Research Article Comparison of the Influences of Surface ...

Research ArticleComparison of the Influences of Surface Texture andBoundary Slip on Tribological Performances

Qiyin Lin12 and Baotong Li2

1Fuli School of Food Equipment Engineering and Science Xirsquoan Jiaotong University Xirsquoan Shaanxi 710049 China2State Key Laboratory for Manufacturing Systems Engineering Xirsquoan Jiaotong University Xirsquoan Shaanxi 710049 China

Correspondence should be addressed to Baotong Li baotongmemailxjtueducn

Received 17 May 2015 Revised 15 July 2015 Accepted 15 July 2015

Academic Editor Rama S R Gorla

Copyright copy 2015 Q Lin and B Li This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Close attentions have been widely paid to the engineering textured and slip surfaces for improving bearing tribologicalperformances Comparison studies on the tribological characteristics of slip and textured surfaces are carried out in this workTheanalysis results point out that the influences of surface texture and boundary slip on tribological performances of slider bearingare strongly similar For the determinate surface textures there is one and only value of slip velocity to make the tribologicalperformances of textured and slip surfaces in agreement The corresponding relation between the slip velocity and the texturestructure parameters is also obtained and the size of slip velocity is directly related to the texture geometry parameters includingits position parameters This study will help us to further understand the relationship between boundary slip and surface textureand also the slip phenomenon

1 Introduction

In order to enhance bearing performance some innovativeapproaches are developed for instance engineering texturedand slip bearing as well as novel-configuration bearing [1ndash6]Surface texture is growing being used for different purposessuch as the improvement of tribological performances Theincrease in the load-carrying capacity under hydrodynamiclubrication is one of the most successful applications ofsurface texture in tribological field andmany attentions havebeen paid to it

As early as 1950 Salama experimentally and theoret-ically investigated the influence of macroroughness (canbe regarded as surface texture) on the performance ofthrust bearing and indicated that it owns two functionsfeeding lubricant and generating hydrodynamic film [7]Hamilton et al pointed out that the improvement of load-carrying capacity resulted from the asymmetric pressuredistribution induced by surface texture [8] Due to thecavitation at the front-end of dimple the negative pressureis prevented and the asymmetric pressure distribution isobtained The high fluid pressure at the back-end of dimple

(convergence zone) counteracts the low fluid pressure atthe front-end of dimple (divergence zone) finally resultingin the enhancement of load-carrying capacity Anno et alfoundout thatmicroasperity acted like amicrohydrodynamicbearing and generated load support [9] Toslashnder indicatedthat texturing (roughening) the inlet region of plane padcould obtain hydrodynamic action and positive lift and theinlet roughnesstexture produced an equivalent virtual effectof a Rayleigh step [10] Meanwhile microgeometries in inletzone make the flow into the bearing meet less resistancethan it out of the bearing thus the available lubricant withinthe pressure-generating zone is increased Brizmer et alpointed out that two concepts could be used to produceload-carrying capacity in parallel sliding bearing a collectivedimples effect in partial textured surface and an individualdimple effect in full textured surface [11] For partial texturedsurface each dimple strongly affects its neighboring dimplesresulting in a collective effect of the dimples and a step-like pressure distribution over the textured zone For fulltextured surface the dimples do not interact (individualeffect) resulting in a periodic pressure distribution An inletsuction mechanism was demonstrated by Olver et al [12]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 126824 10 pageshttpdxdoiorg1011552015126824

2 Mathematical Problems in Engineering

The pressure reduction from the inlet to the front-end ofdimplepocket results in suction of lubricant into bearing andthus enhances load support This inlet suction mechanismmakes it possible for parallel or low convergence bearingto generate a lubricant film and support an imposed loadArghir et al were enlightened that the pressure generationas well as load support within surface texture is because ofthe convective-inertia effect [1] The transient behavior oftextured bearing got the attention of Gherca et al and theyindicated that the load support produced by moving texturedsurface was based on the squeeze effect of the leading edge ofsurface texture [13] Another mechanism named as balancingwedge action was proposed by Yagi and Sugimura [14] Thismechanism suggests that surface texture on one of thematingsurfaces increases the convergence ratio between the surfacesresulting in the promotion of the whole wedge action in thecontact area

