In blood, the concentration of red blood cells varies with the arterial diameter. In the case of narrow arteries, red blood cells concentrate around the centre of the artery and there exists a cell-free plasma layer near the
arterial wall due to Fahraeus-Lindqvist effect. Due to non-uniformity of the fluid in the narrow arteries, it is preferable to consider the two-phase model of the blood flow. The present article analyzes the heat transfer
effects on the two-phase model of the unsteady blood flow when it flows through the stenosed artery under an external pressure gradient. The direction of the artery is assumed to be vertical and the magnetic field is
applied along the radial direction of the artery. Blood is considered as a non-Newtonian Casson fluid with uniformly distributed magnetic particles. Both the blood and magnetic particles are moving with distinct velocities.
This two-phase problem is modelled using the Caputo-Fabrizio derivative approach and then solved for an exact solution using joint Laplace & Hankel transforms. Effects of pertinent parameters such as Grashoff number,
Prandtl number, Casson fluid parameter and fractional parameters, and magnetic field on blood velocity and particle velocity have been shown graphically for both large and small values of time. Both velocity profiles
increase with the increase of Grashoff number and Casson fluid parameter and reduce with the increase of magnetic field and Prandtl number. The behaviour of temperature is studied for different values of the fractional
parameter.