Magneto-Rheological Fluids (MRFs) are smart materials whose physical properties can be controlled by an exciting magnetic field. MRFs are described as Bingham plastics with variable magnetic field dependent yield stress. Thanks to their particular features, MRFs have been largely employed to realize controllable power dissipating devices and, among them, regulable valves without moving parts. The most commonly configuration used for MRF based valves consists on fluid flow through an annular duct. The conception of such valves implies to deal with different physics. In particular, the magnetic circuit is usually designed and verified by mean of FE (Finite Element) analysis, while the duct geometry is usually dimensioned using an approximated formula based on fluid flow between parallel plates. In the presented work, a complete and detailed derivation of the analytical model is discussed in order to describe the flow of MRFs through an annulus using an approximate parallel plate geometry. Successively, the Bingham-Papanastasiou regularization is chosen as the mean to accurately describe the continuous non-linear yield stress and shear dependent viscosity of a commercially available MRF and it is then implemented into a FE software. This step allows to built a complete multiphysics problem for the design of MRFs based devices. Results obtained from the analytical model and FE analysis are then compared and the different steps in the proposed approaches are validated.