A study of magnetohydrodynamics model of blood flow is made with single walls carbon nanotubes, copper (Cu), tin $({\rm TiO}_2),$ and alumina $({\rm Al}_2{\rm O}_3)$ and Cu as base nanoparticles through a circular cylinder. The fluid inside the tube is acted by an oscillating pressure gradient and an external constant magnetic field. The whole study is based on a mathematical model that includes Caputo fractional-order derivatives. Solutions for the blood velocitie, blood temperature distribution, and blood concentration distribution are obtained through the Laplace transform and expressed by the Wright function. Effects of the fractional-order parameter, magnetic field, the magnetic parameter $M,$ the Grashof numbers Gr and Gm, the dimensionless time $t,$ and the Prandtl parameter $Pr$ are addressed using numerical simulations. Results show that the applied magnetic field reduces the velocities of the fluid and particles. However, under long time intervals, particles seem to be accelerated, but their velocity is suitably controlled by the fractional-order parameter.