On the role of second Newtonian viscosity in EHL point contactsusing double Newtonian shear-thinning model

Puneet Katyal n, Punit KumarDepartment of Mechanical Engineering, National Institute of Technology Kurukshetra, Haryana 136119, India

a r t i c l e i n f o

Article history:Received 29 August 2013Received in revised form7 November 2013Accepted 14 November 2013Available online 28 November 2013

Keywords:EHLShear thinningPoint contactCarreau

a b s t r a c t

The influence of second Newtonian viscosity on the behavior of film thickness in EHL point contacts isinvestigated numerically. The shear-thinning behavior of the lubricants is modeled using the well-recognized and experimentally proven double Newtonian Carreau viscosity equation. The computedresults illustrate substantial increase in the film thickness with increasing second Newtonian viscosities.The effect of second Newtonian viscosity on the variation of EHL film thickness with respect to load,scale, rolling and sliding speed is also discussed herein. The results indicate the importance ofconsidering the effect of second Newtonian viscosity for accurate prediction of film thickness pertainingto shear-thinning lubricants. The comparison of film thickness predictions with published experimentalresults validate both the numerical approach and rheological model employed.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Over the last five decades, rapid developments in the experi-mental techniques and computational methods have made itpossible to measure the behavior of EHL films in real systemsunder a variety of operating conditions. These advancements leadto a deeper understanding of EHL point contact characteristics,which is of great relevance in the successful operation of impor-tant mechanical components. For reliable design, it is necessary toevaluate the lubrication performance under a wide range ofoperating conditions with various types of lubricants. In particular,understanding lubricant rheology and using realistic rheologicalmodels supported by experiments is fundamental for an accurateprediction of EHL characteristics. Many of these realistic lubricantssuch as polymer thickened mineral oils and synthetic oils – wellknown for enhanced lubrication performance in a wide variety ofEHL applications – often exhibit shear-thinning behavior leadingto appreciable thinning of EHL films. The rheology of these shear-thinning lubricants is much more complex and requires moreadvanced rheological models for accurate prediction of thechanges in viscosity and density with respect to the variations inpressure and shear stress. Thus, the inclusion of realistic shear-thinning behavior in EHL analysis is a major concern.

Several efforts have been made in the past to model the non-Newtonian behavior using different rheological models available

in the literature. For instance, Eyring's hyperbolic sine law basedmodel [1] has been the most widely used and universally acceptednon-Newtonianmodel [2–6] to describe the shear-thinning behaviorof lubricants within the EHL community. However, recent numericalstudies [7] and experimental work [8] demonstrated the incapabilityof hyperbolic sine law to describe the experimentally observed film-thinning behavior over a wide range of shear rate. Besides shear-thinning, other aspects related to lubricant rheology have also beenstudied extensively. Bair and Winer [9] presented a nonlinearconstitutive equation incorporating limiting shear strength of lubri-cants. Gecim and Winer [10] proposed a simplified form of Bair-Winer model to predict the film thickness in EHL contacts. Severalresearchers [11–13] carried out EHL analyses in an attempt toincorporate the effect of limiting shear strength.

This paper focuses only on the shear-thinning aspect oflubricant rheology. Among all the models, Carreau viscosity model[7,14–19] best demonstrates the shear-thinning behavior exhib-ited by EHL lubricants. Bair and Qureshi [20] concluded that theshear-thinning response which can be described by a generalizedNewtonian model is of power-law form such as Carreau viscositymodel. Later, Bair [21] employed Carreau-Yasuda model in EHLanalysis and developed approximate correction factor to accountfor the effect of shear-thinning on film thickness. Furthermore,Chapkov et al. [15], Liu et al. [16], Habchi et al. [22] and Katyal andKumar [23] have shown good agreement between the experi-mental and predicted film thickness values using the modifiedCarreau viscosity models in EHL point contacts. Likewise, it hasbeen established that the free volume based Doolittle's piezo-viscous relation describes the realistic lubricant behavior quite

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Tribology International

0301-679X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.triboint.2013.11.009

n Corresponding author.E-mail addresses: katyalgju@gmail.com (P. Katyal),

punkum2002@yahoo.co.in (P. Kumar).

