The behavior of human blood flowing in arteries is still an open topic for its multi-phase nature and heterogeneity. In large arterial vessels the well-known Hagen–Poisueille law, which main assumption is that the blood is Newtonian, is considered acceptable. In small arterial vessels, instead, this law does not reproduce experimental results that show non-parabolic profiles of velocity across the vessel diameter. For capillary vessels the Casson model of fluids that is nonlinear is used in place of the Newton law, resulting in nonlinear governing equations and difficulties in mathematical manipulation. For these reasons an alternative approach is proposed in this paper. Starting from the micro-mechanics of blood, the Hagen–Poisueille model is enriched with long-range interactions that simulate the interactions of non-adjacent fluid volume elements due to the presence of red blood cells and other dispersed cells in the plasma. These nonlocal forces are defined as linearly dependent on the product of the volumes of the considered elements and on their relative velocity. Moreover, as the distance between two volume elements increases, the nonlocal forces are scaled through an attenuation function; if this function is chosen as a power law of real order of the distance between the volume elements, an operator related to the fractional derivative of relative velocity appears in the resulting governing equation. It is shown that the fractional Hagen–Poisueille law is able to reproduce experimentally measured profiles of velocity with a great accuracy, moreover as the dimension of the vessel increases, nonlocal forces become negligible and the proposed model reverts to the classical Hagen–Poisueille model.

中文翻译：

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小腔动脉血管中非线性血流的分数非局部方法

人体血液在动脉中流动的行为因其多相性和异质性仍然是一个悬而未决的话题。在大动脉血管中，众所周知的 Hagen-Poisueille 定律（其主要假设是血液是牛顿血液）被认为是可以接受的。相反，在小动脉血管中，该定律不会重现显示跨血管直径的非抛物线速度分布的实验结果。对于毛细血管，使用非线性流体 Casson 模型代替牛顿定律，导致非线性控制方程和数学运算困难。由于这些原因，本文提出了一种替代方法。从血液的微观力学出发，Hagen-Poisueille 模型富含长程相互作用，可模拟由于血浆中存在红细胞和其他分散细胞而导致的非相邻流体体积元素的相互作用。这些非局部力被定义为线性依赖于所考虑元素的体积和它们的相对速度的乘积。此外，随着两个体积元素之间距离的增加，非局部力通过衰减函数进行缩放；如果选择该函数作为体积元素之间距离的实阶幂律，则在所得控制方程中会出现与相对速度的分数阶导数相关的算子。结果表明，分数哈根-泊肃叶定律能够非常准确地再现实验测量的速度剖面，