1 Introduction

Early on, it was recognized that the shear dependence of the viscosity of polymer-thickened lubricants must influence film thickness [1] in elastohydrodynamic lubrication (EHL). The development of the classical Newtonian film thickness formulas for EHL circular contacts [2] was one of the shining achievements of the field. In the ensuing enthusiasm, it was often overlooked that these formulas lacked precision for all conditions and were accurate only for mineral oils and other low-molecular-weight base oils under mild conditions. Experimental measurements indicated that, for polymer-blended mineral oil or high-molecular-weight silicone oil, the predicted film thickness may be about twice the measured value [3]. This trend could be observed when employing a pressure-viscosity coefficient obtained from a viscometer [4] rather than a pressure-viscosity coefficient which had been adjusted [5] to yield agreement with the same Newtonian formula.

The shear stress in a steady shear flow for a non-Newtonian liquid is related to the shear rate by

$$ \tau = \eta \dot{\gamma } $$
(1)

where the generalized viscosity, η, is some function of the invariants of stress or strain rate. Many empirical functions have been derived for η and most employ a parameter, μ 2, representing a limit to the viscosity at infinite shear rate or simply a second plateau apart from the first Newtonian plateau, μ, at low shear rate.

$$ \eta = \mu_{2} + (\mu - \mu_{2} )F(\dot{\gamma })\;{\text{or}}\;\eta = \mu_{2} + (\mu - \mu_{2} )F(\tau ) $$
(2)

The function, F, goes to 1 as the argument goes to zero and F goes to 0 as the argument goes to infinity. The expectation is that, for a polymer solution, the contribution of the solvent to the viscosity will be unaffected by shear within the inlet of the EHL contact. This contribution of the solvent is expected to result in a second plateau or, at least, an inflection in the flow curve which will be produced by μ 2 > 0. An extensive list of models which have served the role of Eq. (2) may be found in [6].

The shear dependence of the viscosity of polymer solutions has been of extreme importance to rheology [7] and considerable research effort has been devoted to understanding the shear response. It must be stressed that the generalized Newtonian approach which is used here and throughout tribology is incomplete. In addition to shear-dependent viscosity other, even more profound, effects such as normal stress differences result from the shearing of the polymer solutions [8] which are ubiquitous as lubricants.

Within the tribology literature, it is often recommended to set μ 2 equal to the viscosity of the base oil or the viscosity of the oil less polymer [9]. In the rheology literature, where the second Newtonian viscosity has been measured with precision, the second Newtonian viscosity generally is greater than the viscosity of the solvent [10] and occasionally less, but seldom equal. There has been some work to determine the relationship between the intrinsic second Newtonian viscosity and the intrinsic first Newtonian viscosity [11]; however, the principles obtained do not lead to an estimation of μ 2.

It is usually difficult to observe both the first and second Newtonian plateaus in a single experimental flow curve. Experience also indicates that the approach to the second Newtonian plateau is often interrupted by mechanical degradation of the polymer, complications due to shear heating, or the onset of shear dependence of the base oil. Gear oils are less likely to display a clear inflection than motor oils. When a second Newtonian appears, the ordinary meaning of n, \( = \partial \tau /\partial \dot{\gamma } \) in the power-law regime, does not apply unless \( \mu_{2} /\mu < 0.03 \). For such a low value of \( \mu_{2} /\mu \) the second Newtonian would not appear for any ordinary lubricant since the base oil would usually be shear-thinning before the inflection would be observed.

An example for which the first Newtonian appears, as well as an inflection which may be characterized by a second Newtonian viscosity, is shown in Fig. 1. This flow curve for a multigrade motor oil was obtained with a pressurized, thin-film Couette viscometer [6]. The curves plotted in Fig. 1 represent a remarkably useful modification of the Carreau [12] equation

Fig. 1
figure 1

Flow curves for a motor oil obtained with a pressurized, thin-film Couette viscometer

$$ \eta = \mu_{2} + \left( {\mu - \mu_{2} } \right)\left[ {1 + \left( {\frac{\tau }{G}} \right)^{2} } \right]^{{\frac{{1 - \frac{1}{n}}}{2}}} $$
(3)

Notice that in Fig. 1, the value of G, which establishes the limit of Newtonian response has been held constant, a useful and simple application of the time–temperature–pressure superposition principle. The curves can be superimposed by shifting vertically. A useful relation for estimating the Newtonian limit, G, for a polymer solution [13] is

$$ G \approx \frac{{c\rho R_{\text{g}} T}}{{M_{\text{W}} }} $$
(4)

where M W is the molecular weight of a monodisperse polymer of concentration, c, in a solution of mass density, ρ, and absolute temperature, T in Fig. 1, the value of μ 2/μ has also been held constant with good result. The effective shear modulus, G and the power-law exponent, n only have the usual meanings [6] when μ 2 ≪ μ, that is 1 − 1/n equals the slope on a log–log plot of viscosity versus shear stress.