Tribology International 71 (2014) 140–148

accurately as compared to those used conventionally in EHLinvestigations.

Furthermore, many lubricants attain a nearly constant viscosityabove a certain shear stress level instead of showing a continu-ously decreasing viscosity. In other words, the flow curve of suchlubricants show a second Newtonian region and the correspond-ing viscosity is called as the second Newtonian viscosity [24].Carreau viscosity model takes into account the second Newtonianplateau that occurs at very high shear rates (Fig. 1). However, theeffect of second Newtonian viscosity on lubricant film thicknesshas been largely neglected. Since most of the experimental andnumerical investigations [7,14–19] addressed here represent thecases with zero value of second Newtonian viscosity, it would beinteresting to examine the effect of non-zero second Newtonianviscosity as it is likely to cause considerable changes in the EHLcharacteristics. Recently, Kumar and Anuradha [24] investigatedthe effects of second Newtonian viscosity on EHL characteristicsfor the case of line contact. However, due to lack of experimentaldata pertaining to EHL line contacts; it is not possible to validatethe results, Therefore, the present paper deals with a numericallymore complex but experimentally verifiable case of EHL pointcontacts. Using the published experimental data for shear-thinning lubricants, the simulated results are validated herein.The present work sheds light on the influence of second New-tonian viscosity on lubricant film thickness in EHL point contactsfor a wide range of operating conditions.

2. Analysis

The present analysis of EHL rolling/sliding point contact probleminvolves the simultaneous solution of Reynolds, film thickness andload balance equations using a modified Newton–Raphson algorithmwith due consideration to piezo-viscous, shear-thinning and com-pressibility effects. The governing equations are discretized usingfinite difference scheme as described subsequently. The followingsubsections outline the aforesaid governing equations and thenumerical scheme employed.

2.1. Generalized Reynolds equation

The generalized Reynolds Eq. (A2.14) described in Appendix Bfor the case of EHL point contact [23,26] is given below:

∂∂x

ρF2∂p∂x

� �þ ∂∂y

ρF2∂p∂y

� �¼ ðu1þu2Þ

2∂∂xðρhÞþðu2�u1Þ

2∂∂x

ρ h�2F1F0

� �� �ð1Þ

The generalized Reynolds equation in dimensionless form:

∂∂X

ρH3F2∂P∂X

� �þ ∂∂Y

ρH3F2∂P∂Y

� �¼ K

∂∂X

ðρHÞþKS2

∂∂X

ρH 1�2F1F0

!" #

ð2Þ

where X ¼ x=b, Y ¼ y=b, P ¼ p=ph, ρ¼ ρ=ρ0, K ¼ UðE′=pHÞðRx=bÞ3 is adimensionless constant and S¼ ðu2�u1Þ=u0 is the slide to roll ratiodefined as the ratio of sliding velocity to average rolling velocity.The integral functions in the above Eq. (2) are given by

F0 ¼Z 1

0

1ηdZ; F1 ¼

Z 1

0

ZηdZ and; F2 ¼

Z 1

0

Zη

Z�F1F0

!dZ ð3Þ

2.2. Finite difference formulation

The Reynolds Eq. (2) is discretized using the standard mixedsecond order central and first order backward differencing schemeto obtain the equations f ij ¼ 0 as follows:

f ij ¼ εðiþ1=2ÞjPðiþ1Þj�Pij

ΔX2 �εði�1=2ÞjPij�Pði�1Þj

ΔX2

þεiðjþ1=2ÞPiðjþ1Þ �Pij

ΔY2 �εiðj�1=2ÞPij�Piðj�1Þ

ΔY2

�K½ðρHωÞij �ðρHωÞði�1Þj �

ΔX¼ 0 ð4Þ

where εij ¼ ρijðF2ÞijH3ij and ωij ¼ 1þðS=2Þð1�2ðF1=F0ÞÞij

2.3. Lubricant behavior

The present simulations utilize the well-established Carreauviscosity model to describe the double Newtonian shear-thinningbehavior:

η¼ τ=_γ ¼ μ2þðμ1�μ2Þ 1þ μ1 _γ

Gcr

� �2" #ððn�1Þ=2Þ

ð5Þ

Eq. (5) is rewritten in dimensionless form as

η¼ η=μ0 ¼ μ2þðμ1�μ2Þ 1þ πUμ1 _γ

8WHGcr

� �2" #ððn�1Þ=2Þ

ð6Þ

where μ1 ¼ μ1=μ0 and μ2 ¼ μ2=μ0 are the dimensionless first and

second Newtonian viscosities respectively, _γ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið∂u=∂zÞ2þð∂v=∂zÞ2

qis the dimensionless shear rate, Gcr ¼ Gcr=E′ is the dimensionlesscritical stress and n is the power-law index.

It is assumed that μ2 is a constant fraction of μ1. The viscosity–pressure relationship is described by the following free volumeviscosity model [25]:

μ1 ¼ μoexp BV1V0

1ðV=V0Þ�ðV1=V0Þ

� 11�ðV1=VoÞ

� �� �ð7Þ

where B and V1=V0 may be determined experimentally for thespecific lubricant in use. While V=V0 is substituted from thefollowing Tait's equation of state [25]:

ρ0ρ¼ VV0

¼ 1� 11þK ′

o

ln 1þ pKo

ð1þK ′oÞ

� �( )ð8Þ

where ρ0 and V0 are the density and volume respectively atambient pressure and ρ and V are the density and volume at thelocal pressure. The constants Koand K0

0 in the above equation maybe determined experimentally.

Fig. 1. Typical behavior of Shear thinning lubricant [22].

P. Katyal, P. Kumar / Tribology International 71 (2014) 140–148 141

2.4. Boundary conditions

The following boundary conditions [28] are imposed on thepressure distribution:

2.4.1. Inlet boundaryIt is well known that the fluid pressure is equal to the ambient

value far away from the contact zone. Therefore, if the EHL domainis XinrXrXo; �YorYrYo,

PðXin;YÞ ¼ PðX; �YoÞ ¼ PðX;YoÞ ¼ 0 ð9Þ

2.4.2. Outlet boundaryThe pressure drops to zero at the outlet or cavitation boundary

ðX ¼ XOÞ at a vanishingly small rate. Hence

PðXo;YÞ ¼ ∂P∂ζ

����X ¼ Xo

¼ 0 ð10Þ

where ∂P=∂ζ is the pressure gradient in a direction normal to thecavitation boundary.

2.5. Film thickness equation

HðXÞ ¼HoþX2

2þY2

2þ ∑

M

l ¼ 1∑N

k ¼ 1DijklPkl ð11Þ

The elastic influence coefficients Dijkl are given by [28]:

Dijkl ¼2RxphπbE′

ðE1þE2þE3þE4Þ ð12Þ

where

E1 ¼Ψ 1 lnΛ1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiΛ21 þΨ2

1

pΛ2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiΛ22 þΨ2

1

p� �

; E2 ¼Ψ 2 lnΛ2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiΛ22 þΨ2

2

pΛ1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiΛ21 þΨ2

2

p� �

E3 ¼ Λ1 lnΨ1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ21þΨ2

1

qΨ2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ21þΨ2

2

q0B@

1CA; E4 ¼ Λ2 ln

Ψ2þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ22þΨ 2

2

qΨ1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ22þΨ 2

1

q0B@

1CA

Λ1 ¼ k� iþ12

� �ΔX; Λ2 ¼ k� i�1

2

� �ΔX

Ψ1 ¼ l� jþ12

� �ΔY ; Ψ2 ¼ l� j�1

2

� �ΔY

2.6. Load equilibrium equation

The pressure distribution pðx; yÞ in the fluid film must balancethe applied load, F. Hence

∬ pdxdy¼ F ð13Þ

Substituting dimensionless variables P ¼ p=pH , X ¼ x=b, andY ¼ y=b in the above equation,

∬ PdXdY ¼ F

pHb2 ¼

2π3pH

32

Fπ b2

!ð14Þ

The quantity F=π b2 is the average pressure acting on the circularcontact. Therefore, using the fact that the maximum Hertzianpressure pH is 3/2 times the average pressure [28], the aboveequation is written as

ΔF ¼∬ PdXdY�2π3

ð15Þ

Fig. 2. Flow chart of numerical solution procedure [27].

