effect of transverse surface roughness and additives in tehd contacts

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Tribology International 41 (2008) 502–514

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Effect of transverse surface roughness and additives in TEHD contacts

Punit Kumara,�, S.C. Jainb,1, S. Rayc,2

aMechanical Engineering Department, National Institute of Technology Kurukshetra, Kurukshetra 136 119, Haryana, IndiabDepartment of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, Roorkee 247 667, India

cDepartment of Metallurgical and Materials Engineering, Indian Institute of Technology Roorkee, Roorkee 247 667, India

Received 19 August 2005; received in revised form 28 October 2007; accepted 30 October 2007

Available online 20 December 2007

Abstract

Surface roughness effects in mixed rheological thermal EHL of rolling/sliding line contacts are investigated numerically. The surface

roughness is assumed to be transverse and its profile is generated by a sinusoidal function defined in terms of its amplitude and

wavelength. A homogeneous mixture of Newtonian base oil and power law fluid additive with varying volume fraction, viscosity ratio

and power law index is used to represent polymer-modified oils. The velocity profile for the mixed rheological fluid model is obtained

using perturbation method to derive Reynolds and mean temperature equations. It is found that the surface roughness effects on EHL

characteristics are significantly modified due to the presence of polymeric fluid additives.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: EHL; Mixture; Roughness; Additives; Power law

1. Introduction

Mechanical components such as gears, roller bearingsand cams, which form an integral part of motion controland power transmission systems, involve concentratedEHL rolling/sliding line contacts. In order to ensuresatisfactory working of such components it is importantto be able to predict the EHL characteristics underpractical conditions. Several efforts have been made toreplace the conventional EHL theory based on isothermalNewtonian fluid model by a more realistic approachinvolving the effects of temperature rise [1–3], non-New-tonian fluid behaviour [3–8], surface roughness [9–14] anddynamic load [11]. Despite this, an important aspectremains largely unexplored. This is regarding the influenceof additives in EHL conjunctions. It is a common practiceto add polymeric fluid additives to the base oil as film

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

iboint.2007.10.010

ing author. Tel.: +911744 239623; fax: +91 1744 238050.

esses: [email protected] (P. Kumar),

rnet.in (S.C. Jain), [email protected] (S. Ray).

32 285691.

32 285732.

thickeners and VI improvers. In such cases, the classicalNewtonian as well as non-Newtonian theories fail topredict the correct flow behaviour of lubricants. Hence, amixture theory is utilized to take account of correctrheology of the lubricant.Dai and Khonsari [15] derived the governing equations

for hydrodynamic lubrication involving a mixture of twoincompressible fluids. The base oil was taken as Newtonianand the additive oil was assumed to be a 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.The interaction terms may be dropped, under the assump-tion of homogeneous mixture, to obtain a simplifiedlubrication equation. Based on this, Li [12] presented theanalysis of hydrodynamic lubrication in a journal bearingusing a homogeneous mixture of Newtonian base oil andpower law fluid additive. Kumar et al. [11] derived theReynolds equation and mean lubricant temperatureequation for a mixture of Newtonian and Ree–Eyringfluids to demonstrate the use of mixed rheological fluidmodel in thermal EHL of rough rolling/sliding linecontacts.

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Nomenclature

Dimensional parameters

a amplitude of roughness, mmb half-width of Hertzian contact zone

ð¼ 4RffiffiffiffiffiffiffiffiffiffiffiffiffiW=2p

pÞ, m

cp specific heat of the lubricant, J/(kgK)ca, cb specific heat of the disks, J/(kgK)E0 effective elastic modulus of rollers 1 and 2, Pah film thickness, mk thermal conductivity of the lubricant, W/(mK)ka, kb thermal conductivity of the disks, W/(mK)p pressure, Paph maximum Hertzian pressure ( ¼ E0b/4R), PaR equivalent radius of contact, muo average rolling speed ( ¼ (ua+ub)/2), m/sua, ub velocity of the lower and upper surface,

