ARTICLE IN PRESS

0301-679X/$ - s

doi:10.1016/j.tr

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sjaindme@iitr.e

Tribology International 41 (2008) 482–492

www.elsevier.com/locate/triboint

Influence of polymeric fluid additives in EHL rolling/slidingline contacts

Punit Kumara,�, S.C. Jainb, S. Rayc

aMechanical Engineering Department, National Institute of Technology Kurukshetra, Kurukshetra 136119, IndiabDepartment of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India

cDepartment of Metallurgical and Materials Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India

Received 3 July 2005; received in revised form 5 May 2007; accepted 17 October 2007

Available online 11 December 2007

Abstract

The effect of polymeric fluid additives on EHL behavior of rolling/sliding line contacts is investigated numerically at low as well as

high loads. The polymer-modified oil is represented by a homogeneous mixture of Newtonian base oil and power law fluid with varying

concentration, viscosity ratio and power law index. The Reynolds equation incorporating the mixed rheological fluid model is derived

using perturbation method. The EHL characteristics computed for polymer-modified oils are found to depend upon the effective

viscosity of the lubricant mixture which is governed by the superposition of shear thinning behavior and piezo-thickening effect of the

polymeric fluid additive. Since the reference viscosity of polymeric fluid additives is much higher than that of base oil, therefore, polymer-

modified oils are shown to yield thicker fluid films in most of the cases. The results show a significant variation in maximum fluid pressure

and minimum fluid film thickness with the volume fraction, reference viscosity ratio and power law index of the polymeric fluid additive.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: EHL; Polymeric fluid additives; Power law fluid; Line contacts; Viscosity ratio

1. Introduction

Elastohydrodynamic lubrication of rolling/sliding linecontacts is of great relevance in the successful operation ofthe mechanical components such as gears, roller elementbearings, cams, etc., which form a vital part of most of themachines. In order to meet the growing demand of industryfor highly advanced technologies, the operating loads areincreasing, the fluid films are becoming thinner and specialpurpose lubricants are being employed. Therefore, it isnecessary to develop a better understanding of elastohy-drodynamic lubrication taking account of the rheologicalbehavior of practically used lubricants under a wide rangeof operating conditions.

Although several workers [1–7] have incorporated thenon-Newtonian fluid behavior in EHL analysis, a major

ee front matter r 2007 Elsevier Ltd. All rights reserved.

iboint.2007.10.008

ing author.

esses: punkum2002@yahoo.co.in (P. Kumar),

rnet.in (S.C. Jain), suraydmt@iitr.ernet.in (S. Ray).

aspect remains largely unexplored. This is regarding theinfluence of additives in EHL conjunctions. When poly-meric fluid additives are added to the base oil as filmthickeners and VI improvers, the classical Newtonian aswell as non-Newtonian theories fail to predict the flowbehavior of the lubricants correctly. In these cases, mixturetheory is applied to the lubrication problem to takeaccount of the correct flow behavior of the lubricant. Daiand Khonsari [8] derived the governing equations forhydrodynamic lubrication involving a mixture of twoincompressible fluids. The base oil was taken as Newtonianand the additive oil was assumed to be simple non-Newtonian fluid. The resulting mixture was classified as anon-homogeneous and non-Newtonian fluid. Due to non-homogeneity of the mixture, interaction terms appear inthe conservation laws corresponding to each constituent[8]. The interaction terms may be dropped, under theassumption of homogeneous mixture, to obtain a simplifiedlubrication equation [9]. Based on this, Li [9] presented theanalysis of hydrodynamic lubrication in journal bearing

ARTICLE IN PRESS

Nomenclature

Dimensional parameters

b half width of Hertzian contact zone, b ¼

4RffiffiffiffiffiffiffiffiffiffiffiffiffiW=2p

p, (m)

E0 effective elastic modulus of rollers 1 and 2(Pa)

H film thickness (m)hmin minimum film thickness (m)ho offset film thickness (m)p pressure (Pa)ph maximum Hertzian pressure, ph ¼ E0b/4R, (Pa)R equivalent radius of contact (m)uo average rolling speed, uo ¼ (ua+ub)/2, (m/s)ua, ub velocities of lower and upper surfaces, respec-

tively (m/s)v surface displacement (m)w applied load per unit length (N/m)x abscissa along rolling direction (m)

Greek symbols

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

(s�1)ro inlet density of the lubricant (kg/m3)r lubricant density at the local pressure and

temperature (kg/m3)t shear stress in fluid (Pa)Z fluid viscosity (Pa s)Za viscosity of the additive fluid (Pa s)

Non-dimensional parameters

G non-dimensional material parameter, G ¼ aE0

H non-dimensional film thickness, H ¼ hR/b2

Hmin non-dimensional minimum film thickness, Hmin ¼

hminR/b2

Ho non-dimensional offset film thickness, Ho ¼

hoR/b2

n power law indexN total number of nodesP non-dimensional pressure, P ¼ p/phPmax maximum fluid pressureS slide to roll ratio, S ¼ (ub�ua)/uoU non-dimensional speed parameter, U ¼ Zouo/E0Rv̄ non-dimensional displacement, v̄ ¼ vR=b2

