ORIGINAL PAPER

Numerical Analysis of the Oil-Supply Condition in IsothermalElastohydrodynamic Lubrication of Finite Line Contacts

Xiaoling Liu • Peiran Yang

Received: 29 September 2009 / Accepted: 20 January 2010 / Published online: 7 February 2010

� Springer Science+Business Media, LLC 2010

Abstract In order to investigate the effect of oil-supply

condition on the lubrication performance of machine

components, such as gears and roll bearings, a full

numerical solution of the isothermal finite line contact

elastohydrodynamic lubrication (EHL) problem under

different oil-supply conditions was obtained. The supplied

oil quantity was given with the thicknesses of layers of oil

films on both solid surfaces, and an equivalent thickness of

such supplied oil films was introduced. An algorithm

similar to that proposed by Elrod in 1981 was developed to

determine the pressure starting position automatically. The

pressure field was solved with a multi-grid solver which

enables the difficulty of the huge mesh differences in two

directions be overcome easily. The surface deformation

produced by pressure was calculated with a multilevel

multi-integration method. Based on the Newtonian lubri-

cant assumption, comparisons of solutions between the

starved and fully flooded contacts have been made. Results

show that the pressure starting position and the central and

minimum film thicknesses vary with different oil-supply

thicknesses. In addition, the influence of the thickness of

the oil-supply layer, the end profile radius, the entrainment

velocity, and the maximum Hertzian pressure on the

starved fluid film thickness has been investigated. In con-

clusion, the optimum quantity of the supplied oil is very

important for the discussed problem.

Keywords Starvation � Oil-supply condition �Isothermal EHL � Finite line contacts

List of Symbols

bH Hertzian contact radiusffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8wRx=ðpLE0Þp

(m)

E0 Reduced elastic modulus (Pa)

G Material parameter (aE0)h Total gap between the two contacting

surfaces (m)

ha, hb Thin film thicknesses on each contacting

surface (m)

hf Film thickness (m)

hoil Equivalent thickness of the oil-supply

layers (m)

h00 Rigid central film thickness (m)

hcen Central film thickness (m)

hmin Minimum film thickness (m)

H Dimensionless gap hRx

�

b2H

Hcen Dimensionless central film thickness

hcenRx

�

b2H

Hf Dimensionless film thickness hfRx

�

b2H

Hmin Dimensionless minimum film thickness

hminRx

�

b2H

Hoil Dimensionless equivalent thickness of the

oil-supply layers hoilRx

�

b2H

Hcfullyflooded Dimensionless central film thickness in

fully flooded contacts hcenfullyfloodedRx

�

b2H

l Length of the straight part of roller (m)

L Length of the roller (m)�L Dimensionless length of the roller (L/bH)

p Film pressure (Pa)

pH Maximum Hertzian contact pressure, 2w/

(pbHL) (Pa)

Rx Radius of cylindrical roller (m)

Ry Radius of the end profile of the roller (m)

ua, ub Velocities of surfaces a and b (m/s)

ue Entrainment velocity, ðua þ ubÞ=2 (m/s)

X. Liu (&) � P. Yang

School of Mechanical Engineering, Qingdao Technological

University, 11 Fushun Road, Qingdao 266033,

People’s Republic of China

e-mail: lxl@qtech.edu.cn

123

Tribol Lett (2010) 38:115–124

DOI 10.1007/s11249-010-9580-x

Ue Dimensionless velocity parameter

g0ue=ðE0RxÞw Applied load (N)

W Dimensionless load parameter w=ðE0RxLÞx, y Coordinates (m)

xin, xout, yout Domain boundaries (m)

xsta, xcav Inlet and outlet locations of the pressurized

region at the x-axis (m)

X, Y Dimensionless coordinate (x/bH, y/bH)

a Barus viscosity–pressure coefficient (m2/N)

g Viscosity of lubricant (Ns/m2)

g0 Ambient viscosity (Ns/m2)

q Density of lubricant (kg/m3)

q0 Ambient density of lubricant (kg/m3)

h Fractional film content (hf/h)

