## Abstract

Pressure-viscosity coefficients (PVC) are used in the predictions of elastohydrodynamic lubricated (EHL) componentry. These coefficients are obtained by either viscometry or optical EHL inference. The literature indicates that each method differs in its conclusion. Those who favor viscometry believe optical methods yield a misleading coefficient. Those who favor optical methods suggest low shear viscometric results over-predict the high shear-influenced film thickness. This work compares each method relative to di-(ethylhexyl) sebacate (DEHS), and five MIL and DOD spec lubricants. PVC results from viscometry and two optical methods are presented. Comparisons are made relative to other published measurements. Conclusions show PVCs inferred from optical film thickness measurements, differ from those obtained by viscometry. Viscometry methods are demonstrated as being consistent. Optically inferred results have uncertainty and require ample data to align with classical dimensionless speed exponents. While the optical measurements are truly EHL, the test conditions fall outside the fitted window of classical algebraic film equations, like that of Hamrock and Dowson. The PVC discrepancies, between optical inference and viscometry and for the studied fluids, cannot be explained by the proper account of the refractive index, shear thinning models, and/or film thickness correction models.

## Introduction

Elastohydrodynamic lubrication (EHL) theory demonstrates how relative motion can entrain a viscous liquid into a geometric wedge, giving rise to pressure. If the pressure is sufficient, the relatively moving solids are separated by a thin load-carrying fluid film. The fluid film thickness within the conjunction becomes relatively uniform or centralized. The load-carrying pressures are often sufficient to elastically deform the abutting solids and increase the viscosity of the entrained fluid. Knowledge of the film thickness is of value—be it for design, quantifying efficiency, or assessing fatigue.

There are many isothermal EHL solutions that predict film formation. They collectively solve the Reynolds equation, a geometric expression describing the film, equations of elasticity, and functions for the lubricant’s pressure-dependent properties [1]. Classical property equations include the Dowson and Higginson density model [2] and Roelands pressure-dependent viscosity model [3]. Film thicknesses are grossly underestimated when these pressure-dependent behaviors are overlooked [2].

Altogether, EHL calculations can be computationally expensive and/or time-consuming. A number of researchers have regressed these solutions to algebraic expressions, all of which use atmospheric dynamic viscosity and a pressure-viscosity coefficient (PVC) [4–9]. The well-known Walther viscosity equation is suitable for calculating the temperature-dependent atmospheric viscosity; however, there is no parallel ease for the PVC. Data available to calculate PVCs are limited and specific to individual isotherms. To date, there are no widely accepted temperature-dependent PVC equations.

^{2}Bair [12] offers Eq. (2) as a numerical equivalent to Eq. (1).

*a*)

*b*)

There are two methods for obtaining PVC. Some compute either Eq. (1) or Eq. (2) from viscometry measurements. Others infer the PVC by aligning optically measured films to film thickness models such as Hamrock and Dowson’s.

High-pressure viscometry methods use either a falling body apparatus [15], a rolling ball apparatus [16], a diamond anvil cell [17], or other means [11] to measure lubricant viscosity. The falling body viscometers used in this work utilized an intensifier to increase pressure within the test chamber. The pressurized chamber contains both the test fluid and a sinker. Inversion of the chamber utilizes gravity to pull the sinker through the fluid. As the sinker falls, a linear variable differential transformer (LVDT) records its time of flight. These rigs are calibrated so fall time can be correlated to viscosity. Bair [15] details the falling body assembly referenced herein, and Sadinski et al. [18] discuss the digital signal processing. Published viscosities have been measured up to temperatures of $220\u2218C$, and pressures from near 0 up to 1.4 GPa, [15]. Once fitted to a viscosity model [18], isothermal viscosities can then be inserted into Eq. (2) or (numerically) integrated using Eq. (1).

Optical methods, [19–32], measure the EHL films generated between a transparent glass disk and a highly reflective ball. The disk is commonly coated with a semi-reflective layer containing a targeted reflectivity of ∼20% [19]. Light undergoes a division of amplitude as it passes through the coating. A portion of the light reflects, and a portion refracts through the lubricant and onto the steel ball surface. The light then reflects off the ball and returns to the disk. Some reflected light from the ball passes back through the underside of the coated disk. This causes the light reflected from the interior surface of the glass disk to constructively or destructively interfere with the out-of-phase light reflected from the ball surface.

*λ*is the wavelength,

*N*is the fringe order,

*ϕ*is a phase shift (which, although varies with the transmitting media, is assumed to be significantly less than the fringe order,

*ϕ*≪

*N*),

*h*

_{sp}is the spacer thickness, $noil*$ is the adjusted lubricant refractive index, and

*n*

_{sp}is the spacer refractive index. Optical methods that use Eq. (4) directly are referred to as spectral methods within this paper [26]. A second method, referred to as colorimetry, is anchored in the same principles. Colorimetric methods differ by using a calibration to associate the colors produced by the constructive interference to fluid film thickness values [19].

*α*

_{opt}) by Eq. (5). The inferred value is referred to an “effective PVC” in SAE ARP6157 [33].

*η*

_{W}is atmospheric dynamic viscosity, calculated herein from density and Walther’s kinematic viscosity (

*ν*

_{cSt,W}=

*η*

_{W}/

*ρ*

_{oil}) equation,

*C*

_{HD}is a constant encapsulating correction factor; in the non-lubricant-specific terms of Eq. (3), $CHD=2.69\u22c5\Phi \u223c\u22c5Rx\u22c5(U\eta W)0.67\u22c5$$(G\alpha *)0.53\u22c5W\u22120.067\u22c5(1\u22120.61\u22c5exp(\u22120.73k))$, in which $\Phi \u223c$ denotes the product of all correction factors.

Competing works, [10,24,31,34–47], claim pros and cons for each approach. Published data show the two methods can yield differing results. At $25\u2218C,$ viscometry measurements of di-(ethylhexyl) sebacate (DEHS) yield a PVC of 14.97 GPa^{−1} [48], while optical measurements infer 7.5 GPa^{−1} [30]. The work of Westlake and Cameron [24] is often cited and offers a seemingly favorable comparison. However, some doubt has been cast by the paper’s appended discussion, written by Winer and Sanborn, and a published correction [28].

Those in favor of viscometry claim establishing PVC from viscosity measurements is emphatic. Their measurements are consistent and not beholden to complex tribological events. In addition, the pressure- and temperature-dependent viscosity measurements allow one to assess common EHL assumptions.

