Numerical Approximations of Non-Newtonian Fluid Flows with Applications

本文主要研究物理应用中非牛顿流体流动的数值逼近问题。这种流体流动在我们的日常生活中大量存在，从我们体内的血液流动到塑料等聚合物材料的生产。该项目考虑了两个原型问题：(I)缺陷边界条件的最优控制；(Ii)弹性介质中的非牛顿流动。血液流动是与非牛顿流体与弹性血管壁相互作用等情况有关的最重要的例子之一，在这种情况下，每个流入和流出边界上只指定了流量或平均压力。本研究中的模型问题要么涉及表示多物理行为的耦合区域，要么涉及耦合状态-伴随系统。这增加了数值复杂性，因为应力和速度都必须在区域中求解，并且控制方程之间的强相互作用要求求解算法在分离算子的同时实现最佳收敛速度。此外，由于需要近似的未知数很多，因此需要开发有效的解算器来解决这些问题。所提出的研究涉及解耦方案及其稳定性和收敛问题。研究的主要贡献是发展了耦合系统中非牛顿流动的稳健数值格式，并对非牛顿流动的最优控制进行了分析和数值研究。关于牛顿流动的多学科问题已经有了广泛的研究，但到目前为止，对非牛顿流动的数学和数值研究还远远落后。由于许多重要的生物和工程过程涉及非牛顿流体流动，在这些应用中对数学支持有很大的需求。该研究为物理环境下非牛顿流体流动问题的数值模拟拓展了数学基础。此外，这项研究还为重要过程的数值模拟提供了改进的算法，从而使生物医学和聚合物行业受益。

This research is focused on numerical approximation of non-Newtonian fluid flows in physical applications. Such fluid flows are abundant in our everyday lives, from the flow of blood in our bodies to the production of polymeric material such as plastics. There are two prototypal problems considered in the project: (i) optimal control for defective boundary conditions, and (ii) non-Newtonian flow within an elastic medium. Blood flow is one of most important examples related to such situations as a non-Newtonian flow interacts with an elastic vessel wall, where only flow rate or mean pressure is specified on each inflow and outflow boundary. The model problems in this research involve either coupled domains representing multi-physics behavior or coupled state-adjoint systems. This increases the numerical complexity as both stress and velocity must be resolved in the domains, and the strong interaction between the governing equations requires solution algorithms that achieve optimal convergence rates while splitting the operators. Additionally, because of the large number of unknowns to be approximated, there is a need to develop efficient solvers for these problems. The proposed research addresses issues on decoupling schemes, and their stability and convergence. The primary contribution of the research is the development of robust numerical schemes for non-Newtonian flows in coupled systems, and analytical and numerical study of optimal control for non-Newtonian flows.There have been extensive studies on multidisciplinary problems involving Newtonian flows, but to date mathematical and numerical investigations of non-Newtonian flows are still far behind. Because of the many important biological and engineering processes involving non-Newtonian fluid flow, there is a great demand for mathematical support in these applications. The proposed research broadens the mathematical basis for the numerical simulation of non-Newtonian fluid flow problems in physical settings. Also the research benefits biomedical and polymer industries by providing improved algorithms for the numerical simulation of important processes.