1 Introduction

Early experimental work studying the starved mechanism

was conducted by Wedeven et al. [1], Chiu [2], and

Wolveridge et al. [3] using the optical interferometry

techniques. They measured starved film thickness and

linked it with the distance between the inlet meniscus and

the Hertzian contact radius. This distance was used as the

parameter governing the condition of starvation. Theoret-

ical models for point contact-starved lubrication were

investigated by Ranger et al. [4] and Hamrock and Dowson

[5], they used small inlet region boundaries to simulate the

effect of the reduced pressure generation compared with a

fully flooded contact. Based on the study of Elrod [6] and

Chevalier et al. [7, 8] presented a steady-state solution for

circular EHL contact with the multilevel method that

facilitated the simulation of starvation/cavitation problems

as a free boundary problem. And later, Damiens et al. [9]

studied the starved elliptical EHL contact. Wijnant [10]

studied the dynamic smooth EHL contacts. Using an Elrod

algorithm, Yin et al. [11] achieved a numerical solution for

the starved thermal EHL in elliptical contacts, and the

lubricant was assumed to be Newtonian fluid.

Recently, Yang et al. [12] investigated the mechanism of

starvation and the thermal and non-Newtonian behavior of

starved EHL in line contact using a non-Elrod algorithm.

In the past decades, most researches in line contact EHL

were focused on the infinitely long line contact, i.e., the

general line contact problems. However, no element in

contacts has an infinite dimension. Therefore, the general

line contact assumption does not satisfy the practical situ-

ations and the axial contact dimension should be consid-

ered to be finite, and the lubricant side leakage, therefore,

should not be ignored. The finite line contact EHL

problems, e.g., the cylindrical roller bearing, a pair of

involute gear teeth, a disk cam and its roller follower, etc.,

are of great importance. For the isothermal finite line EHL

contact, Wymer and Cameron [13] obtained the first

experimental EHL result of the finite line contact using

optical interferometry. Bahadoran and Gohar [14] offered a

solution for an unprofiled cylindrical roller from modifi-

cation of the infinite solution. The first complete numerical

analysis for the EHL of the axially profiled roller was

performed by Mostofi and Gohar [15]. Park and Kim [16]

obtained a numerical solution of the EHL of a finite line

contact using a finite difference and the Newton–Raphson

method, whose numerical procedure is fully systematic.

Recently, Liu and Yang [17] carried out the thermal finite

line contact analysis. All above studies assumed that the

contact was fully flooded. In many practical contacts, this

is not the case. Starvation, which is caused by an insuffi-

cient lubricant supply in the inlet of the contact, is usually

appeared in the finite line contacts. As we know, the study

on starvation in finite line EHL contacts has never been

reported before.

In order to investigate the effect of oil-supply condition

on the lubrication performance of the finite line contact, a

full numerical solution of the isothermal finite line contact

EHL problem under different oil-supply conditions has

been obtained with the Newtonian fluid.

2 Theory

Similar to that under the fully flooded lubrication in Ref.

[17], the contacting geometry model is shown in Fig. 1. A

finite line contact is formed between a roller with the end

profile radius Ry and an infinite plane. The infinite plane is

called solid a and the roller solid b. Assume that the

velocities in the x direction of surfaces a and b are ua and

ub, respectively. Figure 2 shows the starved and pressur-

ized regions and the cross-section of y = 0 in a starved

EHL finite contact. The supplied oil quantity can be given

R

l

L

Rx

y

h00

b

a

y

Fig. 1 Contacting geometry model for finite line contact

116 Tribol Lett (2010) 38:115–124

123

with the thicknesses of the oil film layers on both surfaces

and an equivalent thickness of the oil-supply layer hoil can

be introduced as

hoil ¼ ðhaua þ hbubÞ=ue ð1Þ

where ha and hb are the thin film thicknesses on each

contacting surface, while ue is the entrainment velocity.

Gas is laid between the two layers of the supplied films.

The same as Wijnant [18], the whole computing domain

was divided into two parts: the pressurized area and the

starved area (see Fig. 2). It is assumed that in the pres-

surized region the oil completely filled in the gap h

between solid a and b; while in the starved region, the oil

partly filled in the gap. The position of the meniscus along

the x-axis is defined as xsta in the inlet region, while xcav is

the cavitated position in the outlet region. An algorithm

similar to that proposed by Elrod was developed to deter-

mine the pressure starting position automatically.