The same emphasis cannot be placed upon optical methods. van Leeuwen [31] tabulated optically inferred PVCs (*α*_{opt}) for a reference oil, HVI60. *α*_{opt} varies depending on which of the 11 film thickness equations is aligned with the measured film thickness. If one wishes to use the Archard and Cowking equation [9], the suggested *α*_{opt} is 28.4 GPa^{−1}. For Hamrock and Dowson [6], Chittenden [7], or Nijenbanning et al. [8], the suggested *α*_{opt} values are 18.1, 19.3, and 15.5 GPa^{−1}, respectively. This difference results from the unique exponents within each film thickness model.

Jackson [49] adds another approach that uses a lubricant parameter (LP = *η* · *α*) to apply equal exponential (*δ* = 0.74) weight to velocity, viscosity, and the PVC [50]. Despite these differences, optical results have long been labeled as giving “very precise data” [20] and they “correlate very well with values obtained from conventional viscometers” [24].

Those in favor of optical measurements suggest the results are tribologically inclusive. Both shear and thermal effects, which are not included in basic film thickness models, are used to explain the differences between viscometry and optically inferred PVCs. Optically inferred values include these effects, while low shear viscometry neglects them. Westlake and Cameron [24] state “it is unlikely the conventional low-shear, high-pressure viscometers will give meaningful *α*-values for fluids other than those of the highest shear stability.”

SAE [51] labels high-pressure viscometers as being “benign” to shear rate. Cheng [50] notes early film thickness equations were “grossly inadequate” when compared to experimental results. Cheng listed three possible causes for the differences: (1) the heating effect in the inlet, (2) the inability of the viscosity to rise with pressure according to the static pressure-viscosity experiment during a short time interval, and (3) loss of viscosity due to high shear rates. Studies in thermal [52,53], starvation [23,25,54,55], and shear thinning effects [56–59] resulted in the development of many daisy-chained film thickness correction models, $\Phi \u223c$. Use of these compartmentalized models within Eq. (5) alters the reported PVC values. An “effective” PVC, determined with or without correction models, holds to the view that it supersedes outstanding model errors.

This paper offers a comparison by studying one reference fluid (DEHS) and five commercially available ester-based fluids. DEHS, an SAE APR6157 [33] recommended confirmation lubricant, has been heavily studied in viscometry. Two of the commercial fluids conform to MIL-PRF-23699 (Military Specification), and three conform to DOD-PRF-85734 (Department of Defense). Viscometry [18,60,61] and optical EHL measurements were performed in the authors’ laboratory. A discussion of the results addresses the differing methods, assumptions, and references in the existing literature. Published measurements are referenced. Unreferenced measurements are the results of the authors.

## Background

SAE APR6157 [33], an aerospace recommended practice, suggests optical interferometric methods can be used to obtain a PVC. SAE ARP6157 supports SAE AS5780 [62], which requests that aero and aero-derived gas turbine engine lubricants have their PVCs optically inferred from films measured at 40 °C, 70 °C, 100 °C, 130 °C, and 150 °C. SAE defines these temperatures as being the “average temperatures of two trailing thermocouples on the ball, 1 mm from the inlet zone.”

SAE further recommends using one of two different instruments manufactured by either PCS Instruments Ltd. (London, UK) or Wedeven Associates Inc. (Edgmont, PA, United States). These two machines operate with different ball sizes (19.05 mm diameter for PCS and 20.6 mm diameter for Wedeven). The PCS instrument uses a spectral method consistent with the work reported by Johnston [29]. The Wedeven Associates Inc. manufactured machine utilizes colorimetry consistent with Foord [21]. Tests are performed over incremental speeds. SAE expects a PVC accuracy of ±1.5 GPa^{−1}.

Figure 1 plots the PVC as it applies to Grade 5 aviation propulsion and drive system lubricants, which include MIL-PRF-23699 and DOD-PRF-85734 oils. The red curve represents the manufacturer’s published [63] Mobil Jet II optically inferred PVC values. Optically inferred values from both SAE ARP6157 () and SAE AIR5433 () are also included. The optically inferred SAE data were fit using Eq. (7) [64]. Viscometry results () are a combination of published [34] and author-measured data. The black curve is a fit of the viscometry data to a Vogel, Fulcher, and Tamman (VFT)-like equation (Eq. (6), [18]). The shaded area represents a best fit of the maximum and minimum spread of all Fig. 1 data. At 40 °C, the mean gray area PVC is 13.2 GPa^{−1}, while the manufacture’s PVC is 16.1 GPa^{−1}. At 200 °C, the mean gray area PVC is 8.3 GPa^{−1}, while the manufacture’s PVC is 5.6 GPa^{−1}. At 100 °C, the viscometry and optical SAE measurements, which span the gray region in Fig. 1, differ by $22%$ or 1.9 GPa^{−1}. As predicted by Hamrock and Dowson [6, Eq. (33)], this variation equates to $1\u2212(\alpha */\alpha opt)0.53$, $11%$ difference in the central film thickness.

## Viscometry and Optical Pressure-Viscosity Coefficients of DEHS

DEHS viscometry data can be found in the 1953 ASME [65] report. DEHS measurements have also been made by Vergne [48], King [17], Paredes [16], and Sadinski [18]. Table 2 summarizes Eq. (6) constants for the referenced data sets, and Table 3 lists the $25\u2218C$ PVCs. ASME [65] data were analyzed in three ways: (i) isothermal data were inserted into Eq. (2), (ii) isothermal data were fit to the McEwen (Eq. (8)) model [15,48], and (iii) Eq. (1) was integrated via Gauss–Legendre quadrature for a fit to the modified Yasutomi Eq. (9) [18].

*P*

_{∞}to 500 MPa. Table 3 viscometry-based PVCs span 68 years of measurements, reflect on purity [17] and were computed using three methods. The average of all $25\u2218C$ Table 3 PVCs is 14.09 GPa

^{−1}, and its uncertainty is

*t*

_{9,95}·

*S*

_{x}= 0.37 GPa

^{−1}.

*t*

_{ν,c}represents the student

*t*with

*ν*degrees-of-freedom and

*c*% confidence.

*S*

_{x}represents the standard error.