W non-dimensional load parameter, W ¼ w/E0RX non-dimensional abscissa, X ¼ x/bXin inlet boundary co-ordinateXo outlet boundary co-ordinateDX grid size of meshzo Roelands parameter

Greek symbols

m coefficient of frictionr̄ non-dimensional fluid density, r̄ ¼ r=roZ̄ non-dimensional viscosity of Newtonian fluid,

Z̄ ¼ Z=ZoZ21 ratio of additive and base oil viscosities,

Z21 ¼ Za/Zx viscosity modification factorðZ�av:Þinlet average inlet zone effective viscosity

P. Kumar et al. / Tribology International 41 (2008) 482–492 483

using a homogeneous mixture of Newtonian base oiland power law fluid additive. Similarly, Kumar et al. [10]derived the Reynolds equation and mean lubricanttemperature equation for a mixture of Newtonian andRee-Eyring fluids to demonstrate the use of mixedrheological fluid model in thermal EHL of rough rolling/sliding line contacts.

It was shown by Wu et al. [11] that the flow behavior ofpolymer-modified oils can be approximated by a doubletruncated power law fluid model. Therefore, the rheologyof polymeric fluid additives is represented more closely bypower law type of non-Newtonian fluid as compared toRee-Eyring fluid model. Hence, in the present work, theeffect of polymeric fluid additives on isothermal EHLbehavior of rolling/sliding line contacts is investigatedusing a mixture of Newtonian fluid as base oil and powerlaw fluid as additive. The effect of temperature rise onlubricant viscosity and density is neglected in order tostudy the superposition of shear thinning and piezo-thickening effects of the polymeric fluid additive in theabsence of thermal effect. The Reynolds equation incor-porating the mixed rheological fluid model is derived using

perturbation method under the assumptions used by Li [9]and Kumar et al. [10].

2. Mathematical model

2.1. Rheological model of lubricant

A mixture of Newtonian and power law fluids has beenconsidered in the present work. The mixture is homo-geneous as it has been assumed that no chemical reactiontakes place and the constituent fluids retain their originalmechanical properties after being mixed. Hence, the totalshear stress is shared by the two fluids in the proportion oftheir volume fractions [9,10] as follows:

t ¼ ð1� cÞtn þ cta (2.1)

where c, ta and (1�c)tn are the volume fraction and shearstress of the power law fluid additive and the Newtonianbase oil, respectively. The respective constitutive relation-ships are:

tn ¼ Zg and ta ¼ Zajgjn�1g (2.2)

ARTICLE IN PRESSP. Kumar et al. / Tribology International 41 (2008) 482–492484

where g=qu/qy is the shear-strain rate, Za is the viscosity ofthe additive fluid and n is the power law index. Substitutingthese relations in Eq. (2.1) gives

t ¼ ð1� cÞZgþ cZajgjn�1g (2.3)

where Z is the viscosity of the Newtonian base oil.

2.2. Reynolds equation

The Reynolds equation incorporating the effect of usingthe mixture of two fluids is derived in Appendix byperturbation method, as used by Li [9] for a mixture ofNewtonian and power law fluids and Kumar et al. [10] fora mixture of Newtonian and Ree-Eyring fluids under theassumptions laid down by Dien and Elrod [12]. It is givenbelow in non-dimensional form:

@

@X

r̄H3@P=@X

Z̄x

� �� K

@

@Xðr̄HÞ ¼ 0 (2.4)

where

K ¼3Up2

4W 2

x ¼ ð1� cÞ þ Z21cnpSUE0

8WHZ0

� �n�1

(2.5)

where Z21 ¼ Z21/Z is the ratio of additive to base oilviscosity.

2.3. Finite difference formulation

The Reynolds Eq. (2.4) is discretized by using a mixedsecond-order central and first-order backward differencingscheme in space to obtain the equations fi ¼ 0 (2 to N) asfollows:

f i ¼ �iþ1=2Piþ1 � Pi

DX 2� �i�1=2

Pi � Pi�1

DX 2� K½ðr̄HÞi � ðr̄HÞi�1�

DX

(2.6)

where

�i ¼r̄H3

Z̄x1=2

!i

2.4. Boundary conditions

Inlet boundary:

P ¼ 0 at X ¼ X in (2.7)

Outlet boundary:

P ¼@P

@X¼ 0 at X ¼ X 0 (2.8)

Since the first node lies at X ¼ Xin, P1 is kept fixed at 0 inorder to satisfy the inlet boundary condition imposed by

Eq. (2.7). The outlet boundary co-ordinate X0 is determinedby following the procedure used by Kumar et al. [10].

