Fractional derivatives have been proven to showcase a spectrum of solutions that is useful in the fields of engineering, medical, and manufacturing sciences. Studies on the application of fractional derivatives on fluid flow are relatively new, especially in analytical studies. Thus, geometrical representations for fractional derivatives in the mechanics of fluid flows are yet to be discovered. Nonetheless, theoretical studies will be useful in facilitating future experimental studies. Therefore, the aim of this study is to showcase an analytical solution on the impact of the Caputo-Fabrizio fractional derivative for a magnethohydrodynamic (MHD) Casson fluid flow with thermal radiation and chemical reaction. Analytical solutions are obtained via Laplace transform through compound functions. The obtained solutions are first verified, then analysed. It is observed from the study that variations in the fractional derivative parameter, $\alpha$, exhibits a transitional behaviour of fluid between unsteady state and steady state. Numerical analyses on skin friction, Nusselt number, and Sherwood number were also analysed. Behaviour of these three properties were in agreement of that from past literature.

As the leading cause of death worldwide, cardiovascular disease underscores the urgent need for effective therapies and diagnostic tools. The use of magnetic fields and nanoparticles has demonstrated potential for creating cutting-edge treatments. To analyse blood flow in an artery with stenosis and the impact of an external magnetic field on blood flow infused with hybrid nanoparticles, this study is conducted. A generalised power law is used to model the flow of a hybrid blood nanofluid comprising silver (Ag) and gold (Au) nanoparticles. This study focuses on a deeper level of the magnetic field with hybrid nanoparticles in a non-Newtonian fluid, which extends from previous studies on nanoparticles in Newtonian blood. In a straight artery, the blood flow through a cosine-shaped stenosis is simulated using COMSOL Multiphysics software. The physical controlling parameters, including velocity profiles and wall shear stress, are illustrated through graphs. The external magnetic field significantly reduces shear stress and the velocity profile. The addition of gold and silver nanoparticles allows for smooth blood flow in the diseased artery. The findings show a decline in aberrant behaviour and recirculation in the post-stenotic area. The combination of a hybrid nanofluid with an external magnetic field presents a practicable method for improving blood flow in stenosed arteries. The results have implications for targeted drug delivery in stenotic arteries and advancements in nanomedicine.

The effect of the thermal radiation on the MHD Casson fluid along with the fractional derivative in an oscillating vertical plate is elucidated. More exactly, the Caputo fractional model is utilized in developing the governing equations. Besides, the influence of the buoyancy force due to the temperature gradient has also been considered. The derived fractional partial differential equations are converted into ordinary differential equations by using the Laplace transform technique and then are solved for analytical solutions via the characteristic method. The inversion of the Laplace transformation is obtained through the numerical approach of Zakian. The effects of various physical parameters on the velocity and temperature profiles, Nusselt number, and skin friction have been analyzed and depicted in graphs and tables. The distribution of the velocity and temperature either in viscous or Casson fluid do enhance by the fractional parameter.