The feasibility of reducing the viscosity of crude petroleum flows by the application of magnetic fields is examined using an ensemble of Hookean dumbbells. Results predict that a magnetic field cannot be both time-invariant and homogeneous if it is to have any effect. A magnetic field produced by an alternating electric current will have no effect upon the shear stress at the wall of the flow. The normal stress differences at the wall will oscillate 90 degrees out of phase with the magnetic field. They will exhibit damped oscillations within the bulk of the flow. A wave will propagate in the shear stress as well as in the normal stress differences within the bulk of the flow. The shear stress will not oscillate.