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Tribology International 42 (2009) 1522–1530

Contents lists available at ScienceDirect

Tribology International

0301-67

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/triboint

On the role of lubricant rheology and piezo-viscousproperties in line and point contact EHL

Punit Kumar, M.M. Khonsari �

Department of Mechanical Engineering, Louisiana State University, 2058 Patrick Taylor Hall, Baton Rouge, LA 70803, USA

a r t i c l e i n f o

Article history:

Received 25 August 2008

Received in revised form

16 November 2008

Accepted 21 November 2008Available online 6 December 2008

Keywords:

EHL

Shear thinning

Carreau

Generalized Newtonian

Free volume

9X/$ - see front matter & 2008 Published by

016/j.triboint.2008.11.006

esponding author.

ail address: khonsari@me.lsu.edu (M.M. Khon

a b s t r a c t

Accurate prediction of elastohydrodynamic lubrication (EHL) characteristics, i.e., film thickness and

traction coefficient, requires the knowledge of lubricant rheology. The EHL analyses based on Newtonian

fluid model may be useful primarily for film thickness prediction of mineral oils. However, for

applications involving mineral oil/polymer blends and synthetic oils that exhibit shear-thinning

behavior, the use of an appropriate non-Newtonian fluid model is required to predict the EHL behavior

correctly. It is, thus, no surprise that characterization of non-Newtonian fluid behavior in EHL studies

has been the focus of attention over last four decades. In this regard, several types of non-Newtonian

EHL models have been presented and solved using different methods. Unfortunately, many of these

studies suffer from various drawbacks such as use of inappropriate non-Newtonian model,

approximations in the numerical solution and lack of sufficient experimental data pertaining to

lubricant rheology as well as piezo-viscous properties. Over the years, some of these shortcomings have

been transferred erroneously from one generation of researchers to the next. For example,

notwithstanding the fact that the sinh-law lubricant model (commonly referred to as the ‘‘Ree–Eyring’’

model) was actually rejected for use as a shear-thinning model by Henry Eyring himself, it is still being

widely used to describe the shear-thinning behavior of EHL lubricants. The present paper discusses the

recent developments clearly indicating the incapability of sinh-law for the EHL applications. This paper

also reviews the perturbation method often used to reduce the actual constitutive equation to a

simplified form in order to derive a Reynolds-type equation to which normal EHL solvers can be applied.

This approach is still in use even though appropriate procedures for implementing the generalized

Newtonian approach are available that can conveniently and accurately allow the use of the exact

constitutive equation in its original form. In the light of above facts, this paper presents a collective

perspective in an effort to emphasize the importance of implementing realistic non-Newtonian and

piezo-viscous models with accurate treatment methods in EHL applications.

& 2008 Published by Elsevier Ltd.

1. Introduction

Elastohydrodynamic lubrication (EHL) describes the mode oflubrication in highly-stressed, non-conformal contacts that oneencounters in many mechanical components such as gears, rollingelement bearings, cams, and the like. Due to the high pressure inan EHL contact, the elastic deformation of the surfaces andpressure dependence of viscosity play the pivotal role. The coreelement that protects the surfaces is a thin film of lubricant on theorder of a few hundred nanometers or even less. Therefore, it isimportant to be able to predict the lubricant film thickness undera given set of operating conditions. Amongst the several earlyattempts to predict the EHL film thickness, the most significant

Elsevier Ltd.

sari).

contribution was made by Dowson and Higginson in 1960s.Dowson and Higginson [1] conducted a series of full EHL analysesgiving due consideration to elastic deformation and pressuredependent viscosity that culminated in the derivation of the first-ever film thickness equation applicable to EHL line contacts for awide range of operating loads, speeds and material properties. Theconvenient dimensionless form and versatility of this filmthickness equation has been the driving force behind its extensiveusage for the last four decades. Continuing this tradition ofsignificant contributions, Hamrock and Dowson [2] presented thecentral and minimum film thickness equations for elliptical EHLcontacts where the computational complexity is much morechallenging than that in line contacts.

Besides the lubricant film thickness, traction coefficient isanother important EHL parameter as it affects the power loss andwear of surfaces. The accurate prediction of the EHL parametersunder a given set of operating conditions requires the knowledge

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Nomenclature

Dimensional parameters

E0

effective elastic modulus of rollers 1 and 2 (Pa)h film thickness (m)hmin, hc minimum and central film thickness, respectively (m)p pressure (Pa)ph maximum Hertzian pressure (Pa)R equivalent radius (m)u0 average rolling speed, u0 ¼ (ua+ub)/2 (m/s)ua, ub velocities of lower and upper surfaces, respectively

(m/s)

Greek symbols

a piezo-viscous coefficient (Pa�1)_g shear strain rate across the fluid film, _g ¼ du=dy (s�1)

r lubricant density at the local pressure and tempera-ture (kg/m3)

t shear stress in fluid (Pa)m0 inlet low shear viscosity (Pa s)m low shear viscosity at local pressure (Pa s)Z generalized Newtonian viscosity (Pa s)