The isothermal Reynolds equation for a finite line con-

tact of a roller against a flat half-space can be written as

o

ox

qh3

gop

ox

� �

þ o

oy

qh3

gop

oy

� �

¼ 12ue

o

oxqhhð Þ ð2Þ

Elrod [6] introduced a numerical model to simulate

starvation/cavitation using a variable referred to the

fractional film content representing the degree to which

the lubricant film is filled in the gap between the contacting

solids. In this article, the fractional film ratio h in Eq. 2 was

introduced to represent the degree to which the gap h

between the two surfaces is filled with oil:

hðx; yÞ ¼ hfðx; yÞ=hðx; yÞ ð3Þ

where hf (x, y) is the film thickness. Note that if 0 \ h\ 1,

the contact is starved and the inlet film thickness is equal to

hoil, meanwhile, the exit film thickness is thinner than hoil,

and the film thickness is thinner than the gap between the

two contacting surfaces, the pressure equals to the cavita-

tion pressure. However, if h = 1, the gap will be com-

pletely filled with lubricant, i.e., hf = h.

In order to obtain a unique solution, a complementary

condition of Eq. 2 is given:

pðx; yÞ 1� hðx; yÞð Þ ¼ 0 ð4Þ

with

pðx; yÞ[ 0 and hðx; yÞ ¼ 1 ð5Þ

and

pðx; yÞ ¼ 0 and 0\hðx; yÞ\1 ð6Þ

The boundary conditions of Eq. 2 are expressed as

pðxin; yÞ ¼ pðxout; yÞ ¼ pðx; youtÞ ¼ 0

pðx; yÞ� 0 ðxin\x\xout; 0� y\youtÞ

(

: ð7Þ

For the finite line contact shown in Fig. 1, the film

thickness equation can be written as

hðx; yÞ ¼ h00 þx2

2Rxþ ðy� l=2Þ2

2RyfD

þ 2

pE0

ZZ

pðx0; y0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðx� x0Þ2 þ ðy� y0Þ2q dx0dy0

ð8Þ

where fD is a symbolic function, fD = 1 if l/2 \ y B L/2

and fD = 0 if y B l/2.

The load balance equation is given byZZ

pdxdy ¼ w: ð9Þ

The viscosity–pressure relationship proposed by Roelands

[19] is adopted

g ¼ g0 exp A1 ð1þ A2pÞz0 � 1½ �f g; ð10Þ

where A1 = ln(g0) ? 9.67, A2 = 5.1 9 10-9 Pa-1, and

z0 = a/(A1 � A2).

Fig. 2 Starved and pressurized regions and the cross-section of y = 0

in a finite line contact with insufficient supplied lubricant

Tribol Lett (2010) 38:115–124 117

123

The variation in the density of the lubricant with pressure

is defined by the Dowson–Higginson’s relationship [20]

q ¼ q0 1þ C1p=ð1þ C2pÞ½ �; ð11Þ

where C1 = 0.6 9 10-9 Pa-1 and C2 = 1.7 9 10-9 Pa-1.

3 Numerical Method

In this article, the algorithm developed by Liu and Yang

[17] for the fully flooded finite line contact EHL problem is

modified for the starved finite line contact problem.

Equations (1)–(11) need to be written in dimensionless

forms to facilitate the numerical analysis. The pressure

field was solved with a multi-grid solver following that of

Venner [21], which enabled the difficulty of the huge mesh

differences in two directions be overcome easily. The

surface deformation produced by pressure was calculated

with a multilevel multi-integration method, which was the

same as that developed by Brandt and Lubrecht [22]. In

order to obtain the converged values of p and h, the

flowchart of the numerical procedure for p and h in Ref.

[11] was adopted.

The calculation was carried out in half of the whole

domain only (see Fig. 2) due to the symmetry with respect

to the x-axis. The real calculating domain for a finite line

contact is xin = -4.5bH, xout = 4.5bH, and yout = 0.5L.