Reynolds equations for Double-Newtonian shear-thinning have been analytically derived for the one-dimensional case [14] and the two-dimensional case [15]. Correction formulas have already been derived from numerical experiments for single-component liquids for which a second Newtonian plateau is not expected [1621]. One of these, [21], is unusual in that the viscosity function takes the form of an empirical expression for thermal softening from viscous heating rather than for shear-thinning. Corrections are generated by calculating film thicknesses for the Newtonian and shear-thinning rheologies over some range of load, geometry and rolling velocity. The correction factor is the ratio of Newtonian to shear-thinned film thickness. Here, the same approach is taken for double-Newtonian shear-thinning lubricants.

2 Numerical Experiments

In this section, the global numerical procedure employed in this work is described. The authors employ the finite element full-system approach described in [22] for solving the EHL problem. The goal is to model a lubricated contact between a sphere and a plane under a prescribed external load. Both contacting bodies are elastic and have constant surface velocities. Surface separation is insured by a complete lubricant film. In this work, only pure rolling conditions are considered under mild mean entrainment speeds. Therefore, thermal effects are neglected.

In the Full-System approach, the generalized Reynolds, linear elasticity and load balance equations are solved simultaneously. The Reynolds equation for a steady state point contact lubricated with a generalized Newtonian lubricant under unidirectional surface velocities in the x-direction is given by Yang and Wen [23]:

$$ \frac{\partial }{\partial x}\left[ {\left( {\frac{\rho }{\eta }} \right)_{\text{e}} h^{3} \frac{\partial p}{\partial x}} \right] + \frac{\partial }{\partial y}\left[ {\left( {\frac{\rho }{\eta }} \right)_{\text{e}} h^{3} \frac{\partial p}{\partial y}} \right] = 12\frac{\partial }{\partial x}\left( {\rho^{*} U_{\text{m}} h} \right) $$
(5)

where: \( \begin{array}{*{20}c} {U_{\text{m}} = \frac{{u_{\text{p}} + u_{\text{s}} }}{2}} \hfill & {\left( {\frac{\rho }{\eta }} \right)_{\text{e}} = 12\left( {\frac{{\eta_{\text{e}} \rho^{\prime}_{\text{e}} }}{{\eta^{\prime}_{\text{e}} }} - \rho^{\prime\prime}_{\text{e}} } \right)} \hfill \\ {\rho^{*} = \frac{{[\rho^{\prime}_{\text{e}} \eta_{\text{e}} (u_{\text{s}} - u_{\text{p}} ) + \rho_{\text{e}} u_{\text{p}} ]}}{{U_{\text{m}} }}} \hfill & {\rho_{\text{e}} = \frac{1}{h}\int_{0}^{h} {\rho \,{\text{d}}z} } \hfill \\ {\rho^{\prime}_{\text{e}} = \frac{1}{{h^{2} }}\int_{0}^{h} \rho \int_{0}^{z} {\frac{{dz^{\prime}}}{\eta }dz} } \hfill & {\rho^{\prime\prime}_{\text{e}} = \frac{1}{{h^{3} }}\int_{0}^{h} \rho \int_{0}^{z} {\frac{{z^{\prime}{\text{d}}z^{\prime}}}{\eta }{\text{d}}z} } \hfill \\ {\frac{1}{{\eta_{\text{e}} }} = \frac{1}{h}\int_{0}^{h} {\frac{{{\text{d}}z}}{\eta }} } \hfill & {\frac{1}{{\eta^{\prime}_{\text{e}} }} = \frac{1}{{h^{2} }}\int_{0}^{h} {\frac{{z\,{\text{d}}z}}{\eta }} } \hfill \\ \end{array} \)

Note that this equation accounts for the variations of viscosity across the film thickness as can be seen in the integral terms. In fact, the changes in viscosity stem from shear rate variations across the lubricant film. Moreover, both density and viscosity are allowed to vary with pressure as described in the following section. Indices p and s correspond to the plane and the sphere, respectively, and η is the generalized Newtonian viscosity. The film thickness h is defined from the film thickness equation:

$$ h(x,y) = h_{0} + \frac{{x^{2} + y^{2} }}{2R} - \delta (x,y) $$
(6)

where h 0 corresponds to the rigid body separation and δ the equivalent elastic deformation of both contacting solids obtained by solving the linear elasticity equations on a large solid representing a semi-infinite medium as described in [22].

