The present work theoretically investigates the natural convection blood flow as a Brinkman-type fluid with uniformly distributed magnetic particles in a circular cylindrical tube with the applied external magnetic field. The classical model for the blood flow is generalized by using the definition of Caputo time-fractional derivative. The exact solutions are obtained by using the Laplace and Henkel transforms. Unlike the classical model, the obtained general results are expressed in the form of "Lorenzo and Hartley's" and "Robotnov and Hartley's" functions. Graphs are plotted to show the effects of different parameters on the blood flow. Furthermore, the velocity and temperature distributions are discussed in terms of memory. The effect of fractional parameter alpha for a long and short time has also been observed. It is noticed that blood velocity can be controlled using the fractional parameter. It is also found that, for tau0, fluid and particles motion increased, and reverse behavior is observed for tau0. It has been noticed that increasing values of particle mass parameter Pm and magnetic parameter M slow down the motion of blood and magnetic particles. These results are helpful for effective drug delivery and regulating blood flow.
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