模擬血液在血管裡的行為有助於醫療人員或研究學者對心血管疾病掌握相關資訊，並降低手術的風險以及協助手術計畫。此篇論文中，我們使用Carreau-Yasuda模型模擬非牛頓流體以及牛頓流體在三維的血流模擬問題，包括long straight artery problem、end-to-side anastomosis probelm，以及針對個別病患所造出的pulmonary
artery probelm。在離散化方面，對於空間上的離散使用stabilized finite element method，而時間上的離散則使用implicit backward Euler finite difference method，於每一離散時間點，我們採用Newton-Krylov-Schwarz algorithm 解非線性系統。此篇
論文我們藉由比較以兩種不同流體為基礎的血流模擬，進而證實非牛頓流體在血流模擬的重要性，更決定了在複雜血管模型之下，非牛頓流體為不可或缺的條件。

Numerical simulation of blood flow in the arteries becomes an invaluable tools to help both of the physicians to plan the surgery procedure to reduce the risk of surgery and the researchers to understand the cardiovascular diseases. To ease the numerical difficulties of blood flow simulation, blood is often assumed to be Newtonian fluid as the first approximation. However, the shear thinning effect is significant in large arteries due to the dramatic change of the shear stress during a cardiac cycle and the non-homogeneous
properties of blood. Moreover, the recirculation happens frequently in the low shear rate region. To compute accurately the wall shear stress that provides more useful information to predict the formation of intimal hyperplasia, it is necessary to take the rheological
effect of blood flows in to account. In this study, the non-Newtonian blood flows in different complexity of artery were numerically investigated by using 3D fully parallel incompressible fluid solver. Our fluid solver is developed based on generalized Newtonian fluid model, where the viscosity is the function of rate of strain tensor. More specifically, the more commonly-used model for blood flow simulation, the Carreau-Yasuda model,
compared with Newtonian model are reported, including the investigation how the wall shear stress distribution and the streamlines and pressure distribution depend on different physiological conditions and arterial geometries.

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