Dimensionless parameters

n power law indexS slide to roll ratio, S ¼ (ub�ua)/u0

W dimensionless load parameter

P. Kumar, M.M. Khonsari / Tribology International 42 (2009) 1522–1530 1523

of lubricant rheology. The Newtonian fluid model assumes thatthe shear stress is directly proportional to shear strain rate forsimplifying the solution of EHL equations. This results in very highshear stresses at high shear strain rates, whereas, many of thelubricants exhibit shear-thinning behavior at high shear strainrates, which causes deviation from the Newtonian model. Hence,the analyses based on Newtonian fluid model, in general, fail toexplain the experimentally measured traction coefficients.Therefore, in order to obtain a fundamental understanding oflubrication performance and failure in various tribo-elements,it is essential to incorporate the effect of non-Newtonianbehavior of fluid in the numerical scheme. In this regard,some of the pertinent papers available in the EHL literature willnow be discussed. The references cited here are by no meansexhaustive, for a comprehensive review is beyond the scope of thispaper. The primary aim is to draw attention to some importantissues related to the study of shear-thinning behavior in EHLapplications.

Some of the earliest attempts to study the non-Newtonianbehavior of lubricants were made by Milne [3], Tanner [4], Burton[5] and Bell [6]. Using a hyperbolic sine (sinh) law, which wasreferred to as ‘‘Ree–Eyring’’ model, to represent the lubricantshear-thinning behavior, Bell [6] concluded that at high shearrates, the effect of velocity on film thickness diminishes, whereasthat of load increase. However, as discussed subsequently, thisinitiated the trend of using sinh-law for shear-thinning of EHLlubricants and attributing it to Ree and Eyring. This so-called‘‘Ree–Eyring’’ fluid model has thus become very popular and usedextensively by several EHL researchers. See, for example [7–13].

Notwithstanding the popularity of sinh-law as a shear-thinning model, the fact remains that it fails to replicate theexperimentally obtained flow curves of shear-thinning lubricants.Recently, in an extensive review of literature on the origin of sinh-law, Bair [14] provided strong evidence from the originalpublished works of Henry Eyring that sinh-law was not intendedfor characterizing shear-thinning fluids [15]. Rather, it wasof some use for materials thought to exhibit thixotropy[16]—a behavior that is quite different from shear-thinning. Thedata upon which Eyring based the application to thixotropy waslater found to result from viscous heating. On the other hand, the‘‘actual Ree–Eyring’’ model, which consists of multiple flow unitswith appropriate relaxation times, is capable of describing thedata obtained from high-pressure rheometers. Following Dai andKhonsari’s [17] theory of hydrodynamic lubrication involving a

mixture of two fluids, Kumar et al. [18] presented a thermal EHLanalysis of rolling/sliding line contacts using a mixed rheologicalmodel consisting of two flow units corresponding to Newtonianand sinh-law fluids. Being similar to the actual Ree–Eyring modelwhere all the flow units are non-Newtonian, the mixed rheolo-gical model [18] could be an improvement over the sinh-law ifused with correct flow properties.

Evans and Johnson [19] expected that the lubricant shouldhave a limiting shear strength which is not described by the sinh-law. Based on extensive laboratory experiments Bair and Winer[20] proposed a non-linear constitutive equation with the limitingshear strength incorporated. Gecim and Winer [21] used asimplified form of this equation to calculate film thickness andreported upto 40% reduction in film thickness below thecorresponding Newtonian model values. However, this simplifica-tion of Bair–Winer model did not gain much popularity.

A more comprehensive treatment of non-Newtonian fluidmodel with the limiting shear strength in EHL was made byJacobson and Hamrock [22]. Lee and Hamrock [23] proposed thecircular fluid model which was claimed to approach the shape ofthe experimentally established Bair and Winer [20] model. Suiand Sadeghi [24] presented a full numerical solution to study thecombined effect of temperature rise and non-Newtonian sinh-lawfluid in EHL line contacts. Salehizadeh and Saka [25] investigatedthe combined effect of thermal and non-Newtonian character inEHL line contacts under pure rolling conditions with sinh-lawtype of fluid. It was found that the reduction in film thickness athigh rolling speeds was entirely due to thermal effect and not dueto the lubricant rheology. Also, Wang et al. [26] examined thelubrication of EHD line contacts under simple sliding conditionsconsidering sinh-law fluid model.

Hsiao and Hamrock [27] investigated the non-Newtonian andthermal effects on film generation and traction reduction in EHLline contact conjunctions using circular fluid model. It was foundthat the inlet zone pressure built-up and piezo-thickeningdetermine the film generation capability, but inlet zone thermal-thinning reduces this capability. Although the inlet zone shearthinning effect caused by non-Newtonian fluid behavior alsoreduces this capability, it was found to be insignificant. Hsiao andHamrock [27] concluded that the predictions of isothermal modelregarding the film formation and traction generation are valid atlow entraining velocities and near-pure rolling conditions only.

