This work presents analytic and numerical results for a unidirectional flow in a circular capillary of a magnetic fluid. A condition that deviates from the standard equation of equilibrium magnetization due to the vorticity of the flow is considered. We show that the model of a magnetic symmetrical fluid with no-coupling between magnetization and flow leads to velocity profiles very close to the parabolic one. This scenario may change drastically when the fluid magnetization is coupled with the flow vorticity resulting in a nonlinear relation between pressure gradient and flow volume rate and consequently a non-Newtonian behavior. The solutions are developed with the fluid undergoing each a pressure gradient and an applied magnetic field gradient. The dimensionless governing equations reveal an important physical parameter denoted magnetic Reynolds number, which measures the relative importance between hydrodynamic and magnetic forces. The resulting equation is solved numerically and by a regular perturbation method. Explicit expressions for the dimensionless velocity profile, effective viscosity and friction factor as function of the dimensionless magnetic and hydrodynamic physical parameters are presented. Numerical and analytic results are compared and a good agreement is verified for the range where the analytic solution can be used. The studies are applied to explore the emerging propose of drag reducing fluids by applying a magnetic field gradient favorable to the flow direction. We have observed that the application of a controllable magnetic field gradient may be responsible for a meaningful drag reduction in a laminar pipe flow Ver menos