*a*)

*b*)

*c*)

Figure 2 represents the authors’ ability to reproduce the SAE recommended measurements (squares) using either the spectral method (stars) or colorimetry (triangles). The SAE results in Fig. 2 have been increased by 1.74% to equate the published (*Q* = 20 N) and measured (*Q* = 25.9 or 44.5 N) results. These loads result in a consistent mean (384 MPa) and maximum (576 MPa) Hertzian pressure. The spectral results are quite similar, but the colorimetry results deviate as temperature increases. It should be noted the disk and ball $E*$ were assumed to be 117.0 GPa for all tests.^{3}

*n*

_{oil}, at atmospheric pressure; however, the refractive index is a function of the lubricant pressure, temperature, and the wavelength of light passing through the film,

*n*

_{oil}(

*T*,

*P*,

*λ*). Both Foord [21,22] and Wedeven [23,25,54] made use of the Lorenz-Lorentz relationship (Eq. (11)) and Hartung’s density expression to adjust the refractive index (and thereby film measurements) with respect to pressure. The Lorenz-Lorentz equation relates the refractive index of a substance to its number of molecules (

*N*

_{mol}) and polarizability (

*α*

_{polar}), the product of which is proportional to density

*ρ*

_{oil}. Table 2 lists the pressure- and temperature-independent constant of proportionality

*c*

_{LL}.

*a*)

*b*)

*c*)

*d*)

*A*

_{C}= 1.4341,

*B*

_{C}= 8.843 × 10

^{3}, and

*C*

_{C}= −9.0176 × 10

^{8}. Equation (14) was used to calculate the corrected index of refraction using the Abbe refractometer measurement as the reference value (λ

_{ref}).

Film thicknesses were measured using both spectral and colorimetric methods. Data from both methods adjusted the refractive index for temperature and pressure. Additionally, the spectral method was corrected for wavelength. Optically inferred values were calculated using a thermal correction factor (Φ_{Thermal}) per [64] and as defined by Gupta [53]. The temperature-viscosity coefficient (TVC) was obtained from the derivative of the modified Yasutomi model. Thermal conductivity^{4} was defined by $kth=0.1467\u22121.588\xd710\u22124\u22c5(T\u221215\u2218C)$. Equation (5) was then used to obtain *α*_{opt} for each entrainment velocity. Table 4 lists the median *α*_{opt}, of *M*_{u} incremental entrainment velocity, for a given load (*Q*) and temperature (*T*).

The optical results were heavily scrutinized. For example, refractive indices were corrected for pressure effects post-test, post-test calibration offsets were subtracted from Eq. (4) deduced film thicknesses, lens cleanliness was ensured by omitting repeated measurements whose individual standard deviation in excess of 20 nm, and the occasional fringe order N error were addressed. At 25 °C, the mean PVC of all repeated tests (spectral method only) is 12.86 GPa^{−1} and the combined uncertainty is 0.88 GPa^{−1}. Table 4 also lists the results of the authors’ viscometry PVC $\alpha *$. On average, the optically inferred PVC is $10.5%$ less than the viscometry PVC.

A least-squares fit of the log of entrainment velocity relative to the log of the measured film thicknesses yields the experimental exponent *δ*. This exponent, which acts on the dimensionless velocity parameter *U*, is plotted in Fig. 3. The annotations align with the Table 4 test numbers. Sixteen DEHS tests were conducted in total. The average exponent of 11 spectral DEHS tests is *δ* = 0.69, and the average of five colorimetry tests is *δ* = 0.79. Figure 3 also includes exponents from two MIL-PRF-23699 and three DOD-PRF-85734 oils, obtained using both spectral and colorimetric methods. The average of all spectrally obtained results is *δ* = 0.68. The average of all colorimetry obtained results is *δ* = 0.70. The respective spectral and colorimetric exponent uncertainties are *t*_{49,95} · *S*_{X} = 0.01 and *t*_{30,95} · *S*_{X} = 0.08.

Foord et al. [21] used a colorimetric method to measure film thickness on 34 different types of oils. The average exponent from their measurements is *δ* = 0.64, with *t*_{33,95} · *S*_{X} = 0.05 uncertainty. The average exponent of the spectral, colorimetry, and Foord et al. data is 0.67 and *t*_{115,95} · *S*_{X} = 0.01. Figure 3 also plots *δ* = 0.67, the theoretical value used in the Hamrock and Dowson film thickness equation. Other film models make use of differing exponents, including Archard and Cowking (*δ* = 0.74) [9], Cameron and Gohar (*δ* = 1.0) [19], Chittenden (*δ* = 0.67) [7], Cheng (*δ* = 0.725) [50], Wedeven, Evans, and Cameron (*δ* = 0.725) [25], and Grubin (*δ* = 8/11 ≅ 0.727) [2].

Figure 4 plots viscometry and optically inferred PVCs relative to temperature. Solid lines represent Eq. (1) integration of fits to the authors’ viscometry with *P*_{∞} = 500 MPa [16,18,65,66]. The authors’ optical PVC values are plotted by () using the atmospheric index of refraction, () adjusting the index to reflect the mean Hertzian pressure, and () adjusting the refractive index throughout the Hertzian domain. The mean pressure-adjusted points used the Lorenz-Lorentz formula and Murnaghan’s equation of state. Commonly referenced [15,37] optical measurements [30,46] are also included and designated with a blue circle ().

In general, the optically inferred values agree best with viscometry when the atmospheric refractive index is used; this observation is in line with Bair [69]. Table 3 summarizes viscometry PVC values at 25 °C, computed in several ways. Four repeat tests of ∼22 measurements were made using the spectral optical EHL method at 25 °C. One test of 216 incremental speed measurements at 25 °C was made using colorimetry. The optically inferred PVCs are: (a) 14.08 GPa^{−1} when using the atmospheric refractive index, (b) 12.86 GPa^{−1} when adjusting the refractive index by the mean Hertzian pressure, and (c) 11.82 GPa^{−1} when adjusting the refractive index throughout the Hertz domain. The respective uncertainties are *t*_{87,95} · *S*_{X} = 0.15, *t*_{87,95} · *S*_{X} = 0.15, and *t*_{215,95} · *S*_{X} = 0.32 GPa^{−1}. The numbers within Figs. 3 and 4 align with Table 4. Tests 2, 3, 5, 6, 8, and 10 have exponents that agree (within 3%) with Hamrock and Dowson. PVC values of tests 2 and 10 agree with viscometry; conversely, the other tests do not.

There are also assumptions of analysis. Here, the spectral method’s refractive index is assumed to be proportional to the mean Hertzian pressure, as opposed to the maximum or distributed pressure. This differs from works such as Johnston [29]. The optical field of view includes the maximum pressure and regions of less pressure. The mean pressure is a comment on the entire domain, and using it causes an underestimation of the field of view. The use of either value is wrong; however, the optically inferred values are slightly less so than those reported when the maximum pressure is used.