Table 1Convergence of central film thickness as a function of the mesh size.n¼ 0:7; Gcr ¼ 30 kPa; pH ¼ 1:0 GPa; uo ¼ 0:1 m=s; μ2=μ1 ¼ 0.

Fig. 7

nx � ny hc(nm)

64� 64 301.53126� 126 296.25251� 251 293.33501� 501 291.78601� 601 291.75

P. Katyal, P. Kumar / Tribology International 71 (2014) 140–148142

where ΔF is the residue of load equilibrium equation computedduring the numerical procedure.

3. Solution procedure

The mathematical model pertaining to the isothermal EHLpoint contact problem with double Newtonian shear-thinninglubricants is used in the present simulations to study the influenceof second Newtonian viscosity on the behavior of film thickness.

The solution domain in the present simulations ranges fromX¼�4 to 1.5 and Y¼�2.75 to 2.75 with a uniform mesh of501�501 points. It has been verified that further mesh refinementcauses negligible change in the results (a sample case is shown inTable 1). In order to initiate the solution procedure, an initial guessis made for the pressure distribution ½Pij� and offset film thicknessH0. These values are used to calculate film thickness and fluidproperties from Eqs. (9)–(12). The value of η defined by Eq. (6) ateach node within the solution domain, is obtained iterativelyacross the fluid film. The pressure at each node is updated andthe load equilibrium Eq. (14) is used to correct the value of H0.These steps are repeated until ΔF and the relative error in pressuredistribution are less than 10�4. The overall solution scheme usedin the present study is shown by Fig. 2.

4. Experimental validation

In order to establish the validity of the simulation results,Figs. 3 and 4 compare the published experimental film thicknessdata [22] with the corresponding simulated values for a mixture ofSqualane and 15 wt% of PolyIsoPrene (PIP). It is representative ofthe polymer blended multigrade gear and engine oils. The operat-ing conditions, ball radii and material properties are specified inTable 2. Fig. 3 shows h�u0 characteristics for comparison ofcentral and minimum film thickness values obtained using thepresent model with those measured experimentally [22] andcalculated using the equation attributed to Habchi et al. [29].On the other hand, Fig. 4 shows hc�S characteristics for tworolling velocities (0.18 and 0.74 m/s) with the slide-to-roll ratiovarying from 0 to 0.6. Clearly, Figs. 3 and 4 establish the validity ofthe present mathematical model and the numerical approachemployed herein.

5. Results and discussion

The EHL model and the solution procedure described in theprevious section are applied for the evaluation of EHL character-istics in rolling/sliding point contacts. The representative values ofvarious input parameters are taken from Table 3.

5.1. Effect of second Newtonian viscosity

Fig. 5(a) and (b) shows the variation of generalized or effectiveviscosity (η) with shear stress (τ) obtained using Eq. (5) at threedifferent values of μ2/μ1 (0, 0.1 and 0.5) with the shear-thinningparameters at n¼(0.5, 0.7) and Gcr¼(3 kPa and 30 kPa). It can beobserved from these figures that the effective lubricant viscositydecreases steadily with increasing shear stress for μ2/μ1¼0;however, for μ2/μ1¼0.1 and 0.5, it becomes nearly constant beyondthe shear-thinning zone thus showing a second Newtonian pla-teau. Another interesting observation is that the effective viscosityeven at μ2/μ1 as low as 0.1 is much higher than that at μ2/μ1¼0.

uo (m/s)101

h (m

)

10-8

10-7

10-6

hmin Simulationshmin Experimental [22]hc Simulationshc Experimental [22]hmin Equation [29]hc Equation [29]

Squalane + PIPn = 0.8, Gcr = 0.01 MPa,α = 20.9 GPa-1

0.1

Fig. 3. Comparison of (h–uo) characteristics obtained using EHL simulations,experimental data [22] and film thickness equation attributed to Habchi et al.[29] for SqualaneþPIP.

