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Research Papers

# The Phan-Thien and Tanner Model Applied to Thin Film Spherical Coordinates: Applications for Lubrication of Hip Joint Replacement

[+] Author and Article Information
John Tichy

Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590

Benyebka Bou-Saïd

Laboratoire de Mécanique des Contacts et des Solides, Institut National des Sciences Appliquées de Lyon, Villeurbanne 69621, France

J Biomech Eng 130(2), 021012 (Mar 31, 2008) (6 pages) doi:10.1115/1.2899573 History: Received November 07, 2006; Revised September 14, 2007; Published March 31, 2008

## Abstract

The Phan-Thien and Tanner (PTT) model is one of the most widely used rheological models. It can properly describe the common characteristics of viscoelastic non-Newtonian fluids. There is evidence that synovial fluid in human joints, which also lubricates artificial joints, is viscoelastic. Modeling the geometry of the total hip replacement, the PTT model is applied in spherical coordinates for a thin confined fluid film. A modified Reynolds equation is developed for this geometry. Several simplified illustrative problems are solved. The effect of the edge boundary condition on load-carrying normal stress is discussed. Solutions are also obtained for a simple squeezing flow. The effect of both the relaxation time and the PTT shear parameter is to reduce the load relative to a Newtonian fluid with the same viscosity. This implies that the Newtonian model is not conservative and may overpredict the load capacity. The PTT theory is a good candidate model to use for joint replacement lubrication. It is well regarded and derivable from molecular considerations. The most important non-Newtonian characteristics can be described with only three primary material parameters.

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## Figures

Figure 6

Effect of eccentricity—profile of radial (load carrying) normal stress. Simple squeezing flow. Relaxation time λ=0.1s. PTT parameter ξ*=0.1. Solid line: ϵ=0.0; short dashed line: ϵ=0.5; long dashed line: ϵ=0.8.

Figure 1

Schematic of eccentric sphere geometry

Figure 2

Effect of edge boundary condition—profile of radial and azimuthal normal stress. Azimuthal normal stress equals ambient pressure at film edge. Simple steady squeezing flow ϵ=0.0. Relaxation time λ=1s; PTT parameter ξ*=0.0 (Maxwell fluid). Dashed line: normal stress in flow direction πθθ. Solid line: load carrying normal stress πrr.

Figure 3

Effect of relaxation time—profile of radial (load carrying) normal stress. Simple squeezing flow ϵ=0.0. PTT parameter ξ*=0.1. Solid line: λ=1s; long dashed line: λ=0.5s; short dashed line: Newtonian, λ=0.0s.

Figure 4

Effect of relaxation time—profile of radial (load carrying) normal stress. Normal stress πθθ set to ambient at film edge. Simple squeezing flow ϵ=0.0. PTT parameter ξ*=0.1. Solid line: λ=1s; long dashed line: λ=0.5s; short dashed line: λ=0.1s.

Figure 5

Effect of PTT shear parameter ξ—profile of radial (load carrying) normal stress. Simple squeezing flow ϵ=0.0. Relaxation time λ=1s. Solid line: ξ*=0.1; short dashed line: ξ*=0.08; long dashed line: ξ*=0.05.

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