This article aims to investigate the combined effects of the heat transfer and magnetic field on the electrically conducting magnetohydrodynamic two-phase free convective flow of dusty Casson fluid between parallel plates. The flow is caused by buoyancy force with heat transfer due to free convection. Moreover, the left plate moves at uniform velocity while the right plate is stationary, and all dust particles with spherical forms are scattered evenly throughout the fluid. Mathematically, the flow is described as partial differential equations. A newly introduced fractional derivative, namely Caputo-Fabrizio fractional derivative, is used to generalize the given derived system of PDEs. The problem is solved using joint applications of finite sine Fourier and Laplace transforms. Exact solutions of velocity and temperature profile are obtained. Moreover, the impact of different parameters like a magnetic parameter, Grashof number, Dusty fluid parameter, Casson parameter, Peclet number, Reynold number, and particle's mass parameter on temperature, and dust particle and Casson fluid velocity have been discussed. The graphical results for Casson fluid, dusty fluid, and temperature profiles are plotted through Mathcad-15. The behavior of Casson fluid and dusty fluid is matching for different embedded parameters. Moreover, the Nusselt number and skin friction are also determined. It is shown in Table 1 that by increasing the value of the peclet number, the rate of heat transfer decreases. Furthermore, Table 2 shows that by increasing the magnetic parameter, the skin friction decreases. Graphical results conclude that fractional Casson fluid model described a more realistic aspect of the both (Fliud and dust particle) velocities profile, temperature profile, rate of heat transfer and skin friction than the classical Casson fluid model.

本文旨在研究传热和磁场对平行板间含尘卡森流体的导电磁流体动力学两相自由对流流动的综合影响。流动是由浮力引起的，并且由于自由对流而传热。此外，左板匀速运动而右板静止，所有球形尘埃粒子均匀地散布在整个流体中。在数学上，流动被描述为偏微分方程。新引入的分数阶导数，即 Caputo-Fabrizio 分数阶导数，用于推广给定的偏微分方程的派生系统。使用有限正弦傅立叶和拉普拉斯变换的联合应用解决了该问题。获得了速度和温度分布的精确解。此外，还讨论了不同参数如磁参数、格拉肖夫数、尘埃流体参数、卡森参数、佩克莱特数、雷诺数和粒子质量参数对温度、尘埃粒子和卡森流体速度的影响。Casson 流体、含尘流体和温度曲线的图形结果通过 Mathcad-15 绘制。Casson 流体和含尘流体的行为对于不同的嵌入参数是匹配的。此外，还确定了努塞尔数和皮肤摩擦。由表 1 可见，随着颗粒数的增加，传热速率降低。此外，表 2 表明，通过增加磁参数，皮肤摩擦减小。