A mathematical model for the description of biomagnetic fluid flow exposed to a magnetic field that accounts for both electric and magnetic properties of the biofluid is presented. This is achieved by adding the Lorentz and magnetization forces in the Navier-Stokes equations. To demonstrate the effects of magnetic fields, we consider the case of laminar, incompressible, viscous, the steady flow of a Newtonian biomagnetic fluid (i) between two parallel plates; and (ii) through a straight rigid tube with a 60% in diameter, 84% on area, axisymmetric stenosis. Two external magnetic fields were investigated: one produced by an infinite wire carrying constant current, and a dipole-like field. We show, numerically and analytically, that the wire produces an irrotational force that, independent of its intensity, only alters the pressure leaving the velocity field unaffected. In contrast, when the fluid is exposed to the dipole-like field, which generates a rotational force, then both pressure and velocity can be strongly influenced even at moderate field strengths. Similar trends were obtained when a time varying flow is simulated through the axisymmetric stenosis in the presence of the dipole-like rotational magnetic field. It is expected that our findings could have important applications in blood flow control. Copyright © 2013 John Wiley & Sons, Ltd.