Numerical methods for non-Newtonian fluid structure interaction problems

本研究项目的重点是描述非牛顿流体与弹性结构相互作用的流动方程的稳定和有效的数值近似。这种流体流动在我们的日常生活中无处不在，从我们体内的血液流动到制药混合和微流体等工业过程中的流动。这一理论研究将为这类系统的数值算法的进一步发展提供坚实的基础。该研究项目还将拓宽非牛顿流体在物理现实环境中数值模拟的数学基础。本研究项目的目标是开发和分析非牛顿流体结构相互作用的稳定和有效的数值格式。对牛顿流体流动问题的数学研究已经有了很大的进展，但迄今为止对非牛顿流体流动问题的研究还很少。本项目研究的系统涉及表示多物理场行为的耦合域。这增加了数值复杂性，因为应力和速度都必须在域中求解，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时达到最佳收敛率以提高效率。由于计算非牛顿流体（特别是粘弹性流体）的精确近似需要大量的未知数，因此需要为这些问题开发有效的求解器。本研究将有助于发展和严格分析非牛顿流体结构相互作用数值方法的稳定性和准确性。

This research project is focused on the stable and efficient numerical approximation of equations describing the flow of a non-Newtonian fluid interacting with an elastic structure. Such fluid flows are ubiquitous in our everyday lives, from the flow of blood in our bodies to flow in industrial processes such as pharmaceutical blending and microfluidics. This theoretical investigation will provide a solid foundation for the further development of numerical algorithms for such systems. The research project will also broaden the mathematical basis for the numerical simulation of non-Newtonian fluid flow in physically realistic settings. The goal of this research project is the development and analysis of stable and efficient numerical schemes for non-Newtonian fluid structure interactions. There has been significant mathematical research for Newtonian fluid flow problems, but to date few investigations of non-Newtonian flows. The systems studied in this project involve coupled domains representing multi-physics behavior. 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 for efficiency while splitting the operators. Because of the large number of unknowns required to compute an accurate approximation of non-Newtonian (in particular viscoelastic) fluids, there is a need to develop efficient solvers for these problems. This research will contribute to the development and rigorous analysis of stability and accuracy properties of numerical methods for non-Newtonian fluid structure interactions.