An Experimental Validation of the Recently Discovered Scale Effect in Generalized Newtonian EHL

Abstract

New quantitative numerical simulations of the elastohydrodynamic lubrication (EHL) film forming ability of generalized Newtonian liquids have elucidated a previously unrecognized property of EHL films. The dependency of the film thickness on the scale of the contact is greater when the viscosity is shear dependent within the inlet. Measurements of film thickness were performed in a ball on disc experiment using balls ranging from 5.5 to 35 mm in diameter. Three liquids were investigated with varying shear dependence in the range of stress important to film forming. The experimental results confirm the previous analytical findings. Numerical simulations using the measured viscosities show that the increased scale sensitivity is substantially the result of shear-thinning. However, the smallest scales produced films thinner than even the shear-dependent prediction, possibly indicating molecular degradation. It is quite likely that some machine components, which were designed using the effective viscous properties derived from a larger scale film thickness measurement, are operating with substantially lower film thickness than the designer had intended.

Appendix

The classical Hamrock and Dowson [20] equation for central film thickness in a circular contact is given in terms of three dimensionless parameters as

$$ h_{\text{c}} = 1.91R\hat{U}^{0.67} \hat{G}^{0.53} \hat{W}^{ - 0.067} $$

(10)

where the dimensionless parameters are

$$ \hat{U} = \frac{{\mu_{0} \bar{u}}}{{E^{\prime}R}},\quad \hat{W} = \frac{F}{{E^{\prime}R^{2} }},\quad \hat{G} = {\alpha}^{*}E^{\prime} $$

(11)

The only parameters available within this formula (10) for the description of the behavior of the liquid are the ambient viscosity, μ ₀, and the pressure–viscosity coefficient, α*. There are no parameters which quantify either the compressibility of the liquid or the shear dependent viscosity. The dimensionless parameters are given in Table 4 for these experiments.

Table 4 Experimental conditions

Compressibility directly affects the central film thickness by the compression of the liquid volume to the Hertz pressure [23]. Compressibility varies somewhat with composition and increases exponentially with temperature. At least one additional parameter, say K ₀, would be necessary to arrive at a film thickness formula which includes the effect of variable compressibility.

In this work, it is seen that shear dependent viscosity plays an important role in film formation when the Newtonian limit, G, is sufficiently low that shear thinning occurs in the inlet. To include the effect of shear-thinning would require at least two additional parameters, G and n, in a film thickness formula.

In practice, variations in compressibility and shear dependent viscosity are accommodated in Eq. 10 by treating the pressure–viscosity coefficient, α*, as an effective parameter, obtained from a film thickness measurement, rather than from the definition of Eq. 9. This work examines only the effect of shear-thinning on extrapolations to different scale, R, using the effective pressure–viscosity coefficient.

In terms of only the load and scale, Eq. 10 shows that the film thickness varies as

$$ h_{\text{c}} \propto F^{ - 0.067} R^{0.464} $$

(12)

for a Newtonian liquid of slight compressibility. A measurement of film thickness for extrapolation to a real machine would more sensibly simulate the pressure, p _H, than the load of the machine component. Therefore, the relation between pressure and load, $F \propto p_{\text{H}}^{3} R^{2}$, can be employed to recast (12) as

$$ h_{\text{c}} \propto p_{\text{H}}^{ - 0.201} R^{0.330} $$

(13)

for a Newtonian liquid of slight compressibility. The complete form of (13) is given in (8). The contact pressure is not varied in this work and variations in compressibility will therefore not affect the results.

Inspection of Eq. 8 shows that the classical Hamrock and Dowson formula predicts that the speed sensitivity, $d(\log h_{\text{c}} )/d(\log \bar{u})$, should be equal to 0.67. Unless the inlet rheology is operating within one of the Newtonian plateaus, a shear-thinning liquid should have speed sensitivity <0.67. Therefore, it would be inappropriate to extrapolate from the classical formula to smaller scales from a large scale measurement which displayed a speed sensitivity of measurably <0.67. In practice, however, this restriction has been difficult to implement. The large scale measurements presented here have speed sensitivities indistinguishable from Newtonian. Polymer blended liquids can display the Newtonian sensitivity to speed while having significant film thinning [24] because of the plateaus. Base oils can display film thinning, as does the PAO100 investigated in [12]. Another PAO100 was investigated at Imperial College [25] where among ten low molecular weight base oils for which the inlet behavior should be Newtonian the speed sensitivities at various temperatures ranged from 0.64 to 0.74. The PAO100 at various temperatures yielded speed sensitivities from 0.62 to 0.65. It would be difficult to assess the shear thinning behavior of the PAO100 from the speed sensitivity alone in this case and effective pressure–viscosity coefficients were reported. Of course, for low values of both G and n, the speed sensitivity will obviously betray the shear-thinning response.