Technical Briefs

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

[+] Author and Article Information
Kurt Beschorner1

Department of Bioengineering, University of Pittsburgh, Pittsburgh, PA 15219keb52@pitt.edu

C. Fred Higgs

Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213

Michael Lovell

Department of Mechanical Engineering,University of Wisconsin-Milwaukee, Milwaukee, WI 53201


Corresponding author.

J. Tribol 131(3), 034501 (May 22, 2009) (5 pages) doi:10.1115/1.3118783 History: Received March 21, 2008; Revised March 02, 2009; Published May 22, 2009

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.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 1

Diagram of the pin-on-disk apparatus with the polar coordinate system labeled. The polar coordinates, r and θ, are relative to the origin located at the center of the pin. Primary inputs for Reynolds equation are the film thickness, h, and the entrainment velocities, νr and νθ. The average linear velocity, U, is determined from the angular velocity and center-to-center distance of the disk to the pin.

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Figure 2

Pin-on-disk hydrodynamic pressure distribution. A 2D plot and contour plot of the normalized pressure (p∗) is shown when solved with (a) polar Reynolds equation without assumptions that tangential velocity is independent of angle, (b) Cartesian form of Reynolds equation, and (c) the traditional form of polar Reynolds equation. The left plots are side views of the pressure distribution, while the right plots are the top views of the pressure distributions. Pressure is normalized to atmospheric pressure, and the x∗ and y∗ coordinates show locations normalized to the pin radius.

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Figure 3

Diagram of the pad and wafer setup. The origin for the coordinate system is the center of the wafer with the polar coordinates: r and θ relative to this location. The entrainment velocities νr and νθ are determined by the pad and wafer angular velocities, as well as the pad to wafer center-to-center distance, ωp, ωw, and d, respectively.

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Figure 4

Chemical mechanical polishing hydrodynamic pressure distribution when solved (a) with polar Reynolds equation without assumptions that the tangential velocity is independent of angle and (b) with the traditional form of polar Reynolds equation. Pressure (p∗) is normalized to atmospheric pressure, while the x∗ and y∗ coordinates show locations normalized to the wafer radius.




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