respectively, m/sv surface displacement, mw applied load per unit length, N/m

Greek letters

a piezo-viscous coefficient, Pa�1

b thermal expansivity of the lubricant, K�1

go temperature coefficient of viscosity, K�1

l surface roughness wavelength, mro inlet density of the lubricant, kg/m3

ra, rb density of the lower and upper surface,respectively, kg/m3

r lubricant density at the local pressure andtemperature, kg/m3

t shear stress in fluid, Pay temperature of the lubricant, Kyo ambient temperature, KySa=Sb surface temperatures, KZo inlet viscosity of the Newtonian fluid, Pa sZo, a inlet viscosity of the additive fluid, Pa sZ fluid viscosity, Pa sZa viscosity of the additive fluid, Pa s

x relative effective viscosity of the mixturem coefficient of frictionr non-dimensional fluid density ð¼ r=roÞy non-dimensional fluid temperature ð¼ y=yoÞZ non-dimensional viscosity of Newtonian fluid

ð¼ Z=ZoÞZ21 ratio of additive and base oil viscosities ð¼ Za=ZÞZ� effective viscosity of polymer modified oilðZ�av:Þinlet average inlet zone effective viscosityc heat source term due to additive fluid in the

expression for mean temperature

Non-dimensional parameters

a non-dimensional amplitude of roughnessð¼ aR=b2

Þ

A0 normalized surface roughness amplitudeð¼ a=H

Dowson�Higginsonmin Þ

A, B coefficients in the expression for dimensionlessvelocity gradient

c volume fraction of Newtonian fluid in themixture

H non-dimensional film thickness ( ¼ hR/b2)Hmin non-dimensional minimum film thickness

(hminR/b2)Ho non-dimensional offset film thickness ( ¼ hoR/b2)K1,2,3 non-dimensional coefficients in the mean tem-

perature equationl non-dimensional wavelength of roughness

( ¼ l/b)n power law indexo1, o2 Integrals appearing in the expression of cP non-dimensional pressure ð¼ p=phÞ

s dimensionless velocity gradient across the films0, s1 dimensionless velocity gradient at the lower and

upper surface, respectivelyS slide to roll ratio ( ¼ (ub�ua)/u0)U non-dimensional speed parameter ð¼ Zouo=E0RÞ

v non-dimensional displacement ð¼ vR=b2Þ

W non-dimensional load parameter ð¼ w=E0RÞ

P. Kumar et al. / Tribology International 41 (2008) 502–514 503

It was shown by Wu et al. [16] that the flow behaviourof polymer-modified oils can be approximated by adouble truncated power law fluid model. Therefore, therheology of polymeric fluid additives is represented moreclosely by power law type of non-Newtonian fluid. Surfaceroughness is found to cause very high local pressureat the asperity tips, which may lead to failure of the EHLfilm. Hence, the primary objective of the present workis to investigate the combined effect of polymeric fluidadditives and transverse surface roughness on thermalEHL behaviour of rolling/sliding line contacts using amixture of Newtonian fluid as base oil and power law fluidas additive.

2. Mathematical model

2.1. Rheological model of lubricant

The lubricant used in the present analysis is a homo-geneous mixture of Newtonian and power law fluids. It hasbeen assumed that no chemical reaction takes place and theconstituent fluids retain their original mechanical proper-ties after being mixed. Hence, the total shear stress isshared by the two fluids in the proportion of their volumefractions [11,12] as follows:

t ¼ ð1� cÞtn þ cta, (2.1)

ARTICLE IN PRESSP. Kumar et al. / Tribology International 41 (2008) 502–514504

where c, ta and (1�c)tn are the volume fraction (concen-tration) and shear stress of the power law fluid additive andthe Newtonian base oil, respectively. The respectiveconstitutive relationships are

tn ¼ Zg and ta ¼ Zajgjn�1g, (2.2)

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

t ¼ ð1� cÞZgþ cZajgjn�1g. (2.3)

2.2. Reynolds equation

The Reynolds equation incorporating the effect of usingthe mixture of two fluids is derived by perturbationmethod. It is given below in non-dimensional form:

qqX

rH3qP=qX

Zx

� �� K

qqXðrHÞ ¼ 0, (2.4)

where K ¼ 3Up2/4W2,

x ¼ ð1� cÞ þ Z21cnpSUE0

8WHZ0

� �n�1

, (2.5)

where Z21 ¼ Za/Z is the ratio of additive to base oil viscosity.The Reynolds equation (2.4) is discretized by using a

mixed second order central and first order backward dif-ferencing scheme to obtain the equations fi ¼ 0 (i ¼ 2–N)as follows:

f i ¼ �iþ1=2Piþ1 � Pi

DX 2� �i�1=2

Pi � Pi�1

DX 2� K½ðrHÞi � ðrHÞi�1�

DX

� K½ðrHÞT � ðrHÞT�DT �

DT, ð2:6Þ

where �i ¼ ðrH3=ZxÞi.