As discussed by Khonsari and Hua [28], most of the studies onnon-Newtonian fluid behavior approximated the constitutive

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P. Kumar, M.M. Khonsari / Tribology International 42 (2009) 1522–15301524

equation with limiting shear stress to a simpler form while othersused a perturbation scheme so as to derive a Reynolds-typeequation. Subsequently, Khonsari and Hua [29] presented ageneralized Newtonian EHL model using Bair–Winer’s constitu-tive equation in its original form and yielded traction coefficientsin close agreement with experimental findings. The generalizedNewtonian model considers only the shear thinning aspect andneglects other aspects of non-Newtonian behavior such as timedependence, normal stress differences and enhanced resistance toelongational deformation. Bair and Qureshi [30] concluded thatthe shear thinning response that can be described by a general-ized Newtonian model and that is necessary for the description ofboth traction and film thickness is of power-law form such asCarreau viscosity model. Mongkolwongrojn et al. [31] calculatedthe performance characteristics of EHL line contacts for varyingCarreau viscosity model parameters. More recently, Jang et al. [32]presented extensive EHL line-contact simulations using theCarreau model that closely agreed with published experimentalresults. In accordance with Bair’s [14] findings, as discussed above,Jang et al. [32] also showed that sinh-law is incapable ofdescribing the experimentally observed film-thinning patterncorrectly.

In this paper, we present a brief review of the salient featuresof non-Newtonian fluid models used in EHL literature and,through examples of computational results for EHL line and pointcontact, present the importance of some key aspects in theprediction of EHL performance. Specifically, we discuss the use ofconstitutive equation valid over a wide range of shear rate, presentan appropriate method to incorporate the constitutive equation inits original form within the EHL solver, and illustrate the role ofthe piezo-viscous properties in EHL simulations.

2. The salient shear-thinning models in EHL

In this section, we focus our attention to a number of non-Newtonian fluid models that have been applied in EHL analyses. Inthis regard, shear-thinning and limiting shear stress behaviors arethe two main aspects. Shear-thinning refers to the decreasingtrend in the effective lubricant viscosity (t=_g) with increasingshear rate, whereas, limiting shear stress behavior involves anearly constant value of shear stress at sufficiently high shearrates.

2.1. Sinh-law model (so-called ‘‘Ree–Eyring’’)

It is believed that Bell [6], in 1961, initiated the use of thefollowing sinh-law to explain the shear-thinning behavior oflubricants:

_g ¼ t0

msinh

tt0

� �(1)

where _g is the shear rate, t is the shear stress, m is the low shearviscosity and t0, representing the Newtonian limit of thelubricant, is referred to as Eyring stress. Bell [6] attributed thisto Ree and Eyring; however, according to Bair and Qureshi [30],‘‘The first shear-thinning model to result from a molecular theoryis the Prandtl–Eyring Eq.’’ [33–35].

One of the reasons for the widespread acceptance of Eq. (1) forEHL is the fact that a plot of traction against the sliding velocityfor small sliding velocities approximately follows the inversehyperbolic sine function [1]. However, the sinh-law model suffersfrom a serious limitation that, according to Eyring [15], it fails todescribe the shear stress–shear rate relationship over a widerange of shear rate. In fact, as per Bair and Qureshi [30], thepower-law type of flow behavior obtained by experimental

measurements can be described by a series of hyperbolic sineterms as put forward in the actual Ree–Eyring equation.

2.2. Generalized Maxwell model

In 1965, Tanner [36] proposed a generalized Maxwell model forlubrication which is written here for simple shear:

tþ lmDtDt̂¼ ZðtÞ_g (2)

where D/Dt is the convected time derivative and lm is a relaxationtime. Tanner [36] chose a simple power law form of the functionZ(t), which while quite accurate for high shear stresses, couldnot capture the linear Newtonian behavior at low shear stresses.Eq. (2) is often attributed to a later work by Johnson andTewaarwerk [8] assuming Z(t) to be of sinh-law form, as in Eq.(1). The time dependent term in Eq. (2) is often found negligible inmany EHL applications and, hence, dropped for simplicity.

2.3. Bair–Winer model

Bair and Winer [20] proposed the following non-linearconstitutive equation based on extensive laboratory experiments:

_g ¼ 1

G1

dtdt̂þtL

mlnð1� t̂Þ�1 (3)

where t̂ ¼ t=tL and tL is the limiting shear strength of thelubricant.