Bair [15] uses the two-parameter McEwen model to compare the viscometry and optical PVCs’ effect on high-pressure viscosity. Bair plots the Vergne [48] viscosity-based McEwen model: *α*_{McE} = 14.97, *q* = 7.06. Bair then exchanges the viscosity-based PVC with the optically inferred conclusion of Guangteng and Spikes [30]: *α*_{McE} = 7.5, *q* = 7.06. Bair demonstrates that *q* is non-influential, albeit with an inferred PVC value that differed greatly from viscometry results. Figure 5 offers a similar comparison using the median PVC values from Table 3 (*α*_{McE} = *α*_{opt}) and Eq. (8) with *q* = 7.06, [15,48]. Viscometry data from [18,48,65], and fits to data from [16,18,65], are also included.

There is a $\u223c15%$ difference between the optically inferred and viscometry PVCs. If *q* = 7.06, the viscometry data and the optical McEwen prediction diverge at pressures above 100 MPa. The predicted viscosity from the optically inferred PVCs can be made to reasonably align with rheology if the McEwen parameter *q* = 14 (spectral) and *q* = 20 (colorimetric). It should be noted that *q* can only be determined from high-pressure viscometry data.

## Optical Pressure-Viscosity Coefficient Uncertainty

The reported spectrally inferred PVC uncertainty, *t*_{87,95} · *S*_{X} = 0.15 GPa^{−1}, quantifies experimental precision, and Eq. (16) defines the total error (the sum of bias and precision). Equation (16) assumes errors in film thickness, viscosity, and the dimensionless speed exponent *δ*. Equation (16) neglects errors in effective moduli, refractive index, load, and geometry. The uncertainty on the measured film is assumed to be 5% of the mean of the measured film heights, $Uhopt=0.05h\xafopt$. At 25 °C, the mean film height of DEHS is on the order of 120 nm.

Uncertainty in viscosity results from inaccuracies of the Walther fit (a density-dependent fit to Paredes data and statistically compared against ASME atmospheric pressure data) and thermocouple uncertainty. The Murnaghan density equation permitted converting kinematic viscosity (*cSt*) to dynamic viscosity (*cP*). Errors in density were omitted given the accuracy of the Murnaghan fit, which gives a root-mean-square prediction error (RMSPE) of 0.08. These effects combine to yield $U\eta W=0.43cP$.

At speeds of 0.5, 1, and 2 m/s, the uncertainty is $U\alpha opt=[2.242.222.13GPa\u22121]$. If the first-order uncertainty on *δ* is excluded, $U\alpha opt=[1.861.861.77GPa\u22121]$, and if all errors in *δ* are omitted, $U\alpha opt=[1.331.331.27GPa\u22121$]. Uncertainty is heavily dependent on *δ*, whether it be precision or bias. It is also dependent on viscosity, be that atmospheric viscosity or the temperature used to compute viscosity. Only when the exponent effects are omitted from the analysis can one arrive at certainty on the order of the ±1.5 GPa^{−1} required by the SAE.

## Comparison of MIL-SPEC Oils

Sadinski [60,61] published a rheological study on aerospace lubricants that includes the parameters necessary to characterize the rheological response of two MIL-PRF-23699 HTS and three DOD-PRF-85734 oils. For comparison, optical film measurements of these fluids were performed in accordance with ARP6157 [33] and processed in the same manner as those of DEHS. Table 5 lists the bounds of the optical film measurements, and Figs. 6 and 7 plot viscometry and optically inferred PVCs. Table 6 compares the differences.

The plots of viscometry are based on Eq. (6) fits of $\alpha *$. The plots of the spectral optically inferred results are based on Eq. (7) fits of *α*_{opt}. 65% confidence bands are shown for both methods of optical inference. The error bars are less for the spectral data than for the colorimetry assembly.

The viscometry results of all five oils are similar regardless of oil type. Viscometry shows a PVC $\alpha *$of AeroShell 555 that is smaller than the other oils, while the fit of the optically inferred data shows the opposite. When the AeroShell 555 spectral measurements were repeated, the reported optically inferred PVCs at 150 °C were at times equal to or larger than those observed at 100 °C; *α*_{opt} (100 °C) = [10.42, 9.12, 9.60 GPa^{−1}] and *α*_{opt}(150 °C) = [8.37, 9.12, 9.39 GPa^{−1}]. Similar high-temperature trends are evident in the colorimetry results as well.

Although this result is inconsistent with high-pressure rheology, the anomaly may be due to the accumulation of atomized liquid vapor on the microscope lenses, which occurs with either optical method at temperatures above 100 °C. Excluding the 150 °C results, the optical conclusions align with the gray bands of Fig. 1. At 40 °C, the optically inferred PVC is $\u223c$20% less than the viscometry PVC (Table 6). The discrepancy diminishes at elevated temperatures.

The experimental exponents of each test are plotted as open symbols within Fig. 3. It should be noted that the above comments are specific to the mean value of many optical measurements. When considering the experimental error, one notes that as temperature increases (above 60 °C), the reported PVCs are seldom statistically different.

## Discussion of Results

The presented optically inferred PVCs are smaller than those produced by viscometry and have less repeatability. Discrepancies may be due to: (i) the general meaning of the PVC, (ii) the refractive index used to convert wavelengths to film thicknesses, (iii) variation in the experimental dimensionless velocity exponent, (iv) optical measurements that are iso-viscous as opposed to piezo-viscous, (v) viscometry measurements that are benign to shear thinning, and (vi) compounded errors from correction factors in conjunction with algebraic film thickness equations.

### General Meaning of the Pressure-Viscosity Coefficient.

Quantitative EHL requires knowledge of the lubricant’s piezo-viscous response, which through the definition of a total derivative can be written as *α*_{G}(*P*, *T*) = 1/*η* · ∂*P*/∂*η*. If one adopts the 1893 Barus’s formula, *η*_{B} = *η*_{o} · exp(*α*_{B}*p*), *α*_{G} = *α*_{G}(*T*) is independent of pressure. This is contrary to the typical log(*η*) versus pressure response of a lubricant, which shows near-linear behavior at low pressures and a “general concave” response at elevated pressures [70]. This observation, well supported in many early textbooks [71–76], has often been overlooked in favor of the ease with which the Barus model aids EHL formulations.