Slide/Roll Ratio(S)0.0

h c(m

)

0

1

2

3

4

EHL SimulationsExperimental[22]

n = 0.8, Gcr = 0.01 MPa, � � = 20.9 GPa-1

Squalane + PIP

um=0.18 m/s

um=0.74 m/s

x 10-7

0.1 0.2 0.3 0.4 0.5 0.6

Fig. 4. Comparison of (hc–S) characteristics obtained using EHL simulations andexperimental data for SqualaneþPIP [22].

Table 2Operating parameters and lubricant properties.

Oil E′ (GPa) PH (GPa) Rx (m) μ1 (Pa s) μ2 (Pa s) Gcr (MPa) n Doolittle–Tait parameters

B V1=Vo K′ Ko/GPa

SqualaneþPIP [22] 123.9 0.47 0.0127 0.0705 0.0157 0.01 0.8 4.2 0.65 11.29 1.007837

P. Katyal, P. Kumar / Tribology International 71 (2014) 140–148 143

Furthermore, it can also be noticed from Fig. 5(a) and (b) thatthe effective low shear viscosity ratio decreases at a much higherrate for n¼0.5 than that for n¼0.7. At a particular shear stress,increasing the value of μ2/μ1 from 0 to 0.5 diminishes thedifference in the values of effective lubricant viscosity for thetwo values of shear-thinning parameters n or Gcr considered here.For instance, at a shear stress of τ¼100 kPa, the effective viscosityfor Gcr¼3 kPa is nearly 3 times lower than that for Gcr¼30 kPa atμ2/μ1¼0 and n¼0.5; whereas, the effective viscosity at μ2/μ1¼0.5for Gcr¼30 kPa is only around 0.8 times of that for Gcr¼3 kPa. Theabove mentioned aspects of second Newtonian viscosity areincluded in the present analysis to study its effect on EHLcharacteristics.

Fig. 6(a) and (b) shows the comparison of pressure distribu-tions and film shapes obtained at μ2/μ1¼0 and 0.5 for u0¼0.1 m/sand 1 m/s respectively with the shear-thinning parameters fixed atn¼0.7 and Gcr¼10 kPa. It can be observed from these figures forthe case of μ2/μ1¼0.5 that the hydrodynamic pressure builds upmore quickly to a higher value within the inlet zone and thispressure build up is more noticeable for the case of higher rollingvelocity. In addition, it is quite apparent from Fig. 6(a) that thepressure profiles for μ2/μ1¼0 and 0.5 almost coincide throughoutthe contact zone, while a significant difference can be observed inFig. 6(b) for the case of u0¼1 m/s. Also, there is a definite pressurespike for μ2/μ1¼0.5 in comparison to an almost unnoticeable spikefor μ2/μ1¼0 at both the values of rolling speed. These differencesin the pressure profile are owing to higher effective viscosity athigher value of second Newtonian viscosity.

Fig. 7 presents the variation of central film thickness (hc) withthe ratio of second to first Newtonian viscosity (μ2/μ1) at pH¼1 GPa

Fig. 6. Comparison of pressure distributions and film shapes obtained at μ2/μ1¼0and μ2/μ1¼0.5 at a rolling velocity (a) u0¼0.1 m/s, (b) u0¼1 m/s.

Table 3Input parameters.

μo ðPa sÞ ρo ðkg=m3Þ R ðmÞ E′ ðPaÞ Doolittle–Tait parameters

B V1=Vo K′ Ko/GPa α¼ d lnðμ=μo Þdp

� p ¼ 0

(GPa�1)

1.0 864 0.01 2:1� 1011 4.325 0.6669 10.859 0.9159 28.3

τ, )

η/μ)

μ2/μ1 =0.5

μ2/μ1 =0.1

μ2/μ1 =0.0

τ, )

η/μ)

μ2/μ1 =0.5

μ2/μ1 =0.1

μ2/μ1 =0.0

Fig. 5. Variation of effective to low shear viscosity ratio with shear stress fordifferent ratios of second to first Newtonian viscosity at (a) Gcr¼3 kPa,(b) Gcr¼30 kPa.