2.5. Film thickness equation

The film thickness in non-dimensional form is given by

HðX Þ ¼ Ho þX 2

2þ v̄ (2.9)

where, v̄ ¼ vR=b2 is the non-dimensional surface displace-ment given by:

v̄ ¼ �1

2p

Z X o

X in

P lnðX � S̄Þ2 dS̄ (2.10)

v̄ ¼ vR=b2 is computed by using fast Fourier transform(FFT) based technique [13].

2.6. Density–pressure relationship

The present analysis uses Dowson and Whitaker [14]density–pressure relationship for the lubricants in thedimensionless form.

r̄ ¼ 1þ0:6� 10�9Pph

1þ 1:7� 10�9Pph

� �(2.11)

2.7. Viscosity–pressure relationship

The viscosity–pressure relationship proposed by Roe-lands et al. [15] has been used because it covers thebehavior of a wide range of lubricants. The Roelands’equation in dimensionless form is:

r̄ ¼ exp½ðln Z0 þ 9:67Þf�1þ ð1þ 5:1� 10�9PphÞzg� (2.12)

The above equation is also used to calculate the non-dimensional viscosity, Z̄a, of the additive fluid by substitut-ing Zo,a in place of Zo.

2.8. Load equilibrium equation

The pressure distribution obtained from the Reynoldsequation should satisfy the following conditionZ Xo

X i

P dX ¼p2

(2.13)

The integral in Eq. (2.13) is calculated using Simpson’srule and it can be written in the following form:

DW ¼XN

j¼2

CjPj �p2¼ 0 (2.14)

ARTICLE IN PRESS

Table 1

Input parameters for EHL analysis

Inlet viscosity, Zo 0.04 Pa s

Inlet density, ro 864 kg/m3

Pressure–viscosity coefficient, a 1.59� 10�8 Pa�1

Thermal conductivity, ka,b 47W/mK

Density, ra,b 7850 kg/m3

Equivalent radius of the disks, R 0.02m

Equivalent elastic modulus, E0 2.2� 1011 Pa

Non-dimensional speed parameter, U 3500

Non-dimensional load parameter, W 10� 10�11

Non-dimensional material parameter, G 0.2� 10�4

Table 2

Ratios of Z�max for mixtures and pure base oil (MBBVR) at c ¼ 0.2

n VR MBBVR (ðZ�maxÞmixture=ðZmaxÞbase oil)

S ¼ 0.1 S ¼ 0.5

0.55 20 0.88 0.82

40 1.20 0.97

80 3.10 1.66

0.70 20 2.16 1.48

40 10.48 5.57

80 138.19 60.06

0.85 20 19.0 12.19

P. Kumar et al. / Tribology International 41 (2008) 482–492 485

where

Cj ¼

DX3

j ¼ N

4DX3

j ¼ 2; 4; 6 . . .2DX3

j ¼ 3; 5; 7 . . .

8>><>>: (2.15)

3. Solution procedure

In order to initiate the solution procedure forthe problem pertaining to EHL of rolling/sliding linecontacts using polymer-modified oil, an initial guessis made for pressure distribution and the offset filmthickness, Ho. These values are used to calculate filmthickness and fluid properties (density and viscosity) fromEqs. (2.9), (2.11) and (2.12), respectively. The simultaneoussystem of the discretized Reynolds Eq. (2.6) are solvedunder the load equilibrium condition given by Eq. (2.14)using Newton-Raphson technique. The system unknownsare: P2, P3, P4, y, PN and Ho. The new values of theseunknowns are used to update the fluid properties and filmthickness for the next iteration. This procedure continuestill convergence is achieved with a relative accuracyof 0.0001.

40 274.98 225.56

80 790.61 781.79

4. Results and discussion

The proposed model is applied to the problem of EHL ofrolling/sliding line contacts with a mixture of Newtonianand power law fluids as the lubricant. The results have beenobtained for the values of slide to roll ratios, speed andload parameters within the practical range subject to theconstraints imposed by the time cost of computation andthe limitations of the numerical scheme. The nominalconcentration of the polymer concentrates varies to amaximum of 20% by weight so that the maximumresultant active polymer concentration in the blend isnearly 2–3% by weight [11]. Since it is assumed that themixture is homogeneous and the constituents are of samedensity, the additive volume fraction is the same as itsweight fraction. Therefore, the results presented herein arein terms of the volume fraction of the additive fluid.Another important parameter is the ratio of the referenceviscosity of the non-Newtonian fluid at unit shear strainrate and the viscosity of the Newtonian base oil underambient conditions, which is referred as viscosity ratio(VR) in the subsequent text. The relative viscosity of thecommercial grades of polymer blends, which is defined asthe ratio of the blend viscosity to the base oil viscosity, isusually kept around 2 [11]. Therefore, the range of VR forpower law type of additive considered in the presentanalysis is from 20 to 80 so that the relative viscosity of theblend is well within the specified range [11].

The following results are obtained using the values ofinput parameters given in Table 1.