Five levels of grids were used for the calculation of pres-

sure and film thickness. The numbers of nodes on the finest

level are 257 along the x-direction, and 513 in the

y-direction for 0 B y B 0.5L.

All errors were checked on the finest grid level, and the

following two errors were calculated for the obtained

solution:

ERRP ¼

P

i

P

j~Pm

i;j � �Pmi;j

�

�

�

�

�

�

P

i

P

j~Pm

i;j

\0:001 ð12Þ

ERRW ¼R R

~PdXdY � 0:5p�L�

�

�

�

0:5p�L\0:001; ð13Þ

where ~Pmi;j and �Pm

i;j are the obtained and starting pressures of

the iteration process, respectively.

4 Results and Discussion

A steel–steel contact was assumed in this study. The input

data common to all results are g0 = 0.08 Pa s,

a = 2.2 9 10-8 Pa-1, E0 = 226 9 109 Pa, G = 4972,

Rx = 12.7 mm, l = 12.7 mm, and L = 14.7 mm. In order

to give various results, Fig. 3 shows the guide contour map

for the finite line contact: y = 0 and x = 0 are through the

central contact domain in the rolling and the axial directions,

respectively, while section B–B is through the side con-

striction, where the minimum film thickness locates.

For Ue = 6.0 9 10-11, Ry = 4 mm, and W = 1.968 9

10-5 (pH = 0.4 GPa), comparisons of the pressure and the

film thickness contour maps and the corresponding profiles

on the plane of Y = 0 and on section B–B between the fully

flooded and the starved EHL are presented in Fig. 4, where

(a) is the fully flooded solution and (b) the starved solution

for hoil = 0.9 lm. It can be observed from Fig. 4a that both

the maximum pressure and the minimum film thickness are

at the end part of the roller. However, the lubricating

performance in the middle part of the roller is quite dif-

ferent from that at the end part of the roller. Compared with

the fully flooded solution, the pressure of the starved

solution gets lower and the pressure starting position moves

toward the Hertzian contacting area, meanwhile, the film

thickness becomes thinner. In addition, both the pressure

and film thickness on section B–B are much lower than

those on the plane of Y = 0, and the film constriction in the

X-direction on section B–B disappears. Therefore, for the

finite line contact, the study on the starved lubrication is

surely required, especially when the components are

working under starved EHL condition. The difference of

the EHL performance between the middle and end parts of

the roller shows that the analysis of the conventional line

contact can only partially simulate that in the middle part

of the roller. Due to the edge effect at the end parts of the

roller, starvation is more severe in finite line EHL contacts.

For the common data of Ue = 6.0 9 10-11, Ry =

4 mm, and W = 1.968 9 10-5 (pH = 0.4 GPa), further

investigation on the oil-supply condition has been made in

Figs. 5, 6, 7, and 8. Figure 5 shows the distributions of the

pressure and film thickness with different hoil on the plane

of Y = 0 and X = 0, respectively. The left part of Fig. 5

gives results on the plane of Y = 0, while the right part on

the plane of X = 0. On the plane of Y = 0, it can be seen

that, as hoil decreases, the film thickness declines, and the

film constriction becomes flatter and moves toward the

outlet zone. In addition, the pressure starting position

Fig. 3 Guide contour map for finite line contact

118 Tribol Lett (2010) 38:115–124

123

moves toward the Hertzian contacting region and the

pressure spike becomes lower. On the plane of X = 0, the

film thickness, especially the side constriction in the axial

direction declines with the decreasing hoil, while pressure

at the end part changes significantly, and it is about two

times as that in the middle part of the roller. It should be

pointed out that, although the starved degree become

severe when hoil is very small, most available oil enters the

contacting area and the efficiency of the supplied oil is very

high. In order to explicit the effect of hoil on the degree of

the starvation at the end part of the roller, distributions of

the film thickness with various hoil on section B–B are

presented in Fig. 6, where the film thicknesses are much

lower than those on the plane of Y = 0 and no film con-

striction along the entrainment direction is observed.

Conclusion can be made that the degree of starvation is

determined by the equivalent thickness of the oil-supply

layer hoil, and for the finite line contact, the influence of the

oil-supply condition on the middle part is very different

from that on the end parts of the contacting solids.

