text: To understand and predict the behavior of the MR fluid it is necessary to model the fluid mathematically, a task slightly complicated by the varying material properties (such as yield stress). 
  As mentioned above, smart fluids are such that they have a low viscosity in the absence of an applied magnetic field, but become quasi-solid with the application of such a field. In the case of MR fluids (and ER), the fluid actually assumes properties comparable to a solid when in the activated ("on") state, up until a point of yield (the shear stress above which shearing occurs). This yield stress (commonly referred to as apparent yield stress) is dependent on the magnetic field applied to the fluid, but will reach a maximum point after which increases in magnetic flux density have no further effect, as the fluid is then magnetically saturated. The behavior of a MR fluid can thus be considered similar to a Bingham plastic, a material model which has been well-investigated.
  However, MR fluid does not exactly follow the characteristics of a Bingham plastic. For example, below the yield stress (in the activated or "on" state), the fluid behaves as a viscoelastic material, with a complex modulus that is also known to be dependent on the magnetic field intensity. MR fluids are also known to be subject to shear thinning, whereby the viscosity above yield decreases with increased shear rate. Furthermore, the behavior of MR fluids when in the "off" state is also non-Newtonian and temperature dependent, however it deviates little enough for the fluid to be ultimately considered as a Bingham plastic for a simple analysis.
  Thus our model of MR fluid behavior in the shear mode becomes:
  Where  = shear stress;  = yield stress;  = Magnetic field intensity  = Newtonian viscosity;  is the velocity gradient in the z-direction.