4.1. Effective lubricant viscosity

The EHL behavior of polymer-modified oil films isgoverned by the effective viscosity of the lubricant mixture.Therefore, in order to quantify the effect of power lawadditive on the maximum effective lubricant viscosity,Table 2 shows the maximum blend to base oil VRs(MBBVR ¼ ðZ�maxÞmixture=ðZmaxÞbase oil) for various additiveparameters. Fig. 1(a) compares the distribution of non-dimensional effective viscosity, Z*, of pure base oil withthat for polymer-modified oils within the contact zone atthree values of viscosity ratio, VR ¼ 20, 40 and 80 for anadditive concentration c ¼ 0.2, power law index n ¼ 0.55and slide to roll ratio S ¼ 0.1. In the beginning of thecontact zone, as seen from Fig. 1(a), the effective viscosityof polymer-modified oils is lower than that of pureNewtonian fluid. This is due to the shear thinning effectof the non-Newtonian additive fluid which reduces itsviscosity and suppresses the effect of higher referenceviscosity of the additive as compared to the base oil. In thehigher pressure region around the center of the contactzone, the values of effective viscosity of polymer-modifiedoil with VR ¼ 80 increase substantially which is evidentfrom a maximum blend to base oil viscosity ratio(MBBVR) value of 3.1, as listed in Table 2. This is dueto the piezo-viscous effect which depends not only upon thefluid pressure but also on its ambient viscosity. The latterbeing much higher for the additive with VR ¼ 80, it results

ARTICLE IN PRESS

S=0.1, n=0.55, c=0.2

S=0.5, n=0.55, c=0.2

0

200

400

600

800

1000

1200

Base oil

VR=20

VR=40

VR=80

Base oil

VR=20

VR=40

VR=80

1

201

401

601

801

1001

-1 -0.5 0 0.5 1

x/b

-1 -0.5 0 0.5 1

x/b

�*�*

Fig. 1. (a–b) Non-dimensional effective viscosity, Z*, for pure base oil andpolymer-modified oils with c ¼ 0.2 and n ¼ 0.55.

P. Kumar et al. / Tribology International 41 (2008) 482–492486

in a much higher viscosity rise than the base oil at highpressures. This effect completely neutralizes the shearthinning effect of the power law fluid additive. Althoughthe piezo-viscous rise is lower at VR ¼ 40, it suffices tosuppress the shear thinning effect, as evident from anMBBVR value of 1.2 (see Table 2). On further decreasingthe value of VR to 20, the shear thinning effect becomesdominant leading to a decrease in the MBBVR value to0.88. Fig. 1(b) shows the same characteristics as in Fig. 1(a)with the slide to roll ratio increased to S ¼ 0.5. It can beseen from Fig. 1(b) that there is an overall reduction in theeffective lubricant viscosity of the polymer-modified oils ascompared to the corresponding values in Fig. 1(a). This isattributed to the increased dominance of shear thinningeffect over the piezo-viscous effect at higher slide to rollratio. The MBBVR value at VR ¼ 80 is reduced to merely1.66, i.e., almost half of that at S ¼ 0.1. At VR ¼ 20 and40, the shear thinning effect completely suppresses thepiezo-viscous effect, as evident from MBBVR ¼ 0.82 and0.97, respectively (see Table 2).

Although the lubricant viscosity in the contact zoneinfluences the fluid pressures to a large extent, as discussedsubsequently, the fluid film thickness is found to bestrongly influenced by the inlet zone viscosity. Since theinlet zone pressures are substantially low, the shearthinning effect is comparable to piezo-viscous effect even

at high values of power law index. Therefore, the variationof average inlet zone effective viscosity, ðZ�av:Þinlet, with the c

at S ¼ 0.1 and 0.5 are shown in Fig. 2(a) and (b),respectively, for various combinations of n (0.55, 0.70and 0.85) and VR (20, 40 and 80). It can be seen fromFig. 2(a) that ðZ�av:Þinlet increases with c by the factors of1.81, 3.42 and 8.10 for VR ¼ 20, 40 and 80, respectively, atn ¼ 0.85. This clearly indicates the predominance of theeffect of higher reference viscosity of the additive fluid overits shear thinning effect at high value of n. On decreasingthe power law index to n ¼ 0.70, the maximum increase inthe value of ðZ�av:Þinlet at VR ¼ 80 is reduced to 35%,whereas, it remains nearly constant with increase in c from0 to 0.2 at VR ¼ 40. This indicates an increasingdominance of shear thinning behavior of the additive fluidwhich completely balances the piezo-viscous effect atVR ¼ 40. As the VR is reduced further to VR ¼ 20, withn fixed at 0.70, the value of ðZ�av:Þinlet shows a decreasingtrend and falls by a maximum of 13.8%. At a still lowervalue of power law index, n ¼ 0.55, ðZ�av:Þinlet decreases withincrease in c for all three values of VR. The maximumreductions observed are 23.4%, 21.9% and 18.7%at VR ¼ 20, 40 and 80, respectively, with n fixed at 0.55.Fig. 2(b) shows the same characteristics as in Fig. 2(a) atS ¼ 0.5. It can be clearly seen from Fig. 2(b) that there isan overall decrease in the values of ðZ�av:Þinlet as compared tothe corresponding values in Fig. 2(a) which is due to anincreased shear thinning at higher value of slide to rollratio. The above discussion indicates that the effectivelubricant viscosity is highly sensitive to changes in theadditive properties as well as its concentration.

