Numerical methods for non-Newtonian fluid structure interaction problems

该研究项目的重点是描述与弹性结构相互作用的非牛顿流体流动的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工业过程中的流动，如药物混合和微流体。这一理论研究将为此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物理现实环境中非牛顿流体流动数值模拟的数学基础。该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方案。牛顿流体流动问题的数学研究已经取得了很大的进展，但对非牛顿流体流动的研究却很少。在这个项目中研究的系统涉及代表多物理行为的耦合域。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需要求解算法，该算法在分裂算子的同时实现最佳收敛速率以提高效率。由于需要大量的未知数来计算非牛顿（特别是粘弹性）流体的精确近似，因此需要开发有效的求解器来解决这些问题。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的数值方法的稳定性和精度特性。

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.