Gecim and Winer [21] simplified the non-Newtonian fluidmodel of Eq. (3) as follows:

_g ¼ tL

m tanh�1ðt̂Þ (4)

2.4. Ivonen–Hamrock model

Due to the difficulty encountered in incorporating theBair–Winer model in the Reynolds equation, Ivonen and Hamrock[37] proposed a simpler rheological model:

_g ¼ tL

m ½ð1� t̂Þ�1� 1� (5)

The above Eq. (5) was further extended to the following generalform [37]:

_g ¼ ntL

m ½ð1� t̂Þ�1=n� 1� (6)

2.5. Circular fluid model

Lee and Hamrock [23] proposed another lubricant rheologicalcircular model:

_g ¼ tLt̂m ð1� t̂

2Þ�1=2 (7)

It was found that the circular model could be convenientlyimplemented in the numerical scheme [23]. A modified Reynoldsequation incorporating the circular fluid model [23] was derivedand results were obtained for a wide range of slide-to-roll ratiosfrom pure rolling to pure sliding. The value of coefficient offriction was found to be within a reasonable range from 0 to 0.1.

ARTICLE IN PRESS

EHL Line Contact SimulationsW = 1x10-4, Eyring stress = 0.3 MPa

Rolling speed (m/s)0.1

Dim

ensi

onle

ss c

entra

l film

thic

knes

s (h

c/R)

10-6

10-5

10-4

10-3

S = 0 (Perturbation method)

S = 0 (Generalized Newtonian)

S = 1 (Perturbation method)

S = 1 (Generalized Newtonian)

1 10 100

Fig. 1. Comparison of perturbation method and generalized Newtonian approach.

P. Kumar, M.M. Khonsari / Tribology International 42 (2009) 1522–1530 1525

2.6. Sinh-law with limiting shear strength

Wang et al. [26] explained that while the limiting shear stressmodel of Bair and Winer [20] gives a good description of the shearbehavior near the limiting shear stress, it uses the limiting shearstress to determine the initial non-Newtonian behavior at stresslevels much below the limiting shear stress. In order to give dueconsideration to the fact that the initial non-Newtonian behaviorof a lubricant is likely to originate from a physical mechanismdifferent from the limiting shear stress, Wang et al. [26] suggestedthe following rheological model:

t ¼to sinh�1

ðm_g=toÞ for jm_gjpto sinhðtL=toÞ

tL for jm_gjXto sinhðtL=toÞ

((8)

2.7. Power-law models

Several models have been proposed to describe the power-lawresponse at high shear rates as well as the limiting secondNewtonian regime. The most general of these is the Carreau–Yasuda [38] equation:

Z ¼ t=_g ¼ m1 þ ðm� m1Þ½1þ ðl_gÞa�ðn�1Þ=a (9)

where mN is the second Newtonian viscosity and l is thetime constant generally expressed as a Maxwell time constant:l ¼ m/Gcr. Gcr is a shear modulus or a critical stress associated withthe longest relaxation time. Moore et al. [39] have shown usingnon-equilibrium molecular dynamics simulations that this long-est relaxation time, l, is nearly equal to the rotational relaxationtime, gr, for the molecule:

lr ¼12Mmp2rRT

(10)

where r is the lubricant density, R the universal gas constant, T

the absolute temperature and M the molecular weight. The aboveequation being an approximation, the numerical coefficients aregenerally dropped and, hence,

Gcr �rRT

M(11)

By setting a ¼ 2, which is the case for monodisperse liquids, inEq. (9), the Carreau viscosity equation [40] is obtained:

Z ¼ t=_g ¼ m1 þ ðm� m1Þ½1þ ð_gm=GcrÞ2�ðn�1Þ=2 (12)

The following modified form of Carreau–Yasuda model is oftenrecommended for EHL film thickness calculations:

Z ¼ t=_g ¼ m½1þ ðt=GcrÞ2�ð1�1=nÞ=2 (13)

Setting n ¼ 13 in Eq. (13) yields the Rabinowitsch’s generalized

Newtonian viscosity model [30]:

Z ¼ t=_g ¼ m½1þ ðt=GcrÞ

2�

(14)

2.8. Actual Ree–Eyring model

According to the research by Bair [14], the constitutiveequation actually proposed for shear-thinning lubricants by Reeand Eyring is of the following form:

t ¼XN

i¼1

xiti sinh�1ðli _gÞ (15)

where li ¼ m/ti is a characteristic or relaxation time, N is thenumber of flow units (N41), xi is a weighting fraction such that

PNi¼1xi ¼ 1 and m is the low shear viscosity. Bair [14] concluded his

study by posing a challenge for the EHL community to carry outfilm thickness calculations using the abovementioned ‘‘actual’’Ree–Eyring model.

3. EHL film thickness prediction

This section dwells upon the comparison of a frequently usedapproximate method and the exact method for predicting the EHLfilm thickness considering the effect of shear-thinning behavior.Also discussed are some recent developments pertaining to theuse of power-law based shear-thinning models with realistic flowproperties, authenticated with experimental results.