4.2. Pressure distribution and film shape

Fig. 3(a) compares the pressure distribution of pure baseoil with that for polymer-modified oils at VR ¼ 20, 40 and80 for c ¼ 0.2, n ¼ 0.70 and S ¼ 0.1. In the beginning ofcontact zone, as seen from Fig. 3(a), the pressure profilesfor polymer-modified oils almost coincide with that forpure base oil. However, the magnified view shown on thetop right corner in Fig. 3(a) shows that pressure generatedin polymer-modified oil films is marginally lower than thatof pure Newtonian fluid. This is due to the lower effectiveviscosity value of the polymer-modified oils caused byshear thinning effect of the non-Newtonian additive fluid.On moving towards the center of the contact zone, the fluidpressure for polymer-modified oils rises above that for purebase oil.Fig. 3(b) shows the variations of maximum pressure,

Pmax, with c at S ¼ 0.1 for various combinations of n

(0.55, 0.70 and 0.85) and VR (20, 40 and 80). It can be seenthat Pmax at n ¼ 0.55 varies only marginally with increasein c for VR ¼ 20 and 40, whereas, a maximum increase of7.5% is observed for VR ¼ 80. This is due to increase ineffective viscosity caused by predominance of piezo-viscouseffect. At n ¼ 0.70, the shear thinning effect is relativelyless leading to a maximum increase of pressure spike by

ARTICLE IN PRESS

S=0.1

1

6

11

16

21

26

(�* av)inlet

(�* av)inlet

VR=20 n=0.55

VR=20 n=0.70

VR=20 n=0.85

VR=40 n=0.55

VR=40 n=0.70

VR=40 n=0.85

VR=80 n=0.55

VR=80 n=0.70

VR=80 n=0.85

VR=20 n=0.55

VR=20 n=0.70

VR=20 n=0.85

VR=40 n=0.55

VR=40 n=0.70

VR=40 n=0.85

VR=80 n=0.55

VR=80 n=0.70

VR=80 n=0.85

S=0.5

1

6

11

16

21

26

0 0.05 0.1 1.15 0.2 0.25 0.3 0.35

c

0 0.05 0.1 1.15 0.2 0.25 0.3 0.35

c

Fig. 2. (a–b) Variation of average inlet zone effective viscosity, ðZ�av:Þinlet, with additive volume fraction, c.

P. Kumar et al. / Tribology International 41 (2008) 482–492 487

5.8%, 15.6% and 33.4% for VR ¼ 20, 40 and 80,respectively. At n ¼ 0.85, the shear thinning effect is somild that Pmax is found to increase by 24% at a value ofVR as low as 20. However, for VR ¼ 40, Pmax increasessteeply with an initial increase in c from 0 to around 0.05,after which the slope of the curve decreases, as seen clearlyfrom Fig. 3(b). This limits the maximum increase in Pmax to32% at c ¼ 0.2. Further, it can be seen that the curveacquires a negative slope after an initial rise for VR ¼ 80 atn ¼ 0.85 resulting in a much lower value of Pmax, i.e.,merely 26% higher than that for pure base oil. Thisdecrease in the value of Pmax with increasing value of c athigher n and VR is attributed to an increase in the filmthickness which is the result of higher inlet zone effectiveviscosity. Since an increase in the film thickness increasesthe area available for flow, the fluid pressure eases out.

Fig. 3(c) compares the film profile of pure base oil withthat for polymer-modified oils at VR ¼ 20, 40 and 80 forc ¼ 0.2, n ¼ 0.70 and S ¼ 0.1. At VR ¼ 20, there is anoverall reduction in film thickness which can be stated interms of a 6.4% decrease in the value of central film

thickness below that for pure base oil. It is interesting tonote that the film thickness of polymer-modified oil atVR ¼ 20 decreases, even though its effective viscosity isfound to be higher than that of pure base oil within thecontact zone, as mentioned above. Similarly, at VR ¼ 40,the film profile almost coincides with that for pure base oildespite a much higher value of contact zone viscosity. Thefilm thickness is found to increase only for VR ¼ 80, whichis expressed by a 10% higher value of the central filmthickness.On the other hand, the trend of variation of central film

thickness with VR is the same as that of the correspondingvalue of average inlet zone effective viscosity. This isevident from the following correlation: a decrease inðZ�av:Þinlet from 3.124 for pure base oil to 2.694 forVR ¼ 20 results in a 6.4% lower value of central filmthickness, whereas, the average inlet viscosity as well ascentral film thickness for VR ¼ 40 are nearly the same asthe respective values for pure base oil and an increase inðZ�av:Þinlet from 3.124 for pure base oil to 4.219 for VR ¼ 80causes a 10% increase in the value of central film thickness.

ARTICLE IN PRESS

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

x/b

P

Base oil

VR=20

VR=40

VR=80

0.42

0.50

-1 -0.95

S=0.1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0.2 0.25 0.3 0.350.1 0.150 0.05

c

Pmax

S=0.1, n=0.7, c=0.2

S=0.1, n=0.7, c=0.2

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

H

VR=20 n=0.55

VR=20 n=0.70

VR=20 n=0.85

VR=40 n=0.55

VR=40 n=0.70

VR=40 n=0.85

VR=80 n=0.55

VR=80 n=0.70

VR=80 n=0.85

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

x/b

1

-0.9

Base oil

VR=20

VR=40

VR=80

Fig. 3. (a) Pressure profiles for pure base oil and polymer-modified oils with c ¼ 0.2 and n ¼ 0.70, (b) variation of maximum pressure, Pmax, with additive

volume fraction, c, (c) film profiles for the cases in part (a), at S ¼ 0.1.

P. Kumar et al. / Tribology International 41 (2008) 482–492488

Fig. 4(a) shows the pressure distributions for the sameinput parameters as in Fig. 3(a) with the slide to roll ratioincreased to S ¼ 0.5. It can be seen from Fig. 4(a) thatpressure at any point in the fluid film for polymer-modifiedoils is lower than the corresponding values at S ¼ 0.1. Thisreduction in fluid pressure at higher value of S is attributedto relatively lower effective viscosity caused by increasedshear thinning effect at high shear strain rates. The pressurespike at VR ¼ 20, 40 and 80 and for S ¼ 0.5 is 3.4%, 12%and 28.5%, respectively, higher than that for pure base oil.Fig. 4(b) shows the variation of Pmax with c for the samecombinations of n and VR as in Fig. 3(b) at S ¼ 0.5. It canbe seen that Pmax values at S ¼ 0.5 are lower than thecorresponding values at S ¼ 0.1 for n ¼ 0.55 and 0.70,whereas, it is the opposite for n ¼ 0.85. The values of Pmax

for n ¼ 0.85 are slightly higher at S ¼ 0.5 than those at

S ¼ 0.1 for VR ¼ 20, 40 and 10% higher for VR ¼ 80. Thefluid pressure is known to increase with increasing viscosityand decreasing film thickness and vice versa. Due to anincrease in slide to roll ratio, the decrease in effectivelubricant viscosity is accompanied with a decrease in thefluid film thickness. This reduction in effective lubricantviscosity, at higher values of n and VR, is relatively less(as clear from Table 2) due to highly dominant piezo-viscous effect. Therefore, the effect of film thicknessreduction at higher slide to roll ratio leads to higher valuesof Pmax at n ¼ 0.85.The comparison of Fig. 3(c) with Fig. 4(c) shows that the

film thickness for S ¼ 0.5 at any point n the film is lowerthan the corresponding value for S ¼ 0.1. It can be seenfrom Fig. 4(c) that at VR ¼ 20, the central film thickness,which is 0.98 times of the same at S ¼ 0.1, is reduced by

ARTICLE IN PRESS

Base oil

VR=20

VR=40

VR=80

S=0.5

S=0.5, n=0.7, c=0.2

S=0.5, n=0.7, c=0.2

VR=20 n=0.55

VR=20 n=0.70

VR=20 n=0.85

VR=40 n=0.55

VR=40 n=0.70

VR=40 n=0.85

VR=80 n=0.55

VR=80 n=0.70

VR=80 n=0.85

Base oil

VR=20

VR=40

VR=80

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

x/b

P

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0.2 0.25 0.3 0.350.1 0.150 0.05

c

Pmax

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

H

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

x/b

1

Fig. 4. (a) Pressure profiles for pure base oil and polymer-modified oils with c ¼ 0.2 and n ¼ 0.70, (b) variation of maximum pressure, Pmax, with additive

volume fraction, c, (c) film profiles for the cases in part (a), at S ¼ 0.5.

P. Kumar et al. / Tribology International 41 (2008) 482–492 489

8.2% of its value for pure base oil. Similarly, at VR ¼ 40,the central film thickness is lower than its value for purebase oil by 4.8%, i.e., 0.96 times its value at S ¼ 0.1. AtVR ¼ 80, it is slightly higher than the base oil value and0.93 times of that at S ¼ 0.1.

4.3. Minimum fluid film thickness

The variations of minimum fluid film thickness, Hmin,with c at S ¼ 0.1 and 0.5 are shown in Fig. 5(a) and (b),respectively, for various combinations of n (0.55, 0.70 and0.85) and VR (20, 40 and 80). It can be seen from Fig. 5(a)that Hmin increases with c by a maximum of 25.3%, 58.4%and 110.5% for VR ¼ 20, 40 and 80, respectively, atn ¼ 0.85. This is clearly due to a much higher value of inlet

zone viscosity of the polymer-modified oils as compared tothe pure base oil caused by the predominance of the effectof higher reference viscosity of the additive fluid over itsshear thinning effect at high value of power law index.On decreasing the power law index to n ¼ 0.70, the

maximum increase in the value of Hmin at VR ¼ 80 isreduced to 10%, whereas, it remains nearly constant withincrease in c from 0 to 0.2 at VR ¼ 40. This indicatesincreasing dominance of shear thinning behavior of theadditive fluid which completely nullifies the piezo-viscouseffect at VR ¼ 40. As VR is reduced further to 20, with n

fixed at 0.70, the value of Hmin shows a decreasing trendand falls by a maximum of 6.6%. At a still lower value ofpower law index, n ¼ 0.55, Hmin decreases with increase inc for all three values of VR. The maximum reductions

ARTICLE IN PRESS

S=0.1

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0 0.05 0.1 0.15 0.2 0.30.25

c

0 0.05 0.1 0.15 0.2 0.30.25

c

Hmin

A B C

c = 0.033 c = 0.076

1.68

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Hmin

1.68

S=0.5

A B

c = 0.042

VR=20 n=0.55

VR=20 n=0.70

VR=20 n=0.85

VR=40 n=0.55

VR=40 n=0.70

VR=40 n=0.85

VR=80 n=0.55

VR=80 n=0.70

VR=80 n=0.85

VR=20 n=0.55

VR=20 n=0.70

VR=20 n=0.85

VR=40 n=0.55

VR=40 n=0.70

VR=40 n=0.85

VR=80 n=0.55

VR=80 n=0.70

VR=80 n=0.85

Fig. 5. (a–b) Variation of Hmin, with additive volume fraction, c.

P. Kumar et al. / Tribology International 41 (2008) 482–492490

observed are 11.2%, 10.5% and 8.9% at VR ¼ 20, 40 and80, respectively, with power law index fixed at n ¼ 0.55. Acomparison of these observations with Fig. 2(a) reveals aremarkable similarity between the trends of variation ofHmin and average inlet zone effective viscosity.

It can be seen that the use of additive fluids with differentvalues of VR and power law index in varying concentra-tions may yield the same value of Hmin. This is illustrated inFig. 5(a) for Hmin ¼ 1.68, which may be obtained by usingadditives with (VR ¼ 80, n ¼ 0.85), (VR ¼ 40, n ¼ 0.85)or (VR ¼ 20, n ¼ 0.85) when added in the concentrationsc ¼ 0.033, c ¼ 0.076 and c ¼ 0.2, respectively. Fig. 5(b)shows the same characteristics as in Fig. 5(a) at S ¼ 0.5.It can be clearly seen from Fig. 5(b) that there is anoverall decrease in the values of Hmin as comparedto the corresponding values in Fig. 5(a). As in Fig. 5(a),it is shown that the values of c required to obtainHmin ¼ 1.68 for additives with (VR ¼ 80, n ¼ 0.85) and(VR ¼ 40, n ¼ 0.85) are increased to 0.042 and 0.1,respectively.

5. Conclusions

On the basis of the results presented in Section 4, thefollowing conclusions are drawn:

1.

The isothermal EHL characteristics of polymer-mod-ified oil film under rolling/sliding line contacts dependupon the effective viscosity of the lubricant mixture,which is governed by the superposition of shear thinningbehavior and the effect of high reference VR of thepolymeric fluid additive.

2.

The effect of polymeric fluid additive on the lubricantviscosity is quantified by MBBVR. The value ofMBBVR less than or greater than unity indicates thedominance of shear thinning or piezo-viscous effect,respectively, under a given set of operating conditions.

3.

The shear thinning effect increases with decreasing powerlaw index, which reduces MBBVR to a great extent. Thehigher reference viscosity of the polymeric fluid additivescauses a much more pronounced piezo-viscous effect as

ARTICLE IN PRESSP. Kumar et al. / Tribology International 41 (2008) 482–492 491

compared to that in the case of pure base oil, which tendsto nullify the shear thinning effect. Therefore, MBBVR isfound to increase with increasing VR at a fixed value ofpower law index for the conditions considered in thepresent study. Hence, a suitable combination of VR andpower law index may be selected to attain a desiredmagnitude of MBBVR.

4.

Due to the reduction in additive fluid viscosity caused bymore pronounced shear thinning effect at higher slide toroll ratio, the value of MBBVR decreases with increasein slide to roll ratio (S) from 0.1 to 0.5. This reductionincreases with increasing VR at low power law index(n ¼ 0.55 and 0.70), whereas, the reverse trend isobserved at high power law index (n ¼ 0.85).

5.

The film thickness of polymer-modified oils is governedby inlet zone viscosity which is found to be a function ofadditive volume fraction (c) and additive properties(n and VR). The average inlet zone effective viscosity(ðZ�av:Þinlet) may increase or decrease with increasing c

depending upon the relative magnitudes of shearthinning and piezo-viscous effects. At low values of n

and VR, the shear thinning effect dominates over thepiezo-viscous effect leading to lower values of ðZ�av:Þinletfor polymer-modified oils as compared to the corre-sponding values for pure base oil. The reverse trend isobserved at high values of n and VR due to increaseddominance of piezo-viscous effect. The trend reversalmay also occur at a fixed value of n and VR by varyingthe additive volume fraction.

6.

Since the fluid pressure, viscosity and film thickness areinterdependent variables, therefore, the effect of poly-meric fluid additives on fluid pressure is a function ofeffective viscosity and film thickness of polymer-modified oils. Hence, the fluid pressure distribution isfound to be influenced by polymeric fluid additives to anextent depending upon the values of c, n and VR.

7.

The minimum fluid film thickness (Hmin) of polymer-modified oils is shown to vary with c, n and VR in amanner similar to that of average inlet zone effectiveviscosity (ðZ�av:Þinlet). In general, it is found that Hmin

increases with increasing n and VR. However, Hmin mayincrease with increasing c (from 0 to 0.2) dependingupon whether or not the values of n and VR are highenough to neutralize the shear thinning effect ofpolymeric fluid additive.

Appendix

The velocity profile of the lubricant mixture is derived byperturbation method, as used by Li [9] for a mixture ofNewtonian and power law fluids and Kumar et al. [10] fora mixture of Newtonian and Ree-Eyring fluids under theassumptions laid down by Dien and Elrod [12]. Let usintroduce equivalent viscosity, Ze, which is given by

Ze ¼tI

(A.1)

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

u ¼ u0 þ �u1 (A.2)

Then

I ¼@u0

@yþ �

@u1

@y(A.3)

or

I ¼ I0 þ �I10 (A.4)

where

I0 ¼@u0

@y; I1 ¼

@u1

@y(A.5)

Expanding the equivalent viscosity Ze in the region near I0into a Taylor series:

Ze ¼ Z0 þ �Z1 (A.6)

where

Z0 ¼ ZeðI0Þ and Z1 ¼ I1@Ze@I

� �I0

(A.7)

The momentum equation is:

@t@y¼@p

@x(A.8)

Using Eqs. (A.1), (A.3) and (A.6) and neglecting e2:

t ¼ ðZ0 þ �Z1ÞðI0 þ �I10 Þ ¼ Z0I0 þ �ðZ1I0 þ Z0I10 Þ (A.9)

Expanding p,

p ¼ 0þ �f̂ (A.10)

Substituting Eqs. (A.9) and (A.10) in Eq. (A.8):

Z0@I0

@yþ �

@ðZ1I0 þ Z0I10 Þ@y

¼ �@f̂@x

(A.11)

) Z0@2u0

@y2¼ 0 (A.12)

and

@ðZ1I0 þ Z0I10 Þ

@y¼@f̂@x

(A.13)

Integrating Eq. (A.12) under the boundary conditions,u0 ¼ ua at y ¼ 0 and u0 ¼ ub at y ¼ h:

u0 ¼ ua þub � ua

hy (A.14)

where ua, ub are the velocities of the lower and uppersurfaces, respectively, and h is the film thickness. Sub-stituting Eq. (3.10) into Eq. (3.16) gives

Z0@2u1

@y2¼@f̂@x

(A.15)

ARTICLE IN PRESSP. Kumar et al. / Tribology International 41 (2008) 482–492492

where

Z0 ¼ I0@Ze@I

� �I0

þ Z0

!(A.16)

Integrating Eq. (A.15) under the boundary conditions,u1 ¼ 0 at y ¼ 0 and u1 ¼ 0 at y ¼ h:

u1 ¼ðy2 � hyÞ

2Z0@f̂@x

(A.17)

From Eqs. (A.2), (A.10), (A.14) and (A.17)

u ¼ ua þub � ua

hyþ

y2 � hy

2Z0@p

@x(A.18)

@u

@y¼

ub � ua

hþ

2y� y

2Z0@p

@x(A.19)

Now, for the mixture of Newtonian and power law fluids:

Ze ¼ ð1� cÞZþ cZajI jn�1

@Ze@I

� �I0

¼ cZaðn� 1ÞjI0j

n�1

I0

Z0 ¼ I0@Ze@I

� �I0

þ Z0

!¼ Zx (A.20)

where

x ¼ ð1� cÞ þ cnub � ua

h

� �n�1 ZaZ

(A.21)

Using the velocity distribution given by Eq. (A.18) in masscontinuity equation, the following Reynolds equation isobtained

@

@x

rh3@p=@x

12Zx

� �¼ uo

@

@xðrhÞ (A.22)

The above Reynolds Eq. (A.22) are written in non-dimensional form:

@

@X

r̄H3@P=@X

Z̄x

� �� K

@

@Xðr̄HÞ ¼ 0 (A.23)

where

K ¼3Up2

4W 2; x ¼ ð1� cÞ þ n21cn

pSUE0

8WHZ0

� �n�1

,

Z21 ¼ZaZ; S ¼

ðub � uaÞ

uo.

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