The study of multiphase flows gained much importance because of its extensive applications in nature and industry. These flows possess two or more thermodynamic phases, for example, one component phase (e.g., water vapors and water flow) or several components phase (e.g., water and oil flow). The most common example of multiphase flow in the context of the oil industry is petroleum. Further blood flow, porous structures, fluidized bed, bubbly flow in nuclear reactors, and fiber suspension in the paper industry are some significant examples of multiphase flows. In this paper, we considered the Couette flow of non-Newtonian (couple stress) fluid with variable magnetic field and thermal conductivity effects between parallel walls of the channel. The upper wall of the channel is in constant motion while the lower wall is in a fixed position. The variable viscosity effects with the suspension of hafnium particles are also discussed by taking Vogel’s viscosity case. The shooting method based on the R–K method is applied to obtain the numerical solution of the current problem. A comparison between Newtonian and non-Newtonian fluids is presented by sketching graphs. The variations in flow and temperature of fluid against various involved factors, including variable viscosity, wall temperature, thermal radiations, variable magnetic field, and thermal conductivity are sketched and also physically described. It is observed that variable viscosity parameter elevated both velocity and temperature profiles while wall temperature parameter decelerated both fields. Further, noticed that the variable thermal conductivity and variable magnetic field impede the velocity of the fluid and also retarded the temperature field. Our attempt is not just useful to investigate the mechanical and industrial multiphase flows but also delivers important results to fill the gap in the existing literature.

中文翻译：

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运动面热影响偶应力流体多相流动的数值分析

多相流的研究因其在自然界和工业中的广泛应用而变得非常重要。这些流动具有两个或更多个热力学相，例如，一个组分相（例如，水蒸气和水流）或几个组分相（例如，水和油流）。石油工业中多相流最常见的例子是石油。进一步的血流、多孔结构、流化床、核反应堆中的气泡流以及造纸工业中的纤维悬浮液是多相流的一些重要例子。在本文中，我们考虑了具有可变磁场和通道平行壁之间的导热效应的非牛顿（耦合应力）流体的库埃特流动。通道的上壁处于恒定运动中，而下壁处于固定位置。还通过 Vogel 的粘度案例讨论了铪颗粒悬浮液的可变粘度效应。应用基于R-K方法的射击方法来获得当前问题的数值解。牛顿流体和非牛顿流体之间的比较通过草图来呈现。流体的流量和温度对各种相关因素（包括可变粘度、壁温、热辐射、可变磁场和热导率）的变化进行了草图和物理描述。据观察，可变粘度参数提高了速度和温度分布，而壁温参数降低了这两个场。进一步，注意到可变的热导率和可变的磁场阻碍了流体的速度并延迟了温度场。我们的尝试不仅有助于研究机械和工业多相流，而且还提供了重要的结果来填补现有文献中的空白。