In 1925, Weibull used a transform, $Piv,as=\u222b0p\u221e\eta o/\eta (p)dp$, to consolidate the piezo-viscous lubricant pressure response *η*(*p*) and momentum pressure gradient ∂*p*/∂*x*. Blok defines *P*_{iv,as} as a “corrected pressure” [77] and a “fictitiously iso-viscous film pressure” [14]. *P*_{iv,as} is also referred to as the “reduced pressure” [2], the “limiting value” that maximizes relative “carrying capacity” [2], and the pressure that would be generated if a fluid viscosity were constant [78]. Roelands [3] states if the viscosity is “exponential [in] variation,” then “the iso-viscous pressure approximates asymptotically to a finite upper limit,” and “the limiting value is the reciprocal … of the viscosity-pressure coefficient,” $\alpha *=1/Piv,as.$ Blok [14] suggests “that with all the published analytical viscosity-pressure relationships … the *P*_{iv,as} value approaches the (asymptote) value fairly closely, within a few percent, when (*P*_{∞}) is greater than something like 3 · *P*_{iv,as} or 4 · *P*_{iv,as}.”

Figure 8 displays Blok’s 1965 [14] plot relative to DEHS. Each isotherm represents Eq. (1) integration of the Yasutomi fit, whereby the graph’s abscissa denotes the upper integration bound *P*_{∞}. The star on each isotherm denotes a “few (3%) percent slope” [14] corresponding to *P*_{∞} ≅ 4 · *P*_{iv,as}. The curves begin at *P*_{∞} = 100 MPa and terminate at *P*_{∞} = 500 MPa. Near the asymptote at 100 °C, the inverse of the iso-viscous pressure is *α*_{B} = 9.3 GPa^{−1}. The reported $\alpha *=9.2GPa\u22121.$ As shown by Bair [79], Dench et al. estimate the optical assemblies’ inlet pressure to be 100 MPa, while Spikes [46] estimates 150 MPa. Films are borne of the inlet, and even assuming 150 MPa, the corresponding 100 °C iso-viscous pressure is 77 MPa and its inverse is 13 GPa^{−1}. Figure 8 also suggests the asymptote is temperature dependent and, excluding the 150 °C isotherm, reached to within a “few percent” at 500 MPa. It is for this reason all Eq. (1) integrations herein have an upper bound of *P*_{∞} = 500 MPa.

The recommended SAE confirmation fluid, when tested under the SAE listed conditions, is incapable of being tested at sufficient inlet pressures to reach the iso-viscous asymptote. Table 8 lists *α*_{opt} at the minimum and maximum entrainment velocities. The maximum entrainment velocity yields the highest inlet pressure, and *α*_{opt} is closest to the viscometry $\alpha *$. Although this agrees with what has been stated above, the trend is actually the opposite. The lowest entrainment velocity yields the lowest *α*_{opt}, but it should in theory [79] overestimate *α*_{opt} and underestimate *P*_{iv,as}.

Many pioneering works, such as Gatecomb 1945 [80], Grubin 1949, and Dowson and Higginson 1962 [81], use various single-parameter exponential viscosity functions [73] and the Weibull transform to import EHL theory. In contrast, McEwen 1952 [82] used a more suitable two-term viscosity power law. In 1966, several pivotal works were published; among them is the extensively referenced Roelands [3] viscosity model.

While the Roelands model is superior to the Barus equation, it overestimates the piezo-viscous “general concave” trend at pressures above 350 MPa [83]. Nonetheless, Hamrock and Dowson model the piezo-viscous response using the Roelands model and Weibull transform. Hamrock and Dowson reference Roelands’ thesis, which shows that PVC can be analyzed in many ways. Two proposed methods include integration of a Roelands fit, or interpolation of the Roelands pressure-viscosity coefficient from Table XIII-2, p. 453 of [3]. Table 7 compares these methods as established at the time of Hamrock and Dowson.

In summary:

There are many definitions of a PVC [12].

When integrating, one should ensure

*P*_{∞}/*P*_{iv,as}≥ 4, and that the sufficient value of*P*_{∞}is temperature dependent.The PVC conclusion of viscometry data is dependent on the method of analysis (Table 7).

The inlet pressures of the optically measured films are significantly less than the viscometry- based iso-viscous pressure asymptote.

Integration of the Roelands equation overestimates the PVC.

Results developed from Roelands’ table are $10%$ less than $\alpha *$.

### Refractive Index.

A lubricant’s refractive index changes with pressure, temperature, and wavelength. The Lorenz-Lorentz relationship models these changes for pressure and temperature. Authors such as Johnston [29,84] neglect such changes by citing Paul and Cameron, who concluded the index changed only 10% through a 1 GPa pressure rise. Foord [21,22] and others [25,54] couple the Lorenz-Lorentz expression to the Hartung [85] density model. Westlake [24] suggests the Hartung equation may not be reliable. The authors used the Murnaghan equation of state in the Lorenz-Lorentz model.

Figure 9 plots the density and refractive index of DEHS using the Murnaghan and Hartung models. Vertical lines annotate the mean (384 MPa) and maximum (576 MPa) Hertzian pressures. For completeness, film thickness measurements were analyzed by both disregarding pressure and adjusting for pressure. Pressure adjustments included studies using the mean Hertzian pressure, the maximum Hertzian pressure, and the Hertzian domain.

Omitting the pressure effects of the refractive index yields a $\u223c5%$ error in the measured film. At 100 °C and 383 MPa, the $2%$ difference between the accurate Murnaghan and inaccurate Hartung predictions translates to a $\u223c1%$ difference in measured film thickness. At 25 °C, the difference between analyzing the results using the mean pressure relative to the maximum pressure equates to a $1%$ difference in film thickness. When the index is adjusted for pressure, the inferred pressure-viscosity coefficient decreases.

In summary:

The errors in modeled density are insufficient to explain the differences between optically inferred and viscometry PVCs.

Proper consideration of the refractive index increases the discrepancy between viscometry and optically inferred PVC values.

### Log-Log Exponent of Velocity on Film Thickness.

Figure 3 summarizes 81 experimentally determined exponents acting on the dimensionless speed parameters. The mean of 50 spectrally determined, 31 colorimetry determined, and 34 published values is 0.67. 115 measurements are required to converge, with normality, to a theoretical value. Table 8 studies variations in the exponent. The difference in measured film height (*h*) relative to the fitted line is, on average, less than $7.5%$. The difference between the fitted and theoretical exponent is less than $6.2%$. Since *δ* is positive and differs from theoretical results, the optically inferred PVC is smaller at low speeds than at high speeds. The minimum adjusted thermal correction factor, Φ_{T}, of each test is near unity.

In summary:

The spectral method gives an exponent in line with Hamrock and Dowson, as well as Chittenden, and its standard deviation is less than demonstrated in other published works.

The theoretical 0.67 is obtained when data are sufficient.

The PVC does not always align with viscometry results, even when the exponent is approximately 0.67 (reference the annotations of Figs. 3 and 4).

Uncertainty in the exponent prevents one from being able to resolve the optically inferred PVCs within ±1.5 GPa

^{−1}.

