The Bingham plastic constitutive model has been widely used to predict the post-yield behavior of electro- and magneto- rheological fluids (ER and MR fluids). However, if these fluids experience shear thinning or shear thickening, the Bingham plastic model may not be an accurate predictor of behavior, since the post-yield plastic viscosity is assumed to be constant. In a recent study, it was theoretically and experimentally demonstrated that the Herschel-Bulkley fluid model can be successfully employed when evaluating non- Newtonian post-yield behavior of ER and MR fluids. In this paper, the Herschel-Bulkley model is employed to include a detailed analysis of ER and MR fluid dynamics through pipes and parallel plates. Simplified explicit expressions for the exact formulation are also developed. It is shown that the proposed simplified model of the Herschel-Bulkley steady flow equations for pipes and parallel plates can be used as an accurate design tool while providing a convenient and generalized mathematical form for modeling ER and MR fluids. Theoretical and experimental analyses are presented for a MR fluid damper, which is designed, developed, and tested at the University of Nevada, Reno (UNR).