We consider the unsteady flow of Burger fluid within a circular cylindrical tube, driven by a time-dependent pressure gradient, a body acceleration and a magnetic field acting normal to the flow direction. The solutions of the fractional constitutive equations governing the unsteady Burger’s fluid flow through arterial walls were obtained via the Laplace transform and the finite Hankel transform. The effects of magnetic field on parameters such as blood temperature and velocity were studied by Caputo time-fractional derivatives. We note that the solutions of many particular models such as fractional Oldroyd-B fluid, fractional Maxwell fluid, fractional second grade fluid and fractional Newtonian fluid models can be recovered from the solutions of the fractional constitutive equations governing the unsteady Burger’s fluid flow by particularizing the material coefficients (i.e. special and limiting cases of the earlier Burger’s fluid model). The numerical computations have been carried out to analyze the effects of fractional parameter α, similarity parameter β, relaxation time λ₁, retardation time λ₃, radius of the circular cylinder R₀ and material parameter λ₂ on the blood velocity and temperature. Some interesting flow and temperature characteristics are presented graphically and discussed. The study reveals that blood velocity, temperature and fractional parameters are reduced in the presence of magnetic field. The importance of this study can be found in the application fields of magnetic field control of biotechnological processes, bio magnetic device technology, biomedical engineering, medicine, etc.

我们考虑到汉堡流体在圆柱管内的不稳定流动，这是由与时间有关的压力梯度，物体加速度和垂直于流动方向作用的磁场驱动的。通过拉普拉斯变换和有限汉克尔变换，获得了控制非定常汉堡流体通过动脉壁的分数阶本构方程的解。用Caputo时间分数导数研究了磁场对诸如血液温度和速度之类的参数的影响。我们注意到许多特定模型的解，例如分数Oldroyd-B流体，分数Maxwell流体，分数二阶流体模型和分数牛顿流体模型可以通过具体化材料系数（例如，较早的汉堡流体模型的特殊情况和极限情况），从控制非定常汉堡流体流动的分数本构方程的解中恢复。已经进行了数值计算以分析分数参数α，相似性参数β，弛豫时间λ的影响₁，延迟时间λ ₃，圆筒半径R ₀和材料参数λ ₂上的血液速度和温度。图形显示并讨论了一些有趣的流量和温度特性。研究表明，存在磁场时血流速度，温度和分数参数会降低。这项研究的重要性可以在生物技术过程的磁场控制，生物磁设备技术，生物医学工程，医学等领域中找到。