The variations in Hcen/Hcfullyflooded and Hmin/Hcfullyflooded

versus the Hoil/Hcfullyflooded are shown in Fig. 7. It can be

(a)

�4 �3 �2 �1 0 1 2 30

20

40

60

80p

Y

X

0.0

0.2

0.4

0.6

0.8 Y = 0 Section B�B

p (G

Pa)

�4 �3 �2 �1 0 1 2 30

20

40

60

80hf

Y

X

0.0

0.5

1.0

1.5

2.0

h f (µm

)

(b)

�4 �3 �2 �1 0 1 2 30

20

40

60

80p

Y

X

0.0

0.2

0.4

0.6

0.8 Y = 0 Section B�B

p (G

Pa)

�4 �3 �2 �1 0 1 2 30

20

40

60

80hf

Y

X

0.0

0.5

1.0

1.5

2.0

h f (µm

)

Fig. 4 Comparison of the

pressure and the film thickness

contour map and the

corresponding profiles on the

plane of Y = 0 and on section

B–B between the fully flooded

and the starvation EHL for

Ue = 6.0 9 10-11, Ry = 4 mm,

and W = 1.968 9 10-5: a fully

flooded solution and b starved

solution for hoil = 0.9 lm

Tribol Lett (2010) 38:115–124 119

123

seen that when Hoil/Hcfullyflooded is larger than about 1.2, the

Hcen/Hcfullyflooded and Hmin/Hcfullyflooded at the central part

(Y = 0) approach constants, however, the minimum film

thickness changes because the position of the side con-

striction changes with various value of Hoil. When Hoil/

Hcfullyflooded is larger than 5, the minimum film thickness of

the finite line contact gets stable. Considering the side

leakage, Hoil/Hcfullyflooded larger than 5 means a quasi-fully

flooded state. For a deeper insight of the relationship

between the starvation and the fully flooded lubrication,

comparisons of the representative film thicknesses between

the starved and fully flooded contacts are made in Table 1.

For hoil = 1.2 lm, the central film thickness and the min-

imum film thickness on Y = 0 are nearly the same as the

values under the fully flooded case and the relative error is

smaller than 2%, however, relative error of the minimum

film thickness is larger than 40%, quasi-fully flooded

reaches only in the middle part of the roller. When

hoil = 5.0 lm, the relative error for the minimum film

thickness is about 17%, and the quasi-fully flooded is

gradually reaching at the end part of the roller. If actual

supplied oil quantity is much larger than this value, the

contact will be over flooded, and the extra oil will thor-

oughly flows as side leakage. The dimensionless results,

shown in Table 2, are similar to those shown in Table 1.

Variations in Hoil/Hcfullyflooded, Hcen/Hcfullyflooded, Hmin/

Hcfullyflooded, and Hmin/Hcfullyflooded at Y = 0 versus Xsta for

Ue = 6.0 9 10-11, Ry = 4 mm, and W = 1.968 9 10-5

(pH = 0.4 GPa) are shown in Fig. 8. As what can be

expected, as the Hoil/Hcfullyflooded decreases, the starting

position of the pressure moves toward the Hertzian

�4 �3 �2 �1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

Y = 0

p (G

Pa)

X

0.0

0.5

1.0

1.5

2.0 hoil = 0.3 µmhoil = 0.6 µmhoil = 0.9 µmhoil = 1.2 µm

h f (µm

)

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

p (G

Pa)

Y

0.0

0.5

1.0

1.5

2.0

X = 0

h f (µm

)

Fig. 5 Distributions of pressure

and film thickness with different

hoil on the planes of Y = 0 and

X = 0, respectively, for

Ue = 6.0 9 10-11, Ry = 4 mm,

and W = 1.968 9 10-5

�4 �3 �2 �1 0 1 20.0

0.5

1.0

1.5

2.0

hoil = 0.3 µmh

oil = 0.6 µm

hoil = 0.9 µmhoil = 1.2 µm

h f (µm

)

