In this work, we focused on the peristaltic unsteady flow of non-Newtonian nanofluid with heat transfer through a non-uniform vertical duct. The flow obeys Herschel Bulkley model through a non-Darcy porous medium under the effects of mixed convection and thermal diffusion. Moreover, the effects ofthermal radiation, heat generation, Ohmic dissipation, chemical reaction and uniform external magnetic field are investigated. The derived equations that describe the velocity, temperature and nanoparticles concentration are simplified under the assumptions of long wave length and low Reynolds number. These equations have been solved by using a numerical technique with the help of shooting method. The obtained solutions are functions of the physical parameters entering the problem. The effects of these parameters and the obtained solutions are explained and discussed through a set of graphs. It is found that the increment in Prandtl number or Thermophoresis parameter reduces the spread of the nanoparticles (concentration increased) within the fluid along with the thermal diffusivity through the fluid layers. Also the non-Darcy effect supports the inertial forces, and in order to maintain Reynolds number, the viscous forces are motivated and the axial velocity is damped. Moreover, for the validation of the current methodology, this model is reduced to power law model (no yield stress) and compared with the work of Eldabe et al. [16].

中文翻译：

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Herschel Bulkley纳米流体通过非达西多孔介质在滑移条件下的传热蠕变

在这项工作中，我们集中于非牛顿纳米流体的蠕动非稳态流动以及通过非均匀垂直管道的热传递。在混合对流和热扩散的作用下，流体通过非达西多孔介质服从Herschel Bulkley模型。此外，还研究了热辐射，热量产生，欧姆耗散，化学反应和均匀外部磁场的影响。在长波长和低雷诺数的假设下，简化了描述速度，温度和纳米粒子浓度的推导方程。这些方程式已经通过使用数字技术借助射击方法来求解。所获得的解是输入问题的物理参数的函数。这些参数的影响和获得的解决方案将通过一组图形进行解释和讨论。发现普朗特数或热泳参数的增加减小了纳米粒子在流体内的扩散（浓度增加）以及通过流体层的热扩散率。非达西效应也支持惯性力，并且为了保持雷诺数，会激励粘性力并衰减轴向速度。此外，为了验证当前的方法，该模型被简化为幂定律模型（无屈服应力），并与Eldabe等人的工作进行了比较。[16]。发现普朗特数或热泳参数的增加减小了纳米粒子在流体内的扩散（浓度增加）以及通过流体层的热扩散率。非达西效应也支持惯性力，并且为了保持雷诺数，会激励粘性力并衰减轴向速度。此外，为了验证当前的方法，该模型被简化为幂定律模型（无屈服应力），并与Eldabe等人的工作进行了比较。[16]。发现普朗特数或热泳参数的增加减小了纳米粒子在流体内的扩散（浓度增加）以及通过流体层的热扩散率。非达西效应也支持惯性力，并且为了保持雷诺数，会激励粘性力并衰减轴向速度。此外，为了验证当前的方法，该模型被简化为幂定律模型（无屈服应力），并与Eldabe等人的工作进行了比较。[16]。该模型简化为幂定律模型（无屈服应力），并与Eldabe等人的工作进行了比较。[16]。该模型简化为幂定律模型（无屈服应力），并与Eldabe等人的工作进行了比较。[16]。