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。该研究项目将为多相复杂流体的流体动力学开发一套数学结构。复杂流体与粘性流体(如水、油)的不同之处在于，即使在最简单的实验中，它们也需要解析微观结构来解析行为。对于单相复杂流体，粘性流体的Navier-Stokes模型没有捕捉到剪切变稀(粘度随剪切速率的增加而下降)和剪切中法向应力的产生(平行板反向平移时产生沿其相互法线的力)的特征现象。然而，用单相聚合物液体的动力学理论成功地预测了这些特征。当混合在一起形成多相混合物时，无论是在自然界(生物膜或肺部粘膜层)还是在合成(纳米棒或纳米血小板分散在聚合物基质中)，不同的流体相容易分离。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，这只被合理地理解为平衡状态。在河流或管道中的生物膜、通过协调的纤毛和潮气呼吸将肺部呼吸道粘液层推向喉部以及将聚合物纳米复合材料流动加工成薄膜或模具的典型情况下，出现了突出的挑战，预测工具尚不存在。在这项研究项目中，将发展一个数学上一致的通用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。只有当伴随着用于推导适用于基准实验的简化模型的明确协议、每个模型简化的数值算法以及通过盲目实验来测试理论预测能力的直接模拟时，动力学理论才是有用的。这些结构将与推理方法一起开发，以便能够通过实验确定多相复杂流体模型的所有物理参数。该理论的一般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力学的详细特异性来证明。聚合物纳米复合材料是一种非常有前途的新型合成材料，由传统聚合物与性能增强的纳米棒或小片组成的鸡尾酒组成。从一个简单的事实就可以洞察纳米复合材料真正特殊的原因：在聚合物基质的单个雨滴中体积分数为1%的纳米血小板引入了整个足球场的新表面积！颗粒的数量和大小以及颗粒相和聚合物相之间新的表面接触的新特征压倒了目前的实验和理论能力。流动不可能在实验上探测(颗粒太小，太多，无法跟踪方向和位置)，也没有预测理论，因此也没有模拟工具来指导材料设计。该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部呼吸道中空气和组织之间的粘膜层。这项研究项目将为一般多相复杂流体混合物的流体动力学理论和模拟开发一张数学蓝图。在这一策略中，混合物中每一相的细节以及相之间的化学和物理相互作用作为输入，以及检验该理论所需的实验数据。这种形式主义将产生专门针对多相流体的理论，为基准实验推导模型简化的方法，以及数值方法和模拟。一般的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领域提供预测工具：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；以及通过模拟药物和物理治疗之前和期间的粘液传输来改善肺部健康。

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.