The perturbation method described in Appendix A is fre-quently applied in EHL analysis in order to derive a Reynolds-typeequation incorporating the effect of shear-thinning behavior.However, it is based on certain simplifying assumptions and,therefore, it cannot be applied indiscriminately for all conditions.On the other hand, as explained in Appendix B, the generalizedNewtonian approach applied for ordinary shear-thinning exhib-ited by multigrade lubricants allows the derivation of a general-ized Reynolds equation usable for any given constitutive equationin its original form, i.e., without reducing the constitutiveequation to an approximate form. Using this approach, Khonsariand Hua [28,29] solved the EHL line contact problem withBair–Winer shear-banding model and presented results in closeagreement with experiments. In order to visualize the inaccuracycaused by the use of perturbation method, Fig. 1 compares thevariation of dimensionless central film thickness (hc/R) withrolling speed obtained using perturbation method (approximatemethod) and generalized Newtonian approach (exact method) inan EHL line contact under pure rolling (S ¼ 0) as well as rolling/sliding (S ¼ 1) at a typical load of W ¼ 1�10�4. The simulationresults in Fig. 1 are obtained for a sinh-law fluid as this non-Newtonian model and the corresponding perturbation-modifiedReynolds equation (A.20) are in extensive use. It may be notedthat Eyring stress represents the Newtonian limit of a shear-thinning lubricant and 0.3 MPa used here is well within thetypical range. In fact, some shear-thinning EHL lubricants areknown to have a Newtonian limit as low as a few kilo-Pascals.It can be seen from Fig. 1 that the pure rolling film thicknessvalues obtained using perturbation method are higher than theexact values, which is obviously due to the fact that theperturbation-modified Reynolds equation reduces to its classicalNewtonian form in the absence of sliding. This is a well-known

ARTICLE IN PRESS

W = 2.58x10-5

Rolling speed (cm/s)1

Dim

ensi

onle

ss c

entra

l film

thic

knes

s (h

c/R

)

10-6

10-5

10-4

10-3

Dyson & Wilson (S=0) [41]Dyson & Wilson (S=1) [41]

CarreauGcr = 80 kPan = 0.5

sinh law (Eyring stress = 80 kPa)

NewtonianS = 0S = 1

S = 0S = 1

10 100 1000 10000

Fig. 2. Comparison with experimental film thickness by Dyson and Wilson [41].

EHL Point Contact Simulationsph = 0.529 GPa, E' = 123.9 GPa,

10.10.010.001

h (n

m)

10

100

1000

Experimental hc [42]Experimental hmin [42]

Simulated hmin

H-D hmin

Simulated hc

H-D : Hamrock & Dowson Simulated : Full EHL simulations

using Carreau model(n = 0.74, Gcr = 31 kPa)

uo (m/s)

H-D hc

lo = 1.42 Pa.s, S = 0

Fig. 3. Comparison with experimental film thickness using the same input by Liu

et al. [42].

P. Kumar, M.M. Khonsari / Tribology International 42 (2009) 1522–15301526

fact and is restated here for the sake of completeness. A moreimportant conclusion which follows from Fig. 1 is that theperturbation method overestimates the extent of film-thinningdue to shear response of the lubricant at high slide to roll ratios.Therefore, it is apparent that perturbation method, under a givenset of operating conditions, may yield results close to the exactvalues only for a limited range of slide-to-roll ratios.

Another important subject of the present discussion is theexperimental validation of EHL film-thinning predicted by aparticular shear-thinning model over a wide range of speed andslide-to-roll ratio using realistic flow properties. To the best ofauthors’ knowledge, no such work was available for the case ofEHL line contacts which could meet all the aforementionedrequirements together until recently when Jang et al. [32]reported good agreement between the film thickness valuespredicted using Carreau viscosity model and the film thicknessdata obtained experimentally by Dyson and Wilson [41] using ahighly shear-thinning lubricant (polydimethylsiloxane) for rollingspeeds ranging over two orders of magnitude and slide to ratiovarying from 0 to as high as 1. In this study [32], it was also shownthat the most well-accepted sinh-law fails to describe theexperimentally observed film-thinning behavior. This comparisonis reproduced here in Fig. 2 by the present authors using thesecond order Reynolds equation, different from the integratedform of Reynolds equation used in [32]. From a comparison of theNewtonian and Carreau film thickness, it may seem interestingthat a large shear-thinning effect is exhibited even under purerolling and in fact, it is found that sliding does not contributemuch to the film-thinning—a detail that is clearly apparent fromnumerical as well as experimental results.

Similarly, Liu et al. [42] showed an excellent agreementbetween the numerically predicted and experimentally measuredcentral and minimum film thickness in EHL point contact for awide range of rolling speed under pure rolling using theCarreau–Yasuda viscosity model, Eq. (13), for the polyalphaolefin(PAO) lubricant used in the experiments. Fig. 3 shows a similarcomparison simulated by the present authors using the Carreauviscosity model, Eq. (12). It can be seen from Fig. 3 that theCarreau viscosity model also yields good agreement with experi-mental results in this case.

4. Traction coefficient

Historically, EHL theory was born because hydrodynamiclubrication could not predict realistic film thickness for non-

conformal geometries. So, tribology researchers were faced withthe challenge of calculating film thickness consistent with theobservations indicating the existence of a lubricant film. Whilefilm thickness is the key parameter necessary to ensure theprotection of mating surfaces, in a sense, it is not a performance-evaluation parameter. Since traction coefficient affects the powerloss and wear, it better serves as a measure of lubricantperformance. Therefore, in addition to film thickness, tractioncoefficient has been of key interest in EHL applications.