### Transition Map.

Published optical PVC values have at times been so low (e.g., Ref. [30]) that Bair [36] questioned if optical tests are truly piezo-viscous. The earthly transition from iso-viscous to piezo-viscous is smooth, slow, and infinitely differentiable. Algebraic film equations, born of permuted assumptions regarding the viscous behavior of the lubricant and elasticity of the abutting solids, have an unphysical sharp and piecewise transition. Figure 10 plots the algebraically defined transition map [86] for all 1551 reported optical film thickness measurements performed in this work. Values represented by the solid symbols were computed using the reported optically inferred PVCs. Hollow symbol values were computed using results from viscometry. All individual optical tests are piezo-viscous elastic (PVE). The axes are plotted using the Johnson [87] dimensionless parameters *g*_{E} and *g*_{V}, which have been written in terms of Table 1 dimensionless parameters used by Hamrock and Dowson.

Hamrock and Dowson’s fit of 34 solutions to the EHL Reynolds equation sampled four dimensionless material parameters, $2310\u2264(GHamrock=\alpha \u22c5E*)\u22646785$. Hamrock [1,6] categorizes these extrema as being representative of paraffinic oil paired with bronze (*G* = 2310) and Si_{3}N_{4} (*G* = 6785) moduli. The remaining sampled *G* values paired bronze with a naphthenic oil and steel with a paraffinic oil. Hamrock does not detail the Roelands parameters; however, by assuming $Esteel*=207GPa$, one can estimate that Hamrock assumed the PVCs to be 22 GPa^{−1} and 34 GPa^{−1}.

While the effective Young’s modulus of the optical rigs is similar to that of bronze, the PVC values reported in Figs. 4, 6, and 7 are a fraction of those used by Hamrock. The material parameters used herein range of 819 ≤ *G*_{opt} ≤ 1825. This is outside the curve fit range of Hamrock and Dowson. Finally, it should be noted that while *G* is outside Hamrock’s design of experiments, the tested dimensionless films are within the design of experiment (DOE). *H*_{opt} ∈ (1, 21) × 10^{−6}.

In summary:

The tests and inferred computations, when defined by the piecewise transition map, are piezo-viscous.

However, the material parameter

*G*is outside the curve fit window of Hamrock and Dowson.

### Shear Thinning.

Miline [88] demonstrates how the non-Newtonian behavior of a lubricant yields an “appreciably less” load capacity. Bell (1962, [2, p. 50]) shows that a non-Newtonian model predicts a “much smaller film.” While Dowson [2] notes that Grubin’s formula over-predicts relative to experimental results, he also states Bell’s under-predictions are likely a result of poor understanding of the high shear behavior. Amid the early works, “for several decades, progress in understanding film-forming friction in EHL was paralyzed by two unsupported assumptions [10]”: (i) assuming the EHL inlet is Newtonian, and (ii) assuming viscous shear thinning is inseparable from viscous heating.

Bair [40] uses the non-Newtonian response of the inlet [42] to explain an example in which the optically inferred PVC is approximately half that of viscometry [41]. Villechaise et al. [43] found that Newtonian theory significantly overestimates the measured film thickness of Sanotrac S50 and concluded that “it is dangerous to use the theoretical equations to extract the *α* value from central film thickness measurements” [37]. References [37,42,43] suggest shear thinning is responsible for the discrepancy between optically methods and viscometry.

The authors conducted spectral optical film measurements to amplify shear thinning. Figure 11(a) plots 14 repeated tests whose slide-to-roll ratio (SRR) is zero. Each test consisted of 20 incremental speeds, 0.1 ≤ *u*_{e} ≤ 2 m/s. Measured films are plotted relative to (subtracted from) the mean of the 14 $SRR=0%$ tests. The repeated measurements were Gaussian, and their relative 95% confidence band window is denoted by the hatched Fig. 11(a) gray area. As the entrainment speed increases, certainty diminishes.

Figure 11(b) shows that the entrainment speed exponents of the $SRR=0%$ tests range of 0.677 ≤ *δ* ≤ 0.706. When $SRR=0%,$ the minimum thermal correction factor is Φ_{Thermal} = 0.998. Tests were then conducted at fixed values for $SRR=$$[5,10,15,$$20,and25%]$. The minimum thermal correction factor at $SRR=25%$ is Φ_{Thermal} = 0.945. The general trend of Fig. 11(b) indicates that the entrainment speed exponent diminishes with increasing SRR; however, Fig. 11(a) shows little difference in relative film thicknesses for $SRR<25%$.

Figure 12 plots histograms of extracted PVC values for $h(SRR=0%)$ and $h(SRR=25%)$. The respective inferred PVCs are $aopt(SRR=0%)=11.56GPa\u22121$ and $\alpha opt(SRR=25%)=$$12.62GPa\u22121$. The $65%$ confidence bands do not overlap, and one is 90% confident the PVCs of the $SRR=0%$ and $SRR=25%$ are statistically significant. The confidence bands of all other tests, $SRR<25%$, overlap.

Oils with no noticeable inlet shear thinning align well with Hamrock and Dowson, while oils with pronounced shear thinning do not (Krupka et al. [89]). Shear dependency within the inlet is proportional to speed and pressure (Krupka et al. [47]). Using X-ray measurements, Bell, Kannel, and Alan [24] demonstrate that as speed increases, the measured film thicknesses diverge from predictions. Bair and Winer, Bair [90], and Kumar and Khonsari [91] all offer high-shear correction factor models (Φ_{Shear}). The model proposed by Bair [57] incorporates the slide-to-roll ratio effects on shear and lends itself to direct comparison to Fig. 11.

*ϕ*as being the ratio of the Newtonian and non-Newtonian film thicknesses. Thus, the non-Newtonian or shear correction factor is the inverse: $\Phi Shear=1/\varphi Bair:$$h(\gamma \u02d9,\u2026)=\Phi Thermal\u22c5\Phi Shear\u22c5hHD$.

*W*

_{e}= (

*η*

_{o}

*u*

_{e})/(

*h*

_{cen}

*G*

_{M,τ}).

Sadinski’s [61] high-shear viscosity measurements of Royco 555 were fit to the double modified Carreau model; however, the single-parameter model has been used in this work to estimate the shear thinning correction factor. Per Sadinski, $n\tau =0.26$ and *G*_{M,τ} = 6900 kPa. At 40 °C and with $SRR=0%$, when *u*_{e} = [0.1, 1, and 2] m/s, the minimum shear correction factors, Φ_{shear}, are 0.982, 0.961, and 0.952, respectively. When the measured exponent *δ* is approximately equal to the theoretical value, and when *u*_{e} = 0.1 *m*/*s*, the discrepancy between the *α*_{opt} and $\alpha *$ is large: $(13.9\u22129.34)/13.9=33%$.

