Solution of Reynolds Equation in Polar Coordinates Applicable to Nonsymmetric Entrainment Velocities

Reynolds equation in polar cylindrical (polar) coordinates is used for numerous tribological applications that feature thin fluid films in sliding contacts, such as chemical mechanical polishing and pin-on-disk testing. Although unstated, tribology textbooks and literary resources that present Reynolds equation in polar coordinates often make assumptions that the radial and tangential entrainment velocities are independent of the radial and tangential directions, respectively. The form of polar Reynolds equation is thus typically presented, while neglecting additional terms crucial to obtaining accurate solutions when these assumptions are not met. In the present investigation, the polar Reynolds equation is derived from the cylindrical Navier–Stokes equations without the aforementioned assumptions, and the resulting form is compared with results obtained from more traditionally used forms of the polar Reynolds equation. The polar form of Reynolds equation derived in this manuscript yields results that agree with the commonly used Cartesian form of Reynolds equation but are drastically different from the typically published form of the polar Reynolds equation. It is therefore suggested that the polar form of Reynolds equation proposed in this technical note be utilized when entrainment velocities are known to vary with either radial or angular position.