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A Quantitative Solution for the Full Shear-Thinning EHL Point Contact Problem Including Traction

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Abstract

We present a realistic elastohydrodynamic lubrication (EHL) simulation in point contact using a Carreau-like model for the shear-thinning response and the Doolittle-Tait free-volume viscosity model for the piezoviscous response. The liquid lubricant modeled is a high-viscosity polyalphaolefin which has been shown by high-pressure viscometry to possess a relatively low threshold for shear-thinning as a single-component liquid lubricant. As a result, the measured EHL film thickness is about one-half of the Newtonian prediction. We derived and numerically solved the two-dimensional generalized Reynolds equation for the modified Carreau model based on Greenwood. In this simulation, viscosity was not treated as an adjustable parameter; the models used for the pressure and shear dependence of viscosity were obtained entirely from viscometer measurements. Truly remarkable agreement is found in the comparisons of simulation and experiment for traction coefficient and for film thickness in both pure rolling and sliding cases.

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Abbreviations

a :

Hertzian contact radii, [m]

B :

Doolittle parameter

\(E^\prime\) :

Reduced Young’s modulus of two surfaces, [Pa]

G :

Modulus or critical stress, [Pa]

h, H :

Film thickness, [m], and dimensionless, H = h/a

h 0, H 0 :

Normal approach, [m], and dimensionless, H 0 = h 0 /a

h cen, h min :

Central and minimum film thickness, [m]

k :

Traction coefficient

K 0 :

Bulk modulus at p = 0, [GPa]

K ′0 :

Pressure rate of change of bulk modulus at p = 0

n :

Power law exponent

p, P :

Pressure, [Pa], and dimensionless, P = p/p h

p h :

Maximum Hertzian pressure, [Pa]

p x , p y :

Dimensionless pressure gradients, \(p_x =\frac{h}{G}\frac{\partial p}{\partial x}, p_y =\frac{h}{G}\frac{\partial p}{\partial y}\)

u, v :

Velocities in two directions, [m/s]

\(\bar{u},\bar{v}\) :

Entrainment velocity, \(\bar{u}=\frac{u_1 +u_2}{2}\), \(\bar{v}=\frac{v_1 +v_2}{2}\), [m/s]

U, V :

Dimensionless velocities, U = uμ/(hG), V = vμ/(hG)

v :

Volume, [m3]

v 0 :

Volume at ambient pressure or p = 0, [m3]

v occ :

Occupied volume, [m3]

w :

Load, [N]

X, Y, ξ:

Dimensionless coordinates X = x/a, Y = y/a, ξ = z/h

\(\dot{\gamma}\) :

Shear rate

\(\dot{\gamma}_x, \dot{\gamma}_y, \dot{\gamma}_e\) :

Shear rate in two directions, and composite, \(\dot{\gamma}_e =\sqrt{\dot{\gamma}_x^2 +\dot{\gamma}_y^2}\)

ɛ:

Dimensionless coefficients for Poiseuille flow

λ:

Characteristic time, [s]

\(\mu,\bar{\mu}\) :

Low-shear viscosity, [Pa*s], and dimensionless, \(\bar{\mu}=\mu/\mu_0 \)

μ o :

Low-shear viscosity at ambient pressure, [Pa*s]

\(\rho, \bar{\rho}\) :

Density of lubricant, [kg/m3], and dimensionless, \(\bar{\rho}=\rho/\rho_0 \)

ρ0 :

Density at ambient pressure, [kg/m3]

σ x , σ z :

Normal stresses in two directions, tensile is positive, [Pa]

\(\tau, \bar{\tau}\) :

Shear stress, [Pa], dimensionless shear stress, \(\bar{\tau}=\tau/G\)

\(\bar{\tau}_x,\bar{\tau}_y, \bar{\tau}_e\) :

Dimensionless shear stresses in two directions and combined, \(\bar{\tau}_e =\sqrt{\bar{\tau}_x^2 +\bar{\tau}_y^2}\)

\(\bar{\tau}_a,\bar{\tau}_b\) :

Dimensionless shear stresses on the middle layer

ϕ x ,ϕ y :

Two flow factors in two directions

Ω:

Solution domain

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Acknowledgments

Yuchuan Liu and Q. Jane Wang would like to express their gratitude for the support from Office of Naval Research and Department of Energy. Scott Bair acknowledges the support of the Timken Company for high-pressure rheology research over many years.

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Authors and Affiliations

  1. Center for Surface Engineering and Tribology, Northwestern University, 2145 Sheridan Rd, #B224, Evanston, IL, 60208, USA

    Yuchuan Liu & Q. Jane Wang

  2. Center for High-Pressure Rheology, Georgia Institute of Technology, George W. Woodruff School of Mechanical Engineering, Atlanta, GA, 30332-0405, USA

    Scott Bair

  3. Laboratoire de Mécanique des Contacts et des Structures, UMR CNRS 5259, INSA-Lyon, 18-20 rue des Sciences, Villeurbanne cedex, 69621, France

    Philippe Vergne

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  1. Yuchuan Liu
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Correspondence to Yuchuan Liu.

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Liu, Y., Wang, Q.J., Bair, S. et al. A Quantitative Solution for the Full Shear-Thinning EHL Point Contact Problem Including Traction. Tribol Lett 28, 171–181 (2007). https://doi.org/10.1007/s11249-007-9262-5

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