At higher velocities where *u* = max(*u*_{e}), the individual *α*_{opt} approaches that of viscometry $\alpha *$: $(13.9\u221211.33)/13.9=18%$. If the discrepancies were solely due to shear thinning, the agreement between $\alpha *$ and *α*_{opt} would lessen with sliding velocity. Further, if the discrepancies were solely due to shear thinning, the median slope of Fig. 3 would be less than 0.67. Neither of these statements is true for Royco 555, nor are they evident in the Table 8 DEHS results.

*η*

_{C.Y.}/

*η*

_{Y}, over shear stress. Relative viscosity, as used herein, represents the ratio of high shear viscosity (as modeled by the double modified Carreau–Yasuda equation, Eq. (18)) to low shear viscosity (as modeled by the modified Yasutomi model, Eq. (9)).

*n*

_{1}= 1.6072,

*n*

_{2}= 1.018,

*a*

_{1}= 1.7663,

*a*

_{2}= 2.3799, $G1M,\gamma \u02d9=4423kPa$, and $G2M,\gamma \u02d9=16.7kPa$.

Westlake [24] offers an expression to estimate the shear rate within the EHL inlet: $\gamma \u02d9\u2245(3ue)/(4h)$. The shear rate equivalent of the double modified Carreau–Yasuda viscosity model has been used to estimate the corresponding inlet zone shear stress: $\tau =\gamma \u02d9\u22c5\eta C.Y.(\gamma \u02d9,P,T)$. Figure 13 plots a band of Poiseuille-driven inlet shear stress at $T=40\u2218C,$ assuming an inlet pressure of *P* = 100 MPa and over velocities of 0.1 ≤ *u*_{e} ≤ 2 m/s. At the extreme test conditions ($ue=2m/s,SRR=0%$), the relative viscosity is estimated to be 0.925. An inlet pressure of *P* = 150 MPa estimates relative viscosity to be 0.895 (not shown in Fig. 13). Figure 13 indicates that the reduction in relative viscosity brought about by inlet shearing is relatively small. This aligns with Fig. 11, which documents the shear thinning correction factor to be 0.951 when *u*_{e} = 2 m/s and $SRR=0%$.

Figure 13 also estimates the sliding or Couette-driven shear at $SRR=25%$. This is calculated assuming simple shear, $\gamma \u02d9\u2245(SRR\u22c5ue)/hcen$. Figure 13 shows a significant reduction in relative viscosity when $SRR=25%$. When $SRR=25%$, Figs. 11 and 12 substantiate the Fig. 13 reduction in relative viscosity. It should be noted that the presence of shear thinning and viscous heating should yield an experimental exponent *δ* less than 0.67: $hopt\u221d\Phi Shear(ue\u2212|a|)\u22c5\Phi Thermal(ue\u2212|b|)\u22c5hHD(ue|c|):a+b+c\u22640.67$. This is not evident in Fig. 11(b) for $SRR<25%$.

In summary:

The variability of 14 $SRR=0%$ tests encapsulate all measurements where $SRR<25%$.

At $SRR=0%$, for the tested fluids the thermal correction factor is similar in magnitude to the shear correction factor.

In the inlet, the estimated ratio of non-Newtonian to Newtonian viscosity is 92.5%.

Even though inlet shear is present, at low SRR percentages there is no difference in the dimensionless speed exponent

*δ*.The optically inferred PVC is closer to the viscometry PVC when $SRR=25%$ than it is when $SRR=0%$.

### Algebraic Elastohydrodynamic Lubrication Modeling Assumptions.

Venner and Bos [92] illustrate how compressibility can alter the central film thickness. Classical film equations built on the Dowson and Higginson equation of state overestimate film thickness. The optical comparison herein includes thermal correction. While only one of the many models was tested, “ … thermal correction … ignores the thermal feedback which alters the temperature distribution.” [10].

Hamrock and Dowson assume the Newtonian Roelands viscosity model is sufficient. Thermal correction factor models omit high-pressure effects on thermal conductivity and assume the TVC is independent of pressure. While optical conclusions can incorporate a shear thinning correction factor, they assume a single power law response. Differing models will arrive at unique inferred PVCs, much like the work of van Leeuwen [31]. The product of correction models, $\Phi \u223c\u22c5h$, may agree with experimental data; however, the inferred PVC will be misleading if one has not properly accounted for all complex tribological phenomena.

The optically inferred PVCs of AeroShell 555 differ from other DOD-PFR-85734 fluids. Spectral measurements of AeroShell 555 were repeated twice at $40\u2218C$ and 70 °C and three times at $100\u2218C$, $130\u2218C$, and 150 °C. At 150 °C, the difference in the optically inferred PVC is 1.1 GPa. The repeatability is half of the uncertainty. The maximum measured film of the AeroShell 555 tests at 150 °C was 64 nm. The product of all correction factors would need to be significant to explain the differences between the fitted AeroShell 555 curve and individual PVC. While the figured analysis includes only thermal corrections, the minimum calculated thermal correction value was 0.988.

In summary:

The concatenation of modeling assumptions can be an impediment to assigning causation of trend in optically measured films.

Optically inferred PVCs can at times be misleading.

## Conclusions

Viscometry methods are, regardless of source, consistent in their PVC conclusions. Optically inferred PVC values (be it Spikes, Foord, SAE, or borne of the authors’ spectral or colorimetry measurements) exhibit significant scatter. The uncertainty of an optically inferred PVC, for DEHS and under the stated assumptions, was determined to be on the order of ±2.2 GPa^{−1}. The entrainment velocity exponent of optically measured film thicknesses can significantly differ from the theory. Variation in the exponent *δ* is a primary reason optically inferred PVCs exhibit uncertainty.

Six studied hypotheses attempted to explain the discrepancy between viscometry and optically inferred PVCs. The outcome, relative to the fluids studied herein, suggests:

A viscometry PVC value is specific to its functional definition and dependent on the method of analysis.

Proper account of the refractive index cannot explain the discrepancies between viscometry and optically inferred PVCs.

The exponent acting on dimensionless speed agrees with theory provided the data is ample, e.g.,

*O*(1000) measurements. PVCs can differ from viscometry even when the exponent aligns with theory.The optically measured film thicknesses are classified as being PVE, but the tested material parameters are outside the fit window of Hamrock and Dowson.

Shear thinning alone cannot explain the discrepancy between viscometry and optically inferred PVCs.

