Modeling and Multiple Fractional Solutions for Two-Phase Blood Flow in a Slant Cylindrical Tube with Memory Effect

α→1MPC(α,β, and γ)This work investigates the magnetohydrodynamic (MHD) flow of blood in an inclined cylindrical tube containing infused magnetic nanoparticles (MNPs). This system has significant implications for hyperthermia treatment and targeted drug administration. It is crucial for comprehending completely the interdependent effects of MNPs, tube inclination, and MHD. The Brinkman type fluid model is employed in the established two-phase flow. The Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu (AB) fractional operators are used in succeeding fractionalization to account for memory effects and allow for a comparison analysis between various fractional frameworks. The joint Laplace-Hankel transforms are used to produce the exact solutions. When α→1 the classical results are restored. The results reveal that velocities caused by Lorentz forces decrease as the magnetic number (M) increases, with CF showing the fastest reaction. At the same time, because of stronger viscous and inertial effects, higher particle concentrations (PC) reduce velocities. The fractional orders (α,β, and γ), display dual time-dependent dynamics, speeding up flows over extended periods (after initially slowing them down. The most physically accurate predictions are produced by AB's Mittag-Leffler kernel, which combines short- and long-term memory effects. The results show the importance of fractional calculus in simulating complex biofluid dynamics, particularly in inclined tubes where gravity, magnetic, and viscous forces interact, and they offer a solid foundation for enhancing magnetic drug delivery devices.