It is well known that the isothermal Newtonian fluid modelyields very high, often unrealistic, values of traction coefficient.The shear-thinning behavior exhibited by a lubricant in thecontact zone is often used to describe the experimentallymeasured traction data. In fact, the sinh-law seems to havegained much of its popularity with the spread of a notion that thetraction tests by Conry et al. [43] confirmed the validity of sinh-law. However, the observations leading to this conclusion mighthave been the result of an incorrect piezo-viscous relationship, theimportance of which is discussed subsequently. The flow proper-ties based on a traction curve may not be able to predict the filmthickness correctly for the entire speed-range of interest. Thecorrect prediction of film thickness together with tractioncoefficient requires an accurate and reliable shear-thinning modelwhich is capable of replicating the experimentally obtained flowcurves for a wide range of shear rates.

Besides the shear-thinning model, an important factorwhich plays a decisive role in traction prediction is thepiezo-viscous response of lubricants. We shall address this issuein more detail since it has not received adequate attention.The majority of available studies on EHL traction prediction oftenuse either of the two well-known pressure–viscosity equationsgiven below:

4.1. Exponential viscosity–pressure equation often attributed

to Barus

m ¼ mo expðapÞ (16)

4.2. Roelands equation

m ¼ m0 exp ðlnm0 þ 9:67Þf�1þ ð1þ 5:1� 10�9pÞzgj k

(17)

The surprising fact is that neither of the above two equationssuccessfully model the real piezo-viscous behavior at pressures as

ARTICLE IN PRESS

2D Graph 2

-30

% e

rror

in tr

actio

n co

effic

ient

-100

0

100

200

300

400

500EHL Line Contact Simulationsph = 0.6 GPa, uo = 1 m/s, �o = 1.42 Pa.sn = 0.74, Gcr = 31 kPa, S = 1

% uncertainty in α-20 -10 0 10 20 30

Fig. 5. Percentage error in the value of traction coefficient versus percentage

uncertainty in the value of pressure coefficient of viscosity (a).

EHL Line Contact Simulationsph = 2 GPa, uo = 1 m/s, �o = 1.42 Pa.s

S0.00

Trac

tion

Coe

ffici

ent

0.00

0.05

0.10

0.15

0.20

0.25

γ = 0.03

γ = 0.04

γ = 0.02

0.02 0.04 0.06 0.08 0.10 0.12

n = 0.74, Gcr = 31 kPa, α = 19 GPa-1

Fig. 6. Variation of traction behavior with temperature coefficient of viscosity (g).

P. Kumar, M.M. Khonsari / Tribology International 42 (2009) 1522–1530 1527

high as the typical EHL pressures. On the other hand, it is a lesserknown fact within the EHL community that the free-volumeviscosity model describes the piezo-viscous behavior of lubricantsquite accurately. In 1951, Doolittle [44] developed the firstfree-volume model based on the physical concept that theresistance to flow in a liquid depends upon the relative volumeof molecules present per unit free volume. To the best of authors’knowledge, no EHL simulations were carried out using the free-volume model until recently when Liu et al. [45] used thefollowing Doolittle’s free-volume model in EHL point contactsimulations:

m ¼ m0 exp BV1V0

1V

V0�

V1V0

�1

1�V1V0

2664

3775

0BB@

1CCA (18)

where V1=V0 and B are constants and V/V0 may be calculatedusing the Tait’s equation of state [42,45,46]:

V

V0¼ 1�

1

1þ K 00ln 1þ

p

K0ð1þ K 00Þ

� �(19)

Also, based on the high pressure viscometer results, Bair [46]confirmed and recommended the use of free-volume viscositymodel for EHL. In order to visualize the effect of pressure–visc-osity relationship on the traction behavior, Fig. 4 shows thevariation of traction coefficient with slide-to-roll ratio obtainedusing the three pressure–viscosity relationships, Eqs. (16)–(18),for an EHL line contact at a rolling speed u0 ¼ 1 m/s and maximumHertzian pressure ph ¼ 0.5 GPa. The lubricant considered here isPAO, the same as in Fig. 3, for which the experimentally measuredCarreau viscosity parameters and Doolittle–Tait parameters areavailable [42]: m0 ¼ 1.42 Pa s, n ¼ 0.74, Gcr ¼ 0.031 MPa, B ¼ 4.422,V1=V0 ¼ 0.6694, K 00 ¼ 12:83, K0 ¼ 1.4252 GPa. The value ofa ¼ dðlnmÞ=dpjp¼0, as obtained using Eq. (18) is 19 GPa�1. It canbe seen from Fig. 4 that the Barus equation yields tractioncoefficients that are too high to be shown in this frame. Incontrast, the Roelands equation gives more reasonable but highervalues as compared to the results pertaining to free volume. Yet,Roelands equation is still in frequent use in many EHL publica-tions. This is one of the sources of error in the calculation oftraction calculation. Moreover, generally the pressure–viscouscoefficient, a, used for the calculation of the exponent z in Eq. (17)is not known accurately due to the lack of high pressureviscometer data. This uncertainty in the value of a further

