In this paper, the first analytical endeavor into the fluid dynamic modeling of an MR polishing process is reported. The velocity and shear stress fields of an MR fluid running through a thin slippery channel with a slightly varying height are analytically solved using a bi-viscosity constitutive model and a Navier slip model. Estimations of the mechanical power density and the total power per unit depth applied onto the channel surfaces are also presented. Analytical solutions for the Couette-Poiseuille flow behavior of a bi-viscous fluid flowing through either parallel or non-uniform channels are obtained, and the associated necessary and sufficient conditions characterizing a total of 5 types of flow are derived. The behaviors of the fluid are examined through the use of a parametric diagram of Bingham number (Bn) and Couette number (Co), i.e., Bn-Co or 1/Bn-1/Co diagram, by changing the geometric and operating conditions. Using these diagrams, variations in the rheological characteristics of the flow are investigated in great detail, with a special focus on the movement of the pseudo-core region. Finally, the mechanical power density field obtained for the flow in a converging-diverging channel is used to explain the wear mechanism in the MR polishing process. The effects of the power density field and the total power on the material removal rate (MRR) and the within-workpiece nonuniformity (WIWNU) with respect to various geometric and operating conditions are evaluated. (C) 2011 Elsevier B.V. All rights reserved.