X

Fig. 6 Distribution of the pressure and film thickness with different

hoil on section B–B, for Ue = 6.0 9 10-11, Ry = 4 mm, and

W = 1.968 9 10-5

0 3 6 90.0

0.3

0.6

0.9

1.2

Hcen

/Hcfullyflooded

Hmin

/Hcfullyflooded

Hmin

/Hcfullyflooded

(Y = 0)H

f /H

cful

lyfl

oode

d

Hoil / Hcfullyflooded

Fig. 7 Variations in Hcen/Hcfullyflooded and Hmin/Hcfullyflooded versus

Hoil/Hcfullyflooded for Ue = 6.0 9 10-11, Ry = 4 mm, and

W = 1.968 9 10-5

120 Tribol Lett (2010) 38:115–124

123

contacting area in Fig. 8a; meanwhile, the central and

minimum film thickness declines in Fig. 8b.

The effect of the end shape is very important for the

finite line contact, therefore the solutions with various end

profile radius are discussed. Figure 9 shows distributions of

the film thickness and pressure with different end profile

radius Ry on the planes of Y = 0 and X = 0, respectively,

for Ue = 6.0 9 10-11, hoil = 0.9 lm, and W = 1.968 9

10-5 (pH = 0.4 GPa). It can be seen that the influence of

the end profile radius of the roller on the pressure and film

thickness is significant at the end part; however, there is

almost no influence in the middle. For the same input data,

variations in Hcen/Hcfullyflooded, Hmin/Hcfullyflooded, and Hmin/

Hcfullyflooded at Y = 0 versus the profile radius Ry are shown

in Fig. 10. Due to the large length of the roller, the central

and minimum film thicknesses at the central region have no

change, however, the minimum film thickness at the end

part increases considerably with the increase in radius Ry.

Therefore, it is important to obtain a better lubricating

(a) (b)

�4.5 �4.0 �3.5 �3.0 �2.5 �2.0 �1.5 �1.00

1

2

3

4

5

6

7

Hoi

l/Hcf

ully

floo

ded

Xsta

�4.5 �4.0 �3.5 �3.0 �2.5 �2.0 �1.5 �1.00.0

0.3

0.6

0.9

1.2

Hcen

/Hcfullyflooded

Hmin

/Hcfullyflooded

Hmin

/Hcfullyflooded

(Y = 0)

Hf /

Hcf

ully

floo

ded

Xsta

Fig. 8 Variations in Hoil/

Hcfullyflooded, Hcen/Hcfullyflooded,

Hmin/Hcfullyflooded, and Hmin/

Hcfullyflooded at Y = 0 versus Xsta

for Ue = 6.0 9 10-11,

Ry = 4 mm, and

W = 1.968 9 10-5: avariations in Hoil/Hcfullyflooded

and b variations in Hcen/

Hcfullyflooded and Hmin/

Hcfullyflooded

Table 1 Comparisons of the film thicknesses between the starved

and fully flooded contacts

hoil = 1.2 (lm) hoil = 5 (lm) Fully flooded

hcen (lm) 0.98 0.98 0.98

hmin on Y = 0 (lm) 0.85 0.85 0.84

hmin (lm) 0.27 0.40 0.48

Table 2 Comparisons of Hcen/Hcfullyflooded and Hmin/Hcfullyflooded

versus different Hoil/Hcfullyflooded

Hoil/

Hcfullyflooded & 1.2

Hoil/

Hcfullyflooded & 5

Hcen/Hcfullyflooded 1.0 1.0

Hmin/Hcfullyflooded on

Y = 0

0.86 0.86

Hmin/Hcfullyflooded 0.27 0.41

�4 �3 �2 �1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

Ry = 4 mm

Ry = 10 mm

Ry = 16 mm

Y = 0

p (G

Pa)

X

0.0

0.5

1.0

1.5

2.0

h f (µm

)

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

X = 0

p (G

Pa)

Y

0.0

0.5

1.0

1.5

2.0

h f (µm

)

Fig. 9 Distribution of the film

thickness and pressure versus Ry

on the planes of Y = 0 and

X = 0, respectively, for

Ue = 6.0 9 10-11,

hoil = 0.9 lm, and

W = 1.968 9 10-5

Tribol Lett (2010) 38:115–124 121

123

performance by changing the end profile radius both in

fully flooded and starved EHL contacts.