EHL Line Contact Simulationsph = 0.5 GPa, uo = 1m/s, μo = 1.42 Pa.s

S0.0

Trac

tion

Coe

ffici

ent

0.00

0.02

0.04

0.06

0.08

0.10

RoelandsFree Volume

Barus

0.5 1.0 1.5 2.0

n = 0.74, Gcr = 31 kPa, α = 19 GPa-1

Fig. 4. Comparison between three pressure–viscosity relationships for traction

behavior.

increases the inaccuracy in the predicted value of tractioncoefficient.

Fig. 5 shows the variation of error in the predicted value oftraction coefficient with the uncertainty in the value of a obtainedusing Roelands equation at S ¼ 1 for the same conditions as inFig. 4. The solid circle on the curve corresponds to the value oftraction coefficient for a ¼ 19 GPa�1, which is taken as thereference for calculating the error. It can be seen from Fig. 4 thattraction coefficient is highly sensitive to a. For example, anuncertainty of 710% in the value of a causes an error ranging from�37% to +75% in the value of traction coefficient. Owing to theaforementioned sources of error, the computed value of tractioncoefficient may be just as good as a wild guess!

Crook [47] showed that a theoretical analysis which treats thefluid as Newtonian and takes the variation of viscosity withtemperature into account can predict traction coefficient withinreasonable accuracy. However, the operating speeds and loads inmany EHL applications exceed those in Crook’s experiments.Several workers (for example, Refs. [48–50]) presented thermalEHL analyses using Roelands equation for pressure-dependence ofviscosity coupled with an exponential viscosity reduction withincrease in temperature assuming a constant temperaturecoefficient of viscosity (g). However, it is well-known that g doesnot remain constant with respect to temperature and pressure.

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Since the film thickness and traction coefficient are quite sensitiveto the value of g, this assumption may cause considerableinaccuracy. In order to estimate the bearing of traction coefficienton g, Fig. 6 shows the EHL line contact simulation results tocompare the variation of traction coefficient with slide-to-rollratio at three different constant values of g. These results areobtained taking account of thermal as well as shear-thinningeffects. The operating conditions and fluid properties for Fig. 6 arethe same as in Fig. 4 except for the maximum Hertzian pressurewhich is raised to 2 GPa so that thermal effect is highlypronounced. It can be seen that the traction coefficient valuesincrease substantially with decrease in the value of g. Therefore, itmay be concluded that it is absolutely necessary to take accountof the variation of g as the pressure and temperature vary withinan EHL contact.

5. Conclusions

A review of the general trends in EHL modeling with non-Newtonian fluids has been presented. The universally acceptedhyperbolic sine law used extensively to describe the shear-thinning behavior of EHL lubricants fails to capture the experi-mentally observed film thinning behavior. However, according tothe recent studies, a power–law based Carreau-type viscositymodel successfully characterizes the experimentally observedflow behavior of shear-thinning lubricants and is found highlysuitable to predict the EHL characteristics over a wide range ofoperating speeds and slide to roll ratios. Using full EHLline contact simulations, a comparison of the perturbationmethod—often used to derive a modified Reynolds equationto incorporate the effect of shear-thinning behavior—and thegeneralized Newtonian approach, which allows the use of thenon-Newtonian constitutive equation in its original form, ispresented in order to highlight the inaccuracies associated withthe use of perturbation method. Comparing the traction behaviorobtained using three different pressure–viscosity relationships;the present paper also draws attention on the importance ofknowing the exact piezo-viscous behavior of EHL lubricants foraccurate prediction of traction coefficient. According to recentexperimental studies using high pressure viscometers, the freevolume based pressure–viscosity relationship closely representsthe realistic piezo-viscous behavior for the high pressurestypically encountered in EHL applications. The traction coefficientis highly sensitive to the value of pressure coefficient of viscosity(a) and the same is demonstrated in the present paperby estimating the traction error caused by an uncertainty in thevalue of a.

Appendix A. Perturbation method

Based on the assumptions used by Dien and Elrod [51], severalresearchers use a perturbation scheme to derive a modifiedReynolds equation incorporating the effect of non-Newtonianlubricant behavior. The basic methodology used in a perturbationscheme is shown below for the case of a line contact. The schemebegins with the introduction of equivalent viscosity, Ze, which isgiven by

Ze ¼ t=I (A.1)

where I ¼ qu=qy. Now, velocity u is expanded in terms of e, whichis a small non-dimensional amplitude parameter:

u ¼ u0 þ �u1 (A.2)

Then

I ¼ I0 þ �I1 (A.3)

where

I0 ¼qu0

qy; I1 ¼

qu1

qy(A.4)