Current film thickness models and accompanying correction factors (be it thermal, starvation, and/or shear thinning) cannot assign causation to why viscometry and optical inference differ.

The analysis of viscometry is somewhat forgiving. Viscometry results are obtained using 3 ml of isolated oil. Viscometry measurements are at constant pressure and thermal equilibrium. The maximum RMSPE in fitted high-pressure viscosity was $5.6%$, and the results have been consistent for decades. In contrast, the optically inferred results are unforgiving. Measurements are dynamic and influenced by setup and the assumptions of the analysis. Analysis of the results is sensitive to the exponential behavior of the experiment. Subtle changes in films, in which thickness is on the order of nanometers, yield large changes in the inferred PVC.

Optical measurements are needed. They offer validation of the EHL models at large. Viscometry measurements are needed. They quantify the rheological behavior of an EHL lubricant. Both rigs serve their craft, but extrapolations and inference (as opposed to comparisons) can be problematic.

“A thick film of fog had long obstructed a satisfactory bird’s eye view of the field of thin-film lubrication. In an attempt to disperse the fog, the author has made an extensive correlational study of the experimental and theoretical evidence on the fundamental mechanical aspects of the field.” (Blok [13]).

## Footnotes

Van Leeuwen defines [31] *E*_{ball,disk} = [207, 75] GPa and *ν*_{ball,disk} = [0.29, 0.22], Foord [21] defines *E*_{ball,disk} = [206.8, 75.8] GPa and *ν*_{ball,disk} = [0.3, 0.25], Wedeven [23] defines *E*_{ball,disk} = [206.8, 74.2] GPa and *ν*_{ball,disk} = [0.3, 0.223], and Westlake [24] defines *E*_{ball,disk} = [206.8, 75.84] GPa and *ν*_{ball,disk} = [0.3, 0.25]. These yield effective moduli of $E*=116.9$, 119.3, 116.2, and 119.3 GPa, respectively. SAE [33] posits $E*=110GPa$.

The thermal conductivity, of the oils used in this work, changes with pressure [61]; however, the thermal correction factor model used herein makes use of the isobaric thermal conductivity.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*k*=ellipticity ratio

*q*=McEwen viscosity exponent

*F*=viscous fragility parameter

*G*=dimensionless material parameter

*N*=optically measured fringe order, unitless

*P*=pressure, Pa

*Q*=applied load, N

*T*=temperature: C

*U*=dimensionless speed parameter

*V*=volume at temperature and pressure, m

^{3}*W*=dimensionless load parameter

*a*=_{v}thermal expansivity, 1/K

- ℎ
_{opt}= optically measured film thickness, m, further subscripts refer to colorimetry and spectral methods

- ℎ
_{sp}= thickness of the optical disk spacer layer, m

*n*_{sp}=optical disk spacer refractive index

*u*=_{e}lubricant entrainment velocity, m/s

*C*_{LL}=Lorenz-Lorentz constant, unitless

*C*_{HD}=simplification constants

*G*=_{M}lubricant shear modulus

*H*=_{C}dimensionless central film thickness

*H*_{H&D}=central film thickness as defined by Hamrock and Dowson, m

*K*_{0}=isothermal bulk modulus at

*P*= 0, Pa*K*_{00}=*K*_{0}at zero absolute temperature, Pa*M*=_{u}number of incremental speed tests

*N*_{oil}=compressed lubricant refractive index

*P*=_{Htz}maximum Hertzian contact pressure, MPa

*P*=_{mean}mean Hertzian contact pressure, MPa

*P*=_{∞}upper limit (terminal pressure) on integration of the pressure-viscosity coefficient, Eq. (1)

*P*=_{iv,as}iso-viscous asymptotic pressure, MPa

*S*=_{x}standard error of the fit

*T*=_{R}reference temperature, C

*T*=_{g}glass transition temperature, C

*T*=_{go}atmospheric glass transition temperature, C

*T*_{v,}_{95}=student

*t*for*v*degrees-of-freedom and 95% two-tail confidence- $Uhopt$ =
uncertainty on measured film thickness

- $U\eta W$ =
uncertainty on Walther's viscosity

*U*=_{δ}uncertainty on the dimensionless velocity exponent

- $U\alpha opt$ =
uncertainty on the optically inferred pressure-viscosity coefficient

*V*_{0}=volume at zero pressure, m

^{3}*V*_{R}=volume at reference conditions, m

^{3}- $noil*$ =
wavelength adjusted refractive index

*K′*_{0}=pressure rate of change in 0 ∼

*f*,*g*=PVC model parameters

*k*_{1},*k*_{2},*k*_{3}=viscometry fit pressure-viscosity coefficient parameters

*E** =effective Young's modulus, Pa,

*E*,_{ball}*E*,_{disk}*v*,_{ball}*v*are elastic Young's and Poisson's properties of the ball and disk_{disk}*A*_{C},*B*_{C},*C*_{C}=Cauchy equation constants

*A*=_{W}, B_{W}Walther viscosity constants

*A*_{1},*A*_{2}=glass transition constants

*B*_{1},*B*_{2}=free volume parameter constants

*C*_{1}*, C*_{1}=Yasutomi viscosity constants

*N*(*λ*) =lubricant refractive index as a function of wavelength

*R*,_{x}*R*=_{y}effective radius parallel (

*x*) and (*y*) perpendicular to rolling, m*α*_{opt}=optically inferred pressure-viscosity coefficient, GPa

^{–1}*α** =pressure-viscosity coefficient defined by viscometry, GPa

^{–1}*α*_{McE}=pressure-viscosity coefficient as defined by the McEwen equation

*α*=_{B}pressure-viscosity coefficient as defined by integer multiples of

*P*_{iv}*α*=_{R}pressure-viscosity coefficient as defined by a fit to Roelands equation

*β*=_{K}bulk modulus temperature slope, 1/C

*δ*=exponent acting on dimensionless velocity

*U*, ∼0.67*η*=dynamic viscosity defined by the subscript:

*o*atmospheric,*i*indexed, W Walther, McE McEwen, Y Yasutomi, C.Y. Carreau-Yasuda*λ*=optically measured wavelength

*ρ*=_{Hartung}lubricant density as defined by the Hartung equation

*ρ*_{oil}=lubricant density as defined by the Murnaghan equation

*ρ*=_{o}density at zero pressure

*ρ*=_{R}density at reference temperature and zero pressure

*ν*=_{cSt, W}kinematic viscosity defined by Walther's equation

- Φ =
film thickness correction factors, unitless; subscripts denote thermal and/or shear thinning corrections

*ϕ*=optical phase shift, unitless