2.3. Boundary conditions

Inlet boundary:

P ¼ 0 at X ¼ X in. (2.7)

Outlet boundary:

P ¼qP

qX¼ 0 at X ¼ X o. (2.8)

Since the first node lies at X ¼ Xin, P1 is kept fixedat 0 in order to satisfy the inlet boundary conditionimposed by Eq. (2.7). The outlet boundary coordinateXo is determined by following the procedure used byKumar et al. [11].

2.4. Film thickness equation

The film thickness in non-dimensional form is given by

HðX Þ ¼ Ho þX 2

2þ vþ a sinð2pX=l þ foÞ, (2.9)

where fo is the phase angle, a ¼ aR=b2 the non-dimen-sional amplitude, l ¼ l/b the non-dimensional wavelengthof surface roughness and v ¼ vR=b2 the non-dimensionalsurface displacement given by

v ¼ �1

2p

Z X o

X in

P lnðX � SÞ2 dS, (2.10)

v ¼ vR=b2 is evaluated using a fast Fourier transform(FFT)-based technique [17].It may be noted that the surface roughness used in the

present analysis is transverse, and so the surface roughnessheight varies only along the fluid film in the rollingdirection (x direction) and remains constant for aparticular X-coordinate in the transverse direction (alongthe film width, i.e., z direction).

2.5. Mean temperature of lubricant mixture

Following the procedure outlined by Kumar et al. [11],the mean fluid temperature is given by

yin ¼

ðySa þ ySbÞ2

þ K3rH2qymqX

ðA� 10Þ

120þ K1ð1� cÞZ

ðA2 þ 20S2Þ

240þ c

1þ K2H2qP

qX

A� 10

120

,

(2.11)

where

K1 ¼ðE0RUÞ2

yoZok; K2 ¼

bðE0RÞ2U

4Zok

8W

p

� �2

;

K3 ¼croE0R2U

Zok

8W

p

� �3=2

;

c ¼ � K1cpUE0

8WZoH

� �n�1 Z21Z

A3ðnþ 2Þðnþ 3Þðnþ 4Þ

� ðo2 � o1Þ,

o1 ¼ Aðnþ 4Þ Snþ31 þ Snþ3

0

� �=2,

o2 ¼

snþ41 � snþ4

0 ; s140; s040;

snþ40 � snþ4

1 ; s1o0; s0o0;

snþ41 þ snþ4

0 ; 0o� B=Ao1;

2664

s ¼ AY þ B; s0 ¼ sðY ¼ 0Þ; s1 ¼ sðY ¼ 1Þ,

A ¼16

ZUxWH

p

� �2 qP

qX; B ¼ S � A=2.

2.6. Surface temperatures

The surface temperatures in non-dimensional form [11] are

ySa=Sb ¼ yðY ¼ 0=1Þ

¼ 1þDa=b

Z X

X i

1

H

qyqY

� �Y¼0=1

dX 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX � X 0p , ð2:12Þ

ARTICLE IN PRESS

Table 1

Input parameters

1. Fluid properties

Thermal conductivity, k 0.14W/mK

Specific heat, cp 2000 J/kgK

Inlet viscosity, Zo 0.04Pa s

Inlet density, ro 864 kg/m3

Temperature-viscosity coefficient, go 0.042K�1

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

Coefficient of thermal expansivity, b 6.5� 10�4K�1

2. Disk properties

Thermal conductivity, ka, b 47W/mK

Specific heat of disk, ca, b 460 J/kgK

Density, ra, b 7850kg/m3

Equivalent radius of the disks, R 0.02m

Equivalent elastic modulus, E0 2.2� 1011 Pa

3. Operating conditions

Inlet temperature, yo 313K

Non-dimensional speed parameter, U 1� 1010

Non-dimensional load parameter, W 2� 10�5

Non-dimensional material parameter, G 3500


where

Da=b ¼�kðZoÞ

1=2ðp=8W Þ3=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pra=bca=bka=b½E0Uð1� S=2Þ�

qR.

