We present a numerical investigation of tapered arteries that addresses the transient simulation of non-Newtonian bio-magnetic fluid dynamics (BFD) of blood through a stenosis artery in the presence of a transverse magnetic field. The current model is consistent with ferro-hydrodynamic (FHD) and magneto-hydrodynamic (MHD) principles. In the present work, blood in small arteries is analyzed using the Carreau-Yasuda model. The arterial wall is assumed to be fixed with cosine geometry for the stenosis. A parametric study was conducted to reveal the effects of the stenosis intensity and the Hartman number on a wide range of flow parameters, such as the flow velocity, temperature, and wall shear stress. Current findings are in a good agreement with recent findings in previous research studies. The results show that wall temperature control can keep the blood in its ideal blood temperature range (below 40°C) and that a severe pressure drop occurs for blockages of more than 60 percent. Additionally, with an increase in the Ha number, a velocity drop in the blood vessel is experienced.