P. Katyal, P. Kumar / Tribology International 71 (2014) 140–148144

and u0¼0.1 m/s for three different combinations of shear-thinningparameters n and Gcr . It can be observed from Fig. 7 that hcincreases with increasing value of μ2/μ1; however, the rate of thischange is a function of n and Gcr . Out of the three lubricantsconsidered here, the most shear-thinning one (n¼0.5 andGcr¼3 kPa) is characterized with the highest rate of increase,while the lubricant with n¼0.7 and Gcr¼30 kPa shows the lowestrate of increase in the value of hc. Besides, it can also be observedfrom Fig. 7 that all the three curves corresponding to threedifferent combinations of shear-thinning parameters n and Gcr

are likely to converge to a common point which clearly shows thatcentral film thickness will reach its Newtonian value if the secondNewtonian viscosity is further increased. These observations leadto the conclusion that the EHL film thickness is highly sensitive tosecond Newtonian viscosity, particularly for more shear-thinninglubricants and therefore, its consideration is of great significancein evaluating EHL characteristics.

5.2. Effect of rolling speed

Fig. 8 presents the variation of central film thickness (hc) withrolling speed (u0) at μ2/μ1¼(0, 0.5) for n¼0.5 and Gcr¼10 kPa. It isquite clear from Fig. 8 that the sensitivity of central film thickness

to rolling speed (slope of log hc vs log u0 plot), is higher (0.626) forμ2/μ1¼0.5 as compared to that (0.568) for μ2/μ1¼0. In other words,for a given change in rolling speed, there is a larger change in EHLfilm thickness pertaining to double Newtonian shear-thinning

μ2/μ1= 0.5

μ2/μ1=0

Fig. 9. Variation of central film thickness with radius.

μ2/μ1=0.5

μ2/μ1=0

Fig. 10. Variation of central film thickness with maximum Hertzian pressure.

μ2/μ1=0.5

μ2/μ1=0.0

Fig. 11. Variation of percentage reduction in central film thickness with slide toroll ratio.

μ2/μ1

Fig. 7. Variation of central film thickness with second to first Newtonianviscosity ratio.

μ2/μ1=0.5

μ2/μ1=0

Fig. 8. Variation of central film thickness with rolling speed at different ratios ofsecond to first Newtonian viscosity.

P. Katyal, P. Kumar / Tribology International 71 (2014) 140–148 145

lubricants in comparison to single Newtonian shear-thinninglubricants. Hence, for such lubricants, the use of film thicknessformulas meant for Newtonian or shear-thinning lubricants withμ2¼0 may lead to significant errors. Therefore, it is recommendedto employ the film thickness formulas for double-Newtonianshear-thinning lubricants such as the one offered by Habchiet al. [29]. It is worth mentioning that the speed sensitivity of hcfor Newtonian fluid (0.659) is nearly the same as that for μ2/μ1¼0.5 (0.626) even though the later yields thinner films.

5.3. Scale and load sensitivities

It is well known within the EHL community that the variationof central film thickness with radius (scale) and maximumHertzian pressure (load) may be characterized by a power-law, i.e., hcpRsp� l

H where s and l are termed as the scale and loadsensitivities respectively. It has been proved experimentally aswell as numerically that the values of s and l for the case of shear-thinning lubricants are substantially higher than those for New-tonian fluids [17–19]. In order to confirm these features, Figs. 9 and10 show the variation of central film thickness with radius andmaximum Hertzian pressure respectively, for lubricants withvarying extents of shear-thinning behavior.

As apparent from Fig. 9, there is an increase in the value ofcentral film thickness with the increase in radius; however, thescale sensitivity (given by the slope of log hc vs log R plot) dependson lubricant rheology. Therefore, out of the lubricants consideredhere, the one with n¼0.5 and μ2/μ1¼0 is the most shear-thinningand yields the thinnest films with highest scale sensitivity of 0.519.With an increase in the value of n to 0.7, there is a considerableincrease (more than three times) in the value of central filmthickness along with a reduction in the scale sensitivity to 0.43.However, for μ2/μ1¼0.5, there is only a marginal change in thevalue of central film thickness with increase in the value of n from0.5 to 0.7 and the scale sensitivity becomes nearly equal to that forNewtonian fluid.