2.7. Density–pressure–temperature relationship

The present analysis uses the Dowson and Higginson[18] density–pressure–temperature relationship for lubri-cants in the dimensionless form

r ¼ 1þ0:6� 10�9Pph

1þ 1:7� 10�9Pph

� �ð1� bðym � 1ÞyoÞ. (2.13)

2.8. Viscosity–pressure–temperature relationship

The viscosity–pressure–temperature relationship proposedby Roelands et al. [19] has been used because it covers thebehaviour of a wide range of lubricants. The Roelands’equation in dimensionless form is

Z ¼ expðln Zo þ 9:67Þ �1þ ð1þ 5:1� 10�9PphÞ

z� ��goðym � 1Þyo

" #. (2.14)

The above equation is also used to calculate the non-dimensional viscosity, Za, of the additive fluid by substitutingZo;a in place of Zo.

2.9. Load equilibrium equation

The pressure distribution obtained from the Reynoldsequation should satisfy the following condition:

DW ¼XN

j¼2

CjPj �p2¼ 0, (2.15)

where

Cj ¼

DX=3; j ¼ N;

4DX=3; j ¼ 2; 4; 6 . . . ;

2DX=3; j ¼ 3; 5; 7 . . . :

8><>:

2.10. Coefficient of friction

The coefficient of friction m is given as follows:

m ¼

ffiffiffiffiffiffiffiffi8

pW

r Z X o

X ini

fB ð1� cÞ þ cZ21E0

ZZofB

n�1

" #( )dX ,

where

f ¼pU Z8WH

. (2.16)

3. Solution procedure

The governing equations presented in the previous sectionare solved using the procedure outlined by Kumar et al. [11]

to obtain the pressure distribution, film shape, meanlubricant temperature distribution and surface temperaturespertaining to thermal EHL of rough rolling/sliding linecontacts.

4. Results and discussion

The results pertaining to thermal EHL of rough rolling/sliding line contacts with a mixture of Newtonian andpower law fluids as the lubricant have been obtained forthe values of slide to roll ratios, speed and load parameterswithin the practical range subject to the constraintsimposed by the time cost of computation and thelimitations of the numerical scheme. The effect oftransverse surface roughness on EHL behaviour isinvestigated for different values of amplitude and wave-length. The surface roughness amplitude is normalized byexpressing it as a fraction of the corresponding minimumfilm thickness given by the Dowson and Higginson [18]formula so as to make it more meaningful. In steady-stateanalysis, the surface roughness profile is assumed to bestationary with respect to the contact zone. It is worthmentioning that the ‘‘stationary surface roughness profile’’in rolling/sliding contacts refers to the simulation at aparticular position of the roughness profile relative to thecontact zone. Since the fluid film thickness at a point is afunction of the corresponding surface roughness height, thefluid film shape varies with the position of the roughnessprofile. It is apparent that the EHL behaviour will bedifferent for various positions of a surface roughnessprofile under the same operating conditions. Hence, inorder to assess the effects of surface roughness with a givenamplitude and wavelength, the simulation is carried out forvarious positions of its profile and the results correspond-ing to the least value of minimum fluid film thickness arepresented. The different positions of the roughness profile


are obtained by varying the phase angle fo in the filmthickness (Eq. (2.9)) between 0 and 2p.

