When the shear stress magnitude of a Bingham material exceeds the yield shear stress, quasi‐Newtonian flow results, otherwise the material is rigid. The Bingham model has been used in tribology to describe the behavior of greases, but may also be used for electrorheological fluids proposed as ‘‘smart’’ lubricants. For two‐dimensional flow, different modified Reynolds’ equations are obtained, depending on the possible local formation of a rigid core which may be attached to either surface, or float between the surfaces. From the modified Reynolds’ equation it is straightforward to predict bearing behavior, patching together the different core formation cases. Results are presented for two geometries: the squeeze film damper and the journal bearing.