Similarly, Fig. 10 indicates the respective load sensitivities(given by the slope of log hc vs log pH plot). As expected, thecentral film thickness decreases with increasing value of Hertzianpressure and hence, the slopes of the log hc vs log pH curves arenegative. Similar to scale sensitivity, the load sensitivity alsodepends on lubricant rheology as discussed earlier. The mostshear-thinning lubricant considered here with n¼0.5 and μ2/μ1¼0yields the highest load sensitivity (0.28) and thinnest films. More-over, with an increase in the value of n to 0.7, there is a reduction inload sensitivity to 0.18 with a considerable increase (more than twotimes) in the value of central film thickness. The higher loadsensitivity of shear-thinning lubricants with zero second Newtonianviscosity is attributed to higher shear stresses within the inlet zoneat higher loads. But, for μ2/μ1¼0.5, there are only marginal variationsin the values of load sensitivity (from 0.09 to 0.07) and central filmthickness (442–475 nm) with the increase in the value of n from0.5 to 0.7. Therefore, it is clear from this figure that load sensitivityfollows a similar trend as speed and scale sensitivities.

5.4. Effect of slide to roll ratio

It is quite obvious that sliding along with rolling yields higher shearstresses and hence, more prominent shear-thinning effect. Fig. 11presents the variation of percentage reduction in central film thickness(hc) with slide to roll ratio (S). It can be observed that there is amaximum reduction of 10.73% in the value of hc with an increase inthe value of S from 0 to 0.7 at n¼0.5 and μ2/μ1¼0. Moreover, with anincrease in the value of n to 0.7, the shear-thinning effect reducessignificantly and a maximum reduction of 5.49% is noticed. However,

at μ2/μ1¼0.5, hc experiences only a marginal reductionwith increasingvalue of S.

6. Conclusions

Full isothermal point contact simulations are carried out to studythe effect of second Newtonian viscosity on EHL characteristics. Thewell established and experimentally proven double Newtonian Car-reau shear-thinning model is employed to describe the lubricantrheology along with Doolittle's free volume based equation to modelthe pressure–viscosity response The variation of central film thicknesshas been examined with rolling speed, radius, load and slide to rollratio for different values of shear-thinning parameters. It has beenestablished that with non-zero second Newtonian viscosity, the filmthickness approaches close to Newtonian values. The effect of shear-thinning parameters, i.e., power law index and Newtonian limit onEHL film thickness becomes less marked with increasing secondNewtonian viscosity. Therefore, it is recommended that its considera-tion is essential for accurate prediction of film thickness pertaining toshear-thinning lubricants and must be carefully calculated beforeapplying a film thickness formula pertaining to shear-thinninglubricants.

Appendix A

NotationDimensional parameters

a semimajor axis of Hertzian contact ellipse,a¼ ð6k2 ξFR=πE′Þ1=3 (m)

b semiminor axis of Hertzian contact ellipse, b¼ a=k (m)k ellipticity, k¼ a=b (¼1 for circular contact)E

0effective elastic modulus of Rollers 1 and 2,E′¼ 2=½ðð1�v21Þ=E1Þþð1�v22=E2Þ� (Pa)

h film thickness (m)hc central film thickness (m)P pressure (Pa)pH maximum Hertzian pressure, PH ¼ 3F=2πab (Pa)Rx;Ry equivalent radii in the x and y directions (m)R equivalent radius of contact, 1=R¼ 1=Rxþ1=Ry (m)u0 average rolling speed, u0 ¼ ðu1þu2Þ=2 (m/s)u1;u2 velocities of lower and upper surfaces, respectively (m/s)F applied load (N)

Greek symbols

η generalized Newtonian viscosity (Pa s)