Finally, the load balance equation insures the correct external load F is applied to the contact by balancing it with the integrated pressure field over the contact area:

$$ \int {p\,{\text{d}}\Upomega } = F $$
(7)

This equation insures load balance by monitoring the value of the rigid body separation variable h 0. The generalized Reynolds, linear elasticity and load balance equations are solved simultaneously using a finite element discretization and a non-linear damped Newton resolution procedure. For more details about the technical implementation of the numerical scheme employed in this work, the reader is referred to [22].

3 Selection of Rheological Parameters

Two representations of the pressure dependence of viscosity of lubricating oils were used in this analysis. Both are based upon the Doolittle free volume relation and utilized the Tait equation of state to supply the specific volume of the liquid. The pressure dependence of the density also comes from the Tait equation. The parameters of the Tait equation of state are the universal parameters proposed in Ref. [6, p. 70]. The two sets of Doolittle parameters are the model strong liquid and model fragile liquid in Ref. [6, p. 123]. The temperature was assumed to be 60 °C for the model strong liquid and 80 °C for the fragile liquid. The reference viscosity was 0.3 Pa s, resulting in ambient pressure viscosities of 0.0302 and 0.0123 Pa s for the strong and fragile liquids, respectively, and reciprocal asymptotic isoviscous pressure coefficients of 14.6 and 18.3 GPa−1, respectively. The fragility classification of glass-forming liquids [24] provides a useful means of describing the viscosity dependence on temperature and pressure at high pressure.

The shear dependence of viscosity was given by Eq. (2). The various combinations of parameters of this viscosity function should be representative of the behavior of real lubricants. The combinations of G, n, and μ 2/μ plotted as the solid points in Fig. 2 were obtained from curve fitting of flow curves of motor oils and gear oils. Twelve motor oils were investigated in this work and eight clearly showed an inflection which allowed the determination of μ 2/μ. The remaining data were not used. Five additional oils were included in Fig. 2 from references [14, 2527] and these include two gear oils. The combinations chosen for the numerical experiments are shown as the open points in Fig. 2. A total of 25 individual combinations of G, n, and μ 2/μ (as summarized in Table 1) were investigated numerically.

Fig. 2
figure 2

Experimentally measured values of the second to first Newtonian viscosity ratio and the power-law exponent versus Newtonian limit plotted as solid points and values selected for analysis plotted as open points

Table 1 Combinations of parameters of Eq. (2) employed in the numerical experiments

Both Newtonian and non-Newtonian solutions for central and minimum film thicknesses were generated for 36 permutations involving rolling velocities of 0.1, 0.3, 1, and 3 m/s, reduced radii of 0.005, 0.015, and 0.05 m and Hertz pressures of 0.5, 1, and 1.5 GPa. For the non-Newtonian case, each of these permutations was investigated for 25 combinations of the shear-dependent viscosity parameters shown in Table 1.

4 Film Thickness Results and Derivation of Correction Formula

Each of the 900 results for central and minimum film thicknesses using the pressure-viscosity response of the model strong liquid were treated by dividing into the corresponding Newtonian result to yield values of φ.

$$ \varphi = \frac{{h_{\text{Newt}} }}{{h_{\text{nonNewt}} }} $$
(8)

The task of deriving a correction formula amounts to find an expression that approximates φ. In many past works [1619], an inlet Weissenberg number was used to quantify the shear stress of the inlet flow relative to the Newtonian limit for the liquid.

$$ \Upgamma = \mu_{0} \bar{u}/h_{\text{cNewt}} G $$
(9)

where h cNewt is the Newtonian solution for central film thickness. In the past [1619], it has been useful to employ, as the correction formula, the functional form of the viscosity law, with Γ substituted for the usual Weissenberg number. This is not surprising since the film thickness should vary roughly with viscosity raised to the 2/3 power. This technique is used here. Seven trial functional forms were tested and the form which yielded a combined low standard deviation and simplicity is

$$ \frac{1}{\varphi } = \left( {\frac{{\mu_{2} }}{\mu }} \right)^{a} + \left( {1 - \left( {\frac{{\mu_{2} }}{\mu }} \right)^{a} } \right)[1 + b\Upgamma ]^{(n - 1)} $$
(10)

where n is simply the power-law exponent in the constitutive law. The two parameters a and b and the standard deviations are listed in Table 2 for central and minimum thicknesses obtained from a least squares regression.

Table 2 Parameters and standard deviations of Eq. (10)

The formula (10) was derived for the case of the strong liquid. Applying it to the results for the fragile liquid resulted in standard deviations of 5.0 and 6.1 % for central and minimum film thicknesses, respectively. Fragility appears at high pressures, pressures greater than the inflection pressure, where it has a profound effect on friction; however, the effect on film thickness is not significant.