The common input parameters for the solutions in

Fig. 11, 12, and 13 include: Ry = 4 mm, hoil = 0.9 lm,

and W = 1.968 9 10-5 (pH = 0.4 GPa). Figure 11 shows

the distributions of the film thicknesses and pressures on

the planes of Y = 0 and X = 0, respectively, with dif-

ferent values of the velocity parameter Ue. It can be

observed that the film thickness gets thicker as the

velocity parameter increases, however, the quantity of the

supplied oil is specified, therefore the thicker the film

thickness, the smaller the pressurized area. Therefore,

with the increasing velocity, the position of the inlet

meniscus moves toward the Hertzian contacting region,

the exit constriction becomes flatter, the supplied oil

nearly completely enters the gap between the contacting

solids, and the pressure spike almost disappears. Consid-

ering the compressibility of the lubricant, which is equal

to 87% at the contact center for W = 1.968 9 10-5

(pH = 0.4 GPa) approximately. When the oil supplied

thickness is 0.9 lm, the maximum value of the central

film thickness is 0.78 lm. Therefore, no matter how fast

is the entrainment velocity, the central film thickness is

always lower than 0.78 lm, i.e., if the velocity reaches a

certain value, the central film thickness becomes a con-

stant, as shown in Fig. 12a, in which the relationship

between the film thickness and the velocity parameter is

plotted with double-log coordinates. It can be seen that, in

fully flooded EHL contact, the central and minimum film

thickness is approximately linearly to the velocity, the

difference between the central, and minimum film thick-

ness is very little as the velocity approaches a certain

value. The reason is that the film thickness gets thicker as

the velocity parameter increases under fully flooded case.

However, in a starved contact, the increasing rate of the

central and minimum film thickness is lower than those of

fully flooded condition, especially in the high velocity

case. Both the central and the minimum film thickness of

starved condition are lower than the fully flooded ones.

Variations in Hcen/Hcfullyflooded, Hmin/Hcfullyflooded, and

Hmin/Hcfullyflooded at Y = 0 versus velocity parameter are

shown in Fig. 12b, conclusion can be made that the

higher the velocity parameter, the more severe the star-

vation in finite line contacts.

The variation in the pressure starting position Xsta versus

the velocity parameter is shown in Fig. 13. It can be seen

0 4 8 12 160.0

0.2

0.4

0.6

0.8

Hcen/Hcfullyflooded

Hmin

/Hcfullyflooded

Hmin

/Hcfullyflooded

(Y = 0)

Hf

/Hcf

ully

floo

ded

Ry (mm)

Fig. 10 Variations in Hcen/Hcfullyflooded, Hmin/Hcfullyflooded, and Hmin/

Hcfullyflooded at Y = 0 versus Ry for Ue = 6.0 9 10-11, hoil = 0.9 lm,

and W = 1.968 9 10-5

�4 �3 �2 �1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

Y = 0

p (G

Pa)

X

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Ue = 2x10�11

Ue = 4x10�11

Ue = 2x10�10

h f (µm

)

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

X = 0

p (G

Pa)

Y

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

h f (µ m

)

Fig. 11 Distributions of the

film thickness and pressure with

different velocity parameter on

the planes of Y = 0 and X = 0

for Ry = 4 mm, hoil = 0.9 lm,

and W = 1.968 9 10-5,

respectively

122 Tribol Lett (2010) 38:115–124

123

that the position of the inlet meniscus moves toward the

Hertzian contacting area with the increasing velocity

parameter. Conclusion can then be made that, when the

velocity parameter is fixed, the degree of starvation in finite

line EHL contact is determined by the thickness of the

oil-supply layer, while when the thickness of the oil-supply

layer is fixed, the degree of starvation is mainly determined

by the velocity parameter, in addition, the larger the

velocity parameter, the more the position of the inlet

meniscus close to the Hertzian contacting region.

For Ry = 4 mm, hoil = 0.9 lm, and Ue = 6.0 9 10-11,

the effects of the dimensionless load parameter on the

film thickness and pressure have been investigated (see

Fig. 14). It should be pointed out that, when the dimen-

sionless load parameter changes, the Hertzian contacting

radius bH changes, too. However, the dimensionless

coordinates X and Y are the ratios between the dimen-

sional coordinates x, y, and bH, respectively. Therefore, if

X and Y are used in Fig. 14, the positions of coordinates

are different under various dimensionless load parameters.