ξ second elliptic integral, ξ¼ R π=20

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ð1�k�2Þ sin 2ςdς

qα piezo-viscous coefficient (Pa�1)_γ shear strain rate (s�1)ρ0 inlet density of the lubricant (kg/m3)ρ lubricant density at the local pressure and temperature

(kg/m3)μo inlet viscosity of the Newtonian fluid (Pa s)μ1 first Newtonian viscosity at low shear rates (Pa s)μ2 second Newtonian viscosity at high shear rates (Pa s)τ shear stress in fluid (Pa)

Dimensionless parameters

G dimensionless load parameter, G¼ αE′H dimensionless film thickness, H ¼ hRx=b

2

n power-law index

P. Katyal, P. Kumar / Tribology International 71 (2014) 140–148146

P dimensionless pressure, P ¼ p=phU dimensionless speed parameter, U ¼ μ0u0=E′Rx

W dimensionless load parameter, W ¼ ðF=E′R2x Þ

Greek symbols

η dimensionless generalized Newtonian viscosity, η¼ η=μ0ρ dimensionless fluid density, ρ¼ ρ=ρo

Appendix B

Generalized Reynolds equation

For the equilibrium of pressure and viscous forces in rollingdirection (x) and transverse direction (y) [23]:

∂τzx∂z

¼ ∂∂zðη_γzxÞ ¼

∂p∂x

ðA2:1Þ

∂τzy∂z

¼ ∂∂zðη_γzyÞ ¼

∂p∂y

ðA2:2Þ

where _γzx ¼ ∂u=∂z and _γzy ¼ ∂v=∂zIntegrating Eqs. (A2.1) and (A2.2) gives

τzx ¼ η∂u∂z

¼ τzx1þz∂p∂x

ðA2:3Þ

τzy ¼ η∂v∂z

¼ τzy1þz∂p∂y

ðA2:4Þ

Integrating Eqs. (A2.3) and (A2.4) under the boundary conditionsuðz¼ 0Þ ¼ u1, uðz¼ hÞ ¼ u2, vðz¼ 0Þ ¼ 0; vðz¼ hÞ ¼ 0, the followingequations are obtained for the velocities along x and y directions [30]:

u¼ u1þðu2�u1ÞG0

F0þ∂p∂x

G1�G0F1F0

� �ðA2:5Þ

v¼ ∂p∂y

G1�G0F1F0

� �ðA2:6Þ

Eqs. (A2.5) and (A2.6) in dimensionless form is given below:

u¼ uu0

¼ 1�S2

� �þ S

F0G0þA0 G1�

F1F0

G0

!ðA2:7Þ

v¼ vu0

¼ B0 G1�F1F0

G0

!ðA2:8Þ

Velocity gradients in rolling and transverse directions are obtained bydifferentiating Eqs. (A2.7) and (A2.8):

∂u∂Z

¼ S

ηF0þA0

ηZ�F1

F0

!ðA2:9Þ

∂v∂Z

¼ B0

ηZ�F1

F0

!ðA2:10Þ

where u0 ¼ ðu1þu2Þ=2 is the average rolling speed and

A0 ¼16U

WHπ

� �2∂P∂X

; B0 ¼16U

WHπ

� �2∂P∂Y

ðA2:11Þ

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

F0 ¼Z 1

0

1η:dZ; F1 ¼

Z 1

0

Zη:dZ; G0 ¼

Z Z

0

1ηdZ; and; G1 ¼

Z Z

0

ZηdZ ðA2:12Þ

Applying the continuity of mass flow

∂∂x

Z h

0ρudz

!þ ∂∂y

Z h

0ρvdz

!¼ 0 ðA2:13Þ

Using Eqs. (A2.5), (A2.6) and (A2.13), the following Reynolds equationis obtained:

∂∂x

ρF2∂p∂x

� �þ ∂∂y

ρF2∂p∂y

� �¼ ðu1þu2Þ

2∂∂xðρhÞþðu2�u1Þ

2∂∂x

ρ h�2F1F0

� �� �ðA2:14Þ

where

F2 ¼Z h

0

zη

z�F1F0

� �dz ðA2:15Þ

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