Another approach is taken to benefit from the extensive data obtained from the numerical experiments. A complete expression for the film thickness can be written as the product of a Newtonian solution and the correction Eq. (10). If the Newtonian solution is put in terms of the three Blok [28] dimensionless numbers

$$ H = \frac{h}{R}\left( {\frac{ER}{{2\mu_{0} u}}} \right)^{\frac{1}{2}} ,M = \frac{F}{{ER^{2} }}\left( {\frac{ER}{{2\mu_{0} u}}} \right)^{\frac{3}{4}} ,L = \alpha E\left( {\frac{{2\mu_{0} u}}{ER}} \right)^{\frac{1}{4}} , $$
(11)

The full formula reads

$$ H_{\text{Newt}} = AL^{B} M^{ - C} $$
(12)
$$ H = AL^{B} M^{ - C} \left\{ {\left( {\frac{{\mu_{2} }}{\mu }} \right)^{a} + \left( {1 - \left( {\frac{{\mu_{2} }}{\mu }} \right)^{a} } \right)[1 + b\Upgamma ]^{(n - 1)} } \right\} $$
(13)

The same 900 results for central and 900 results for minimum film thicknesses were employed in a least squares regression. The parameters and the standard deviations are listed in Table 2 for central and minimum thicknesses. Although Eq. (13) yields greater standard deviations, it has the advantage of not requiring an independent Newtonian prediction. Here, the exponent, B, is similar to the classical solutions while C is greater. The Tait equation yields a greater compressibility than the equation of state assumed in the classical formulas (Table 3).

Table 3 Parameters and standard deviations of Eq. (13)

5 Experimental Validation

For experimental validation of the new correction formulas both rheological data and film thickness data are required for the same material, and there are few examples available. Fortunately, the film thicknesses have been measured for one of the reference liquids of reference [29], a solution of 15 % by weight cis-polyisoprene (M = 4 × 104 Daltons) in squalane. New viscosity data at 450 MPa pressure from [30] are shown in Fig. 3 along with data from Ref. [29]. Curves plotted in Fig. 3 are Eq. (2) with G = 23 kPa, n = 0.65, and μ 2/μ = 0.28. In Fig. 3, the curves do not shift vertically because the horizontal axis is shear rate.

Fig. 3
figure 3

Experimentally measured values of the viscosity of SQL + PIP compared with Eq. (3)

Film thicknesses for this liquid, obtained from optical measurements, have been reported in reference [31] for a circular contact formed by a 12.7 mm radius ball against plane with combined elastic modulus of 124 GPa. The load was 23 N to give a maximum Hertz pressure of 0.47 GPa. The test temperature, at 40 °C, results in μ 0 = 0.0711 Pa s and α = 18.53 GPa−1.

The Hamrock and Dowson Newtonian film thickness formulas [2] were used for validation of the correction formulas (10) in Figs. 4 and 5. The corrected film thicknesses are shown to improve the film thickness predictions. The central prediction improved from an average deviation of 41 to −9 % and the minimum prediction improved from an average deviation of 44 % to −11 %.

Fig. 4
figure 4

Experimentally measured central film thicknesses compared to the Hamrock and Dowson prediction and the corrected prediction

Fig. 5
figure 5

Experimentally measured minimum film thicknesses compared to the Hamrock and Dowson prediction and the corrected prediction

Next, the full film thickness formulas (13) are compared to the Newtonian formulas (12) in Figs. 6 and 7. The full film thickness formulas are shown to improve the film thickness predictions over the Newtonian predictions. The central prediction improved from an average deviation of 49 to −6 % and the minimum prediction improved from an average deviation of 74 to 1 %.

Fig. 6
figure 6

Experimentally measured central film thicknesses compared to the present Newtonian prediction and the present full prediction

Fig. 7
figure 7

Experimentally measured minimum film thicknesses compared to the present Newtonian prediction and the present full prediction

6 Conclusions

1. A correction formula has been developed from numerical experiments for a range of parameters of the double-Newtonian modified Carreau equation. The parameters of this shear-thinning model were selected from measurements for real lubricants obtained in Couette viscometers and a capillary viscometer.

2. In addition, a full EHL film thickness formula has been derived from the same numerical experiments. These formulas should not be applied to liquids with a single Newtonian plateau or to cases of μ 2/μ < 0.2.

3. The correction formula and the full formula were successfully validated using published film thickness data and published viscosity data for an EHL reference liquid, a polymer solution.

4. Clearly, viscometer measurements of shear-dependent viscosity which contain the inflection leading to the second Newtonian are essential for a film thickness calculation when a high-molecular-weight component of the lubricant is present.