Most magnetorheological (MR) fluid devices are fixed-pole valve mode devices where the fluid flows through a magnetically active valve. Controlling the strength of the magnetic field inside the valve allows the rheological properties of the MR fluid to be varied. Upon the application of a magnetic field, MR fluids develop a yield stress, which must be overcome before any flow is possible. This behavior can be represented mathematically by models of fluid with a yield stress like the Bingham plastic model. MR dampers have utilized this property of the MR fluids to provide controllable, semi-active vibration control. The most effective and widely used configuration of MR dampers incorporates an annular gap through which the MR fluid is force to flow. This paper presents a solution for annulus flows, derived from fundamental equations of fluid mechanics, of any general model of fluid with a yield stress. An example of the application of the general analytical expressions using the Herschel-Buckley model is given, and the limitations of the parallel plate approximation is illustrated for configurations whereby the size of the annular gap relative to the mean radius is large. Finally, the flow solution is incorporated into the mathematical model of an MR damper designed at the University of Manchester, and simulation results incorporating the effects of compressibility in the modeling procedure are presented. It was shown that this model can describe the major characteristics of such a device — nonlinear, asymmetric and hysteretic behaviors — successfully.