In order to avoid misleading, therefore, dimensional

coordinates x and y are used in this figure. As the

dimensionless load parameter increases, the contact area

(a) (b)

1E�11 1E�10 1E�9

0.1

1

10

hcen�starvedhmin�starvedh

cen�fully flooded

hmin�fully flooded

h f (µm

)Ue

1E�11 1E�10 1E�9

0.1

1

10H

cen/H

cfullyflooded

Hmin/Hcfullyflooded

Hf/H

cful

lyfl

oode

d

Ue

Fig. 12 Comparison of the

central and minimum film

thickness with velocity

parameter between the fully

flooded and the starved

condition for Ry = 4 mm,

hoil = 0.9 lm, and

W = 1.968 9 10-5: adimensional solutions and bdimensionless solutions

1E�11 1E�10 1E�9�3.5

�3.0

�2.5

�2.0

�1.5

�1.0

Xst

a

Ue

Fig. 13 Variation in the pressure starting position versus velocity

parameter for Ry = 4 mm, hoil = 0.9 lm, and W = 1.968 9 10-5

�0.4 �0.3 �0.2 �0.1 0.0 0.1 0.20.0

0.5

1.0

1.5

y = 0

p (G

Pa)

x (mm)

0.0

0.5

1.0

1.5

2.0

W = 1.107x10�5

W = 2.491x10�5

W = 4.429x10�5

h f (µm

)

0 2 4 6 80.0

0.5

1.0

1.5

x = 0

p (G

Pa)

y (mm)

0.0

0.5

1.0

1.5

2.0

h f (µm

)

Fig. 14 Effects of the

dimensionless load parameter

on the film thickness and

pressure on the planes of y = 0

and x = 0 for Ry = 4 mm,

hoil = 0.9 lm, and

Ue = 6.0 9 10-11, respectively

Tribol Lett (2010) 38:115–124 123

123

gets larger, the film thickness gets thinner on the plane of

y = 0, and the position of the inlet meniscus moves

toward the inlet region, while the film exit constriction

moves toward the outlet area. Meanwhile, the film

thickness on the plane of x = 0 also decreases with the

increasing dimensionless load parameter. At the ends of

the roller, the minimum film thickness declines signifi-

cantly. In addition, the pressure, especially the end pres-

sure, becomes higher. Therefore, the high dimensionless

load parameter can results in the starvation in finite line

EHL contact more easily.

The former results are all for the roller length of

L = 14.7 mm, therefore, the influence of Hoil/Hcfullyflooded

on Hcen/Hcfullyflooded for different finite lengths has been

discussed for Ue = 6.0 9 10-11, Ry = 4 mm, and

W = 1.968 9 10-5 (see Fig. 15). It can be seen that the

differences of the starvation caused by the three finite

lengths are very little.

5 Conclusions

A full numerical solution of the isothermal finite line EHL

contacts under different oil-supply conditions has been

obtained. Main conclusions can be made as follows:

(1) Due to the edge effect at the end parts of the roller,

the starvation in finite line contact is more severe than

that in infinite line contact.

(2) Compared with the solution of the starved point EHL

contacts, the performance in the starved finite line

EHL contact is more remarkable, especially at the end

parts of the roller.

(3) In general, the variation tendency in starved finite line

contact is similar to that in infinite line contact and

point contact problems.

Acknowledgment This study was financially supported by the

National Natural Science Foundation of People’s Republic of China

through grant 50705045.

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0 1 2 3 4 5 6 7 80.0

0.2

0.4

0.6

0.8

1.0

1.2

L = 8 mmL = 14.7 mmL = 20 mmH

cen

/Hcf

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floo

ded

Hoil /Hcfullyflooded

Fig. 15 Effects of Hoil/Hcfullyflooded on Hcen/Hcfullyflooded for different

finite lengths for Ue = 6.0 9 10-11, Ry = 4 mm, and W = 1.968 9

10-5

124 Tribol Lett (2010) 38:115–124

123