Expanding the equivalent viscosity Ze in the region near I0 into aTaylor series:

Ze ¼ Z0 þ �Z1 (A.5)

where

Z0 ¼ ZeðI0Þ and Z1 ¼ I1qZe

qI

� �I0

(A.6)

The momentum equation is

qtqy¼qp

qx(A.7)

Using Eqs. (A.1), (A.3), (A.5) and neglecting e2,

t ¼ ðZ0 þ �Z1ÞðI0 þ �I1Þ ¼ Z0I0 þ �ðZ1I0 þ Z0I1Þ (A.8)

Expanding p,

p ¼ 0þ �p0 (A.9)

Substituting Eqs. (A.8) and (A.9) in Eq. (A.7),

Z0

qI0

qyþ �

qðZ1I0 þ Z0I1Þ

qy¼ �

qp0

qx(A.10)

) Z0

q2u0

qy2¼ 0 (A.11)

and

qðZ1I0 þ Z0I10 Þ

qy¼qp0

qx(A.12)

Integrating Eq. (A.11) under the boundary conditions, u0 ¼ ua aty ¼ 0 and u0 ¼ ub at y ¼ h:

u0 ¼ ua þðub � uaÞ

hy (A.13)

where ua, ub are the velocities of the lower and upper surfaces,respectively, and h is the film thickness. Substituting Eq. (A.6) intoEq. (A.12) gives

Z0 q2u1

qy2¼qp0

qx(A.14)

where

Z0 ¼ I0qZe

qI

� �I0

þ Z0

!(A.15)

Integrating Eq. (A.14) under the boundary conditions, u1 ¼ 0 aty ¼ 0 and u1 ¼ 0b at y ¼ h:

u1 ¼ðy2 � hyÞ

2Z0qp0

qx(A.16)

From Eqs. (A.2), (A.9), (A.13) and (A.15),

u ¼ ua þðub � uaÞ

hyþðy2 � hyÞ

2Z0qp

qx(A.17)

qu

qy¼ðub � uaÞ

hþð2y� hÞ

2Z0qp

qx(A.18)

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P. Kumar, M.M. Khonsari / Tribology International 42 (2009) 1522–1530 1529

Applying the above perturbation scheme to sinh law fluid, forexample, gives

Z0 ¼ mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ mub � ua

ht0

� �2s (A.19)

Therefore, modified Reynolds equation is

qqx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ mub � ua

ht0

� �2s

rh3qp=qx

12m

0@

1A ¼ u0

qqxðrhÞ (A.20)

It is evident from the above Reynolds equation that under purerolling, perturbation method does not differentiate between non-Newtonian and Newtonian fluids. However, as shown subse-quently, shear-thinning behavior under pure rolling is quitenoteworthy and in fact, comparable to that under rolling/slidingcondition in practical EHL situations. While the idea behind theperturbation analysis is quite useful and indeed powerful, itslimitations need due consideration before applying it for aparticular set of operating conditions. Probably, application ofthe above form of perturbation-modified Reynolds equationhas led to the false notion among some workers that the purerolling EHL behavior of shear-thinning lubricants is essentiallyNewtonian.

Appendix B. Generalized Newtonian approach

The momentum equations in the rolling direction (x) andtransverse direction (y) are

qtx

qz¼

qqzðZ_gxÞ ¼

qp

qx(B.1)

qty

qz¼

qqzðZ_gyÞ ¼

qp

qy(B.2)

where _gy ¼ qu=qz and _gy ¼ qv=qz.Integrating Eqs. (B.1) and (B.2) yields

tx ¼ Z qu

qz¼ tx1 þ z

qp

qx(B.3)

ty ¼ Zqv

qz¼ ty1 þ z

qp

qy(B.4)

Integrating Eqs. (B.3) and (B.4) under the boundary conditions

uðz ¼ 0Þ ¼ ua; uðz ¼ hÞ ¼ ub; vðz ¼ 0Þ ¼ 0; vðz ¼ hÞ ¼ 0

u ¼ ua þ ðub � uaÞG0

F0þqp

qxG1 � G0

F1

F0

� �(B.5)

v ¼qp

qyG1 � G0

F1

F0

� �(B.6)

where F0, F1, G0, G1 are the integral functions defined as

F0 ¼

Z h

0

1

Z dz; F1 ¼

Z h

0

z

Z dz; G0 ¼

Z z

0

1

Z dz and

G1 ¼

Z z

0

z

Z dz (B.7)

Applying the continuity of mass flow

qqx

Z h

0ru dz

!þ

qqy

Z h

0rv dz

!¼ 0 (B.8)

Using Eqs. (B.5), (B.6) and (B.7), the following Reynolds equation isobtained

qqx

rF2qp

qx

� �þ

qqy

rF2qp

qy

� �¼ðua þ ubÞ

2

qqxðrhÞ

þðub � uaÞ

2

qqx

r h� 2F1

F0

� �� �(B.9)

where

F2 ¼

Z h

0

z

Z z�F1

F0

� �dz (B.10)

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