Much more studies focus on the influences of varioustexture geometries such as shape depth and aspect-ratioon bearing performances The effects of texture geometrysuch as texture depth width number and location of textureon slider bearing performances were studied by Fowell etal [15] In order to discuss optimal texturing propertiesfor minimizing friction or maximizing fluid film thicknessa study on the influence of different texturing parameterson hydrodynamic performance was proposed by Dobrica etal [16] Fesanghary and his coworker paid many attentionson the optimization design of texture shape and geometryfor improving load-carrying capacity based on sequentialquadratic programming algorithm [5] The optimum tex-ture to gain maximum load-carrying capacity is chevron-type shape with flat front for unidirectional sliding and itconsists of pairs of the trapezoid-like shapes for bidirectionalsliding The effects of the parameters of orientation ellipsedimple such as area density and textured fraction on theperformances of hydrodynamic lubrication were analyzed byJi et al [17] Besides Qiu et al investigated the influence ofsuch texture parameters as shape and density on the frictioncoefficient and stiffness of gas-lubricated bearing and theoptimal geometry in terms of the maximum load-carryingcapacity was also obtained [18] Hao et al pointed out thatsurface texture with lower area density and larger size had abetter lubrication performance [19]

Textured surface may be able to induce boundary slipwhen liquid flows on it which means that liquid moleculesdo not adhere to solid surface and there is a relativevelocity at fluid-solid interface and such textured surfacesare commonly known as superhydrophobic surfaces [20ndash24] Superhydrophobic surface can be fabricated from themodification of surface microtopography (eg surface tex-ture) and surface energy Researchers make a lot of work tostudy the influences of slip surface on bearing tribologicalperformances A review on the hydrodynamic lubricationwith slip surface was presented recently by Zhang [25] Spikesdeveloped a hydrodynamic lubrication model based on anextended Reynolds equation and analyzed the applicationof slip surface [26] He showed that slider bearing withslip against its stationary surface could generate both anincrease in load support and a reduction in friction Both

the slider bearing and hydrodynamic journal bearing witha heterogeneous slipnoslip surface were analyzed by Salantand Fortier the results demonstrated that boundary slipwould result in good load-carrying capacity and low friction[2 27] Chen et al demonstrated that the boundary slipcondition would move more fluid at one cross section thanat a cross section without slip [28]The study ofWu indicatedthat the enhanced effect of slip surface on hydrodynamicjournal bearing performance was much greater at a smalleccentricity ratio than at a large eccentricity ratio [29] Thelocation and area of slip surface will significantly affect theload-carrying capacity of both the slider bearing and journalbearing [30 31] Aurelian et al compared the effect of wallslip condition with the effect of surface texture under similarlubrication condition and pointed out that the tribologicalbehavior of slip surface and textured surface was similar[32] Furthermore the influences of boundary slip on heatgeneration were noticed by Mahmoud and Waheed [3]Sharma et al also pointed out that velocity slip would reduceheat transfer rate [33]

Above all most studies focus on the individual influ-ence of surface texture or boundary slip on tribologicalperformances of slider and journal bearings respectivelyand a little other literatures pay attention to the synthesizedimpacts of surface texture and boundary slip There aremany important discoveries demonstrated by these publishedresearches but there are also limitations although someresults indicate that textured surface and slip surfaces havesimilar tribological characteristic the comparison and anal-ysis on their connection are still insufficient for examplethe relationship between slip velocity and texture parametersis less underlined A comparison analysis on the effects ofslip and textured surfaces on tribological performances ofslider bearing is performed in this study and a correspondingrelation between the slip velocity and texture parameters isalso deduced Some interesting results are obtained whichwill help people to further understand surface texture andboundary slip

2 Theory and Model

For continuous isoviscous incompressible fluid medium thecontinuity and Navier-Stokes equations are used to predictthe performances of the fluid domain which are expressed asfollows

120597119906

120597119909+

120597V120597119910

+120597119908

120597119911= 0

120588 (119906120597119906

120597119909+ V

120597119906

120597119910+119908

120597119906

120597119911)

= minus120597119901

120597119909+120583(

1205972119906

1205971199092 +1205972119906

1205971199102 +1205972119906

1205971199112)

120588(119906120597V120597119909

+ V120597V120597119910

+119908120597V120597119911

)

= minus120597119901

120597119910+120583(

1205972V

1205971199092 +1205972V

1205971199102 +1205972V

1205971199112)

Mathematical Problems in Engineering 3

120588(119906120597119908

120597119909+ V

120597119908

120597119910+119908

120597119908

120597119911)

= minus120597119901

120597119911+ 120583(

1205972119908

1205971199092 +1205972119908

1205971199102 +1205972119908

1205971199112)

(1)

where 119906V119908 are velocity components 119909119910119911 are Cartesiancoordinates 120588 is fluid density 119901 is fluid pressure and 120583 isviscosity

A published slider bearing with square textures (dimples)demonstrated by Marian et al in [34] is used to investigateand compare the influences of surