Cardiovascular disease is a major threat to human health. The study on the pathogenesis and prevention of cardiovascular disease has received special attention. In this paper, we have contributed to the derivation of a mathematical model for the nonlinear waves in an artery. From the Navier–Stokes equations and continuity equation, the vorticity equation satisfied by the blood flow is established. And based on the multiscale analysis and perturbation method, a new model of the Boussinesq equation with viscous term is derived to describe the propagation of a viscous fluid through a thin tube. In order to be more consistent with the flow of the fluid, the time-fractional Boussinesq equation with viscous term is deduced by employing the semi-inverse method and the fractional variational principle. Moreover, the approximate analytical solution of the fractional equation is obtained, and the effect of viscosity on the amplitude and width of the wave is studied. Finally, the effects of the fractional order parameters and vessel radius on blood flow volume are discussed and analyzed.

中文翻译：

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细弹性管中复杂粘性流体的建模与分析

心血管疾病是对人类健康的主要威胁。关于心血管疾病的发病机理和预防的研究受到了特别的关注。在本文中，我们为推导动脉中非线性波的数学模型做出了贡献。根据Navier–Stokes方程和连续性方程，建立了血流满足的涡度方程。基于多尺度分析和摄动方法，推导了具有粘性项的Boussinesq方程的新模型，用于描述粘性流体通过细管的传播。为了与流体的流动更加一致，采用半反方法和分数变分原理，推导了具有粘性项的时间分数Boussinesq方程。此外，得到分数阶方程的近似解析解，研究了黏度对波幅和波幅的影响。最后，讨论并分析了分数阶参数和血管半径对血流量的影响。