体相都容易分离。其他作用力（化学键和较弱的吸引势）与相分离竞争以维持混合物，这些作用力只能合理地理解为平衡状态。突出的挑战出现了，而且预测工具还不存在，在远离平衡的条件下，典型的生物膜在溪流或管道中，肺部气道黏液层被协调的纤毛和潮汐呼吸推进到喉部，以及聚合物纳米复合材料在薄膜或模具中的流动处理。本研究项目将建立一个数学上一致的多相复杂流体的动力学理论，包括单个相的物理和化学、它们的混合物和它们的流体动力学。只有当动力学理论有明确的协议来推导适用于基准实验的简化模型，每个模型简化的数值算法，以及通过盲实验直接模拟来测试理论的预测能力时，动力学理论才有用。这些结构将与推理方法一起开发，以便多相复杂流体模型的所有物理参数都可以通过实验确定。该理论的普遍性和多样性将通过对生物膜、粘膜层和聚合物纳米复合材料的流体动力学的详细特异性来证明。聚合物纳米复合材料是一种具有非凡前景的新型合成材料，由传统聚合物和增强性能的纳米棒或血小板混合而成。纳米复合材料的特殊之处可以从一个简单的事实中了解到：单个聚合物基质雨滴中1%的纳米片体积分数会引入整个足球场的新表面积！颗粒数量和尺寸的新特征以及颗粒相和聚合物相之间新的表面接触超出了现有的实验和理论能力。流动是不可能通过实验来探测的（粒子太小，太多，无法跟踪方向和位置），也没有预测理论，因此没有模拟工具来指导材料设计。这个项目将发展必要的理论和计算能力。自然界中产生的多相复杂流体的混合物也存在类似的挑战和限制，例如溪流、池塘和管道中的生物膜以及肺气道中空气和组织之间的粘膜层。本研究计划将为一般多相复杂流体混合物的流体力学理论与模拟发展一个数学蓝图。在这个策略中，混合物中每个相的细节以及相之间的化学和物理相互作用作为输入，以及测试理论所需的实验数据。形式主义将产生多相流体的特定理论，为基准实验推导模型约简的方法，以及数值方法和模拟。一般方法将适用于三种不同的多相材料：生物膜、粘膜层和聚合物纳米复合材料。这些数学结构将为以下方面提供预测工具：用于国防和航空航天工业的新型高性能聚合物纳米复合材料的设计；工业管道中生物膜的修复策略并且，通过模拟药物和物理治疗之前和期间的粘液运输来改善肺部健康。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This research project will develop a suite of mathematical constructs for the hydrodynamics of multiphase complex fluids. Complex fluids are distinguished from viscous fluids (e.g., water, oil) in that they require resolution of microstructure to resolve behavior in even the simplest of experiments. For single-phase complex fluids, the signature phenomena of shear thinning (viscosity falls with increased shear rate) and normal stress generation in shear (oppositely translating parallel plates experience a force along their mutual normal) are not captured by the Navier-Stokes model for viscous fluids. Yet, these features are successfully predicted by the kinetic theory of single-phase polymeric liquids. When combined to form multiphase mixtures, either in Nature (biofilms or lung airway mucosal layers) or synthetically (nano-rods or nano-platelets dispersed in a polymer matrix), the different fluid phases are prone to separate. Other forces (chemical bonds and weaker attractive potentials) compete with phase separation to sustain the mixture, which are only reasonably understood for equilibrium states. Outstanding challenges arise, and predictive tools do not yet exist, in far-from-equilibrium conditions typical of biofilms in streams or pipes, lung airway mucus layers propelled toward the larynx by coordinated cilia and tidal breathing, and flow processing of polymer nano-composites into films or molds. A mathematically consistent kinetic theory for generic multiphase complex fluids, incorporating the physics and chemistry of individual phases, their mixtures and their hydrodynamics, will be developed in this research project. A kinetic theory is only useful when accompanied by clear protocols for the derivation of reduced models applicable to benchmark experiments, numerical algorithms for each model reduction, and direct simulations to test the predictive capability of the theory with blind experiments. These constructs will be developed, along with inference methods so that all physical parameters of a multiphase complex fluid model can be experimentally determined. The generality and diversity of the theory will be demonstrated by detailed specificity to hydrodynamics of biofilms, mucosal layers, and polymer nano-composites. Polymer nano-composites are new synthetic materials with extraordinary promise, consisting of a cocktail of a traditional polymer with property-boosting nano-rods or platelets. Insight into why nano-composites are truly special can be appreciated from a simple fact: one percent volume fraction of nano-platelets in a single raindrop of polymer matrix introduces an entire football field of new surface area! The novel features of number and size of particles together with new surface contact between the particle phase and the polymer phase overwhelm current experimental and theoretical capabilities. Flow is impossible to probe experimentally (particles are too small and too numerous to track orientation and position) and there is no predictive theory, and therefore no simulation tools, to guide material design. This project will develop the requisite theoretical and computational capabilities. Similar challenges and limitations exist for mixtures of multiphase complex fluids arising in Nature, such as biofilms in streams, ponds, and pipes and mucosal layers between the air and tissue in lung airways. This research project will develop a mathematical blueprint for theory and simulation of the hydrodynamics of generic multiphase complex fluid mixtures. In this strategy, details of each phase in the mixture and the chemical and physical interactions between phases serve as inputs, together with the necessary experimental data to test the theory. The formalism will produce the theory specific to the multiphase fluid, the approach to derive model reductions for benchmark experiments, and numerical methods and simulations. The general methods will be brought to bear on three diverse multiphase materials: biofilms, mucosal layers, and polymer nano-composites. These mathematical constructs will provide predictive tools for: the design of novel high performance polymer nano-composite materials for defense and the aerospace industry; remediation strategies for biofilms in industrial pipelines; and, improved pulmonary health through simulations of mucus transport prior to and during drug and physical therapies.