Mathematical modelling and numerical simulation of fluid-structure interaction for complex viscous fluids

工业和医学中遇到的材料通常不属于牛顿粘性流体的经典模型。该项目的目的是深入了解剪切相关的非牛顿流体在随时间变化的运动区域中的复杂行为。因此，我们的目标是双重的：对相应的数学模型进行理论分析，并为实验模拟开发可靠的数值方案。我们所指的非牛顿流体属于剪切变稀/剪切增稠流体，其粘性应力张量是剪切速率的非线性函数。它们被广泛用于工业或生物医学，以模拟技术油、复合材料、聚合物、生物流体等的流动。作为示范性应用，将特别考虑顺应性血管中的血液流动。为了确保导出的数值格式对这种复杂模型是稳健和可靠的，需要进行深入的数值分析，包括收敛和误差分析。

Materials encountered in industry and medicine often fall outside the classical models of Newtonian viscous fluids. The aim of the proposed project is deep understanding of complex behavior of shear-dependent non-Newtonian fluids in time-dependent moving domains. Thus, our aims are two-fold: theoretical analysis of the corresponding mathematical model and the development of a reliable numerical scheme for experimental simulations. Non-Newtonian fluids we have in mind belong to the class of shear-thinning/shear-thickening fluids, for which the viscous stress tensor is a nonlinear function of shear rate. They are extensively used in industry or biomedicine in order to simulate flow of technical oils, composites, polymers, biological fluids, etc. As an exemplatory application blood flow in compliant vessels will be considered in particular. To assure that the derived numerical schemes are robust and reliable for such complex models, an in-depth numerical analysis including convergence and error analysis will be necessary.