Analysis of a viscoplastic flow with field-dependent yield stress and wall slip boundary conditions for a magnetorheological (MR) fluid

Abstract

The present study theoretically analyzes the Couette–Poiseuille flow of a magnetorheological (MR) fluid flowing through a thin channel, where the externally applied magnetic field can be spatially non-uniform. To this end, a magnetic field-dependent (MFD) bi-viscosity constitutive model is newly proposed and employed in conjunction with the Navier wall slip condition. In this analysis, the MFD yield stress and MFD bi-viscosity are considered to be linearly proportional to the strength of the magnetic field, which obeys the inverse cube law of the normal distance from the magnetic tool. Through a succession of detailed analyses, a total of eight types of Couette–Poiseuille flows are found, including three new flow types that cannot be described with a classical bi-viscosity constitutive model. The analytic solutions for all types of these flows are derived along with the restriction conditions for their existence. Furthermore, the possibility of the existence of these eight types of MFD bi-viscous flows is investigated under four separate characteristic conditions that determine the configuration of the induced magnetic field in the channel. The behaviors of the MFD bi-viscous flows are then investigated through the use of a parametric diagram composed of the Bingham number (Bn) and the Couette number (Co), i.e., Bn–Co diagram, for which some parameters such as the effective distance, nominal viscosity ratio, and wall slip condition are changed.