The nominal concentration of the polymer concentratesvaries to a maximum of 20% by weight so that themaximum resultant active polymer concentration in theblend is nearly 2–3% by weight [16]. Since it is assumedthat the mixture is homogeneous and the constituents areof same density, the additive volume fraction is the same asits weight fraction. Therefore, the results presented hereinare in 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 to as the viscosityratio (VR) in the subsequent text. The relative viscosity ofthe commercial grades of polymer blends, which is definedas the ratio of the blend viscosity to the base oil viscosity, isusually kept around two [16]. Therefore, the range of VR

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0 0.1 0.2

Pm

ax

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 0.1 0.2

Pm

ax

Fig. 1. Variation of maximum fluid pressure, Pmax, with normalized surface

n ¼ 0.70 for (a) S ¼ 0.1 and (b) S ¼ 0.5.

for power 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 [16]. The followingresults are obtained using the values of input parametersgiven in Table 1.

4.1. Maximum pressure

Fig. 1(a) shows the variations of maximum fluid pressure,Pmax, with normalized surface roughness amplitude, A0, forpure base oil, c ¼ 0, and polymer modified oils withVR ¼ 40, n ¼ 0.70, c ¼ 0.1 and 0.2, at three different valuesof non-dimensional surface roughness wavelength, l ¼ 0.12,0.20 and 0.28 with the slide to roll ratio fixed at S ¼ 0.1. Itcan be seen from Fig. 1(a) that Pmax increases steeply with aninitial increase in A0 from 0 to 0.1. A subsequent increase inA0 causes a gradual increase in Pmax up to a maximum value.This is followed by a gradual decline to a value slightly below

0.3 0.4 0.5

A'

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

0.3 0.4 0.5

A'

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

roughness amplitude, A0, for polymer-modified oils with VR ¼ 40 and

ARTICLE IN PRESSP. Kumar et al. / Tribology International 41 (2008) 502–514 507

the maximum in some cases. The maximum increments inthe values of Pmax above that for smooth surface are foundto be 131.3%, 119.6% and 89.3% at l ¼ 0.12, 0.20 and 0.28,respectively, for polymer-modified oils with c ¼ 0.2.

The cause of increase in maximum fluid pressure seemsto be the resistance to fluid flow offered by the surfaceroughness. Tonder and Jakobsen [20], in an interferometricstudy on surface roughness effects in EHL contacts, havealso reported that surface roughness causes resistanceagainst fluid flow. The area available for fluid flow isreduced at the asperity tips, and hence the pressure riseslocally in order to maintain the continuity of flow. Thisflow resistance increases with increasing amplitudes ofsurface roughness, which is probably the reason forincreasing values of Pmax with A0. However, an increasein fluid pressure at the asperity tips is accompanied withhigh temperatures, as discussed later in this section. Thehigh temperatures tend to reduce the fluid viscosity which,

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

0 0.1 0.2

Pm

ax

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 0.1 0.2

Pm

ax

Fig. 2. Variation of maximum fluid pressure, Pmax, with normalized surface

n ¼ 0.70 for (a) S ¼ 0.1 and (b) S ¼ 0.5.

in turn, lowers the rise in pressure. This may be a possibleexplanation for the decline observed in the rate of increaseof maximum pressure at higher amplitudes. Further, it isclear from Fig. 1(a) that the maximum pressure decreaseswith increasing value of surface roughness wavelength,which is the same as reported by Sadeghi and Sui [13].Fig. 1(b) shows the same characteristics as in Fig. 1(a) at

S ¼ 0.5. It can be seen that there is an overall reduction inthe value of maximum pressure as compared to thecorresponding values at S ¼ 0.1 on account of lowerlubricant viscosity due to higher temperatures at a higherslide to roll ratio. It can also be seen that the increments inmaximum pressure values are much lesser than that atS ¼ 0.1. This may be due to the fact that the effectiveviscosity of the lubricant is low on account of highertemperatures at higher slide to roll ratio, and hence theflow resistance offered by the surface roughness is reduced.Similar trends are observed at a lower value of power law

0.3 0.4 0.5

A'

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

0.3 0.4 0.5

A'

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28



index, n ¼ 0.55, not shown here due to space constraint. Itis found that the values of maximum pressure as well as theincrements in its value due to surface roughness at n ¼ 0.55are lower than the corresponding values at n ¼ 0.70. This isdue to lower flow resistance offered by surface roughnesson account of lower effective viscosity caused by highershear thinning effect at lower power law index.

Fig. 2(a) shows the variations of Pmax with A0 as inFig. 1(a) with VR increased to 80. On comparison withsimilar characteristics pertaining to polymer-modified oilswith VR ¼ 40, as shown in Fig. 1(a), it is clear that thevalues of Pmax for smooth as well as rough surfacesare higher for higher VR. Further, it is apparent that thereduction in maximum pressure with increasing surfaceroughness wavelength is lower than that at VR ¼ 40. It canbe seen from Fig. 2(b) that an increase in slide to roll ratioto S ¼ 0.5 results in a relatively lower increase in the valueof maximum pressure as compared to that at S ¼ 0.1, asobserved in the case of VR ¼ 40.

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 0.1 0.2

Hm

in

0.58

0.68

0.78

0.88

0.98

1.08

0 0.1 0.2

Hm

in

Fig. 3. Variation of minimum film thickness, Hmin, with normalized surface

n ¼ 0.70 for (a) S ¼ 0.1 and (b) S ¼ 0.5.

4.2. Minimum fluid film thickness

Fig. 3(a) shows the variations of minimum fluid filmthickness, Hmin, with A0 for the same conditions as inFig. 1(a). It can be seen from Fig. 3(a) that Hmin firstdecreases steeply and then gradually with increase in A0. Itis well established that the fluid pressure in EHL falls tozero just beyond the contact zone. Since the pressure nearthe end of contact zone is quite high, it requires highnegative pressure gradients to reduce the pressure to zerowithin a short span. Therefore, the fluid film thicknessdecreases locally to a minimum value, Hmin, in order togenerate high negative pressure gradients. Now, for thecase of rough surface, it is mentioned in the previoussubsection that fluid pressure increases due to the flowresistance caused by the presence of surface roughness.Therefore, it requires much higher negative pressuregradients to reduce the pressure to zero at the outlet.Hence, the local reduction in the fluid film thickness near

0.3 0.4 0.5

A'

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

0.3 0.4 0.5

A'

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28


ARTICLE IN PRESSP. Kumar et al. / Tribology International 41 (2008) 502–514 509

the outlet is much higher as compared to that for smoothsurface. This seems to be the reason for a reduction in thevalue of minimum film thickness with increasing amplitudeof surface roughness. Tonder and Jakobsen [20], in aninterferometric study on surface roughness effects in EHLcontacts, have also reported that flow resistance causedby surface roughness leads to a reduction of fluid filmthickness.

Fig. 3(b) shows the same characteristics as in Fig. 3(a) atS ¼ 0.5. It can be seen that there is an overall reduction inthe value of minimum film thickness as compared to thecorresponding values at S ¼ 0.1 on account of lowerlubricant viscosity due to higher temperatures at higherslide to roll ratio as in the case of smooth surface. It canalso be seen that the reductions in minimum film thicknessvalues are higher than that at S ¼ 0.1. This may be due tothe fact that the overall film thickness at a higher slide toroll ratio is reduced due to increased thermal and shearthinning effects.

0.85

0.95

1.05

1.15

1.25

1.35

0 0.1 0.2

Hm

in

0.58

0.68

0.78

0.88

0.98

1.08

0 0.1 0.2

Hm

in

Fig. 4. Variation of minimum film thickness, Hmin, with normalized surface

n ¼ 0.70 for (a) S ¼ 0.1 and (b) S ¼ 0.5.

Similar trends are observed at a lower value of power lawindex, n ¼ 0.55, not shown here due to space constraint. Itis found that the values of minimum film thickness atn ¼ 0.55 are lower than the corresponding values atn ¼ 0.70 for smooth as well as rough surface conditionsdue to increased shear thinning effect at a lower value ofpower law index. But the percentage reductions in thevalues of minimum film thickness due to surface roughnessare higher than at n ¼ 0.7 despite lower flow resistance atn ¼ 0.55, as mentioned in the previous subsection.Fig. 4(a) shows the variations of Hmin with A0 as in

Fig. 3(a) with VR increased to 80. On comparison withsimilar characteristics pertaining to polymer modified oilswith VR ¼ 40, shown in Fig. 3(a), it is clear that the valuesof Hmin for smooth as well as rough surfaces are higher fora higher VR. It can be seen from Fig. 4(b) that an increasein slide to roll ratio to S ¼ 0.5 results in a higher percentagereduction in the value of minimum film thickness ascompared to that at S ¼ 0.1, as for the case of VR ¼ 40.

0.3 0.4 0.5

A'

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

0.3 0.4 0.5

A'

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28



The above observations lead to the conclusion that theeffect of surface roughness on minimum film thickness inmixed rheological thermal EHL contacts increases withincreasing VR, decreasing power law index and increasingslide to roll ratio for the values of input parameters usedin the present analysis. Therefore, a careful selectionof the additive properties is required to minimize theeffect of surface roughness on the minimum fluid filmthickness.

4.3. Coefficient of friction

Fig. 5(a) shows the variations of coefficient of friction,m, with A0 for pure base oil, c ¼ 0, and polymer-modifiedoils with VR ¼ 40, n ¼ 0.70 and c ¼ 0.1 and 0.2,at l ¼ 0.12, 0.20 and 0.28 for S ¼ 0.1. It is apparentfrom Fig. 5(a) that m increases due to the presenceof surface roughness which is in agreement with the

0.006

0.011

0.016

0.021

0.026

0.031

0.036

0 0.1 0.2

μ

0.006

0.011

0.016

0.021

0.026

0.031

0.036

0.041

0.046

0 0.1 0.2

μ

Fig. 5. Variation of coefficient of friction, m, with normalized surface roughnes

(a) S ¼ 0.1 and (b) S ¼ 0.5.

results presented by Sadeghi and Sui [13]. It can beseen from Fig. 5(a) that m rises steeply to a maximumvalue with an initial increase in A0. Further increase inA0 causes a gradual decline in the value of m slightlybelow the maximum. The probable cause of increase inm is the resistance to fluid flow offered by the surfaceroughness. This flow resistance increases with increasingamplitudes of surface roughness, which seems to be thereason of increasing values of m with A0. The maximumincrements in the values of m above that for smoothsurface are 122.9%, 109.1% and 81.7% at l ¼ 0.12, 0.20and 0.28, respectively, for polymer-modified oils withc ¼ 0.2.Fig. 5(b) shows the same characteristics as in Fig. 5(a)

at S ¼ 0.5. It can be seen that there is an overall increasein the value of coefficient of friction as compared tothe corresponding values at S ¼ 0.1 on account ofthinner fluid films caused by lower lubricant viscosity

0.3 0.4 0.5

A'

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

0.3 0.4 0.5

A'

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

s amplitude, A0, for polymer-modified oils with VR ¼ 40 and n ¼ 0.70 for

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6.00E-03

1.60E-02

2.60E-02

3.60E-02

4.60E-02

5.60E-02

0 0.1 0.2 0.3 0.4 0.5

A'

μ

c=0 l=0.12

c=0.1 =0.12l

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

0.006

0.011

0.016

0.021

0.026

0.031

0.036

0.041

0.046

0.051

0.056

0 0.1 0.2 0.3 0.4 0.5

A'

μ

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

Fig. 6. Variation of coefficient of friction, m, with normalized surface roughness amplitude, A0, for polymer-modified oils with VR ¼ 80 and n ¼ 0.70 for

(a) S ¼ 0.1 and (b) S ¼ 0.5.


due to higher temperatures at higher slide to roll ratio.It is also found that the increments in coefficient offriction are higher than that at S ¼ 0.1. This is probablydue to the fact that the overall fluid film thickness islow on account of higher temperatures at higher slideto roll.

Similar results are obtained for n ¼ 0.55 and it is foundthat the values of coefficient of friction at n ¼ 0.55 arelower than the corresponding values at n ¼ 0.70. On theother hand, the increments in the values of coefficient offriction due to surface roughness are found to be muchhigher than the those at n ¼ 0.70 even though the flowresistance offered by surface roughness is lower on accountof lower effective viscosity caused by higher shear thinningeffect at lower power law index. Fig. 6(a) shows thevariations of m with A0 for VR ¼ 80 and other conditionsbeing the same as in Fig. 5(a). On comparison withFig. 5(a), it is clear that the values of coefficient offriction for smooth as well as rough surfaces are higher forhigher VR.

4.4. Maximum fluid temperature

Fig. 7(a) shows the variations of maximum fluidtemperature, ymax, with A0 for the same conditions as inFig. 5(a). It can be seen from Fig. 7(a) that ymax increasessteeply with an initial increase in A0 from 0 to 0.1.A subsequent increase in A0 causes a gradual increase inthe value of ymax. The probable cause of increase inmaximum fluid temperature is the increase in viscousshear heat due to higher shear stresses and shear strainrates at higher pressure gradients attributed to the flowresistance offered by the surface roughness. Since theflow resistance increases with increasing amplitudesof surface roughness, ymax is found to increase with A0.The maximum increments in the values of ymax abovethat for smooth surface are 43.8%, 41.9% and 37.5%at l ¼ 0.12, 0.20 and 0.28, respectively, for polymer-modified oils with c ¼ 0.2. Fig. 7(b) shows the samecharacteristics as in Fig. 7(a) with the slide to rollratio increased to S ¼ 0.5. It can be seen that there

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320

370

420

470

520

0 0.1 0.2 0.3 0.4 0.5

A'

θm

ax (

K)

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

320

370

420

470

520

0 0.1 0.2 0.3 0.4 0.5

A'

θm

ax (

K)

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

Fig. 7. Variation of maximum fluid temperature, ymax, with normalized surface roughness amplitude, A0, for polymer-modified oils with VR ¼ 40 and

n ¼ 0.70 for (a) S ¼ 0.1 and (b) S ¼ 0.5.

P. Kumar et al. / Tribology International 41 (2008) 502–514512

is an overall increase in the value of maximum temperatureas compared to the corresponding values at S ¼ 0.1 onaccount of higher strain rates at higher slide to roll ratio. Itis found that the percentage increments in maximumtemperature values are nearly the same as at S ¼ 0.1.Similar results obtained for n ¼ 0.55 (not shown in thefigures) reveal that the values of maximum temperature aswell as the increments in its value due to surface roughnessat n ¼ 0.55 are lower than the corresponding values atn ¼ 0.70.

Fig. 8(a) shows the variations of ymax with A0 asin Fig. 7(a) with VR increased to 80. On comparisonwith Fig. 7(a), it is clear that the values of maxi-mum temperature for smooth as well rough surfacesare higher for higher VR. Also, it is deduced fromFig. 8(b) that an increase in slide to roll ratio toS ¼ 0.5 results in a slightly lower percentage increasein the value of maximum temperature as compared tothat at S ¼ 0.1.

5. Conclusions

The presence of transverse surface roughness causesa reduction in the value of minimum fluid film thickness,whereas the coefficient of friction is found to increasein general. These effects are found to increase withincreasing slide to roll ratios, increasing amplitudes anddecreasing wavelengths of surface roughness. Therefore,the process parameters and vibrations during finishingoperations should be controlled so as to eliminate notonly high amplitude but also short wavelength componentsof surface topography. Besides the geometric reasons,surface roughness effects are attributed to abrupt pressurerise at the asperity tips caused by the reduction of areaavailable for flow. The surface roughness effects arefound to be greatly influenced by using a mixture ofNewtonian and power-law fluids as the lubricant.The response of lubricant to the flow resistance causedby the presence of surface roughness is found to be a

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320

370

420

470

520

570

0 0.1 0.2 0.3 0.4 0.5

A'

θm

ax (

K)

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

320

370

420

470

520

570

0 0.1 0.2 0.3 0.4 0.5

A'

θm

ax (

K)

c=0 l=0.12

c=0.1 l=0.12

c=0.2 l=0.12

c=0 l=0.20

c=0.1 l=0.20

c=0.2 l=0.20

c=0 l=0.28

c=0.1 l=0.28

c=0.2 l=0.28

Fig. 8. Variation of maximum fluid temperature, ymax, with normalized surface roughness amplitude, A0, for polymer-modified oils with VR ¼ 80 and

n ¼ 0.70 for (a) S ¼ 0.1 and (b) S ¼ 0.5.


function of the additive parameters and the operatingconditions.

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effect of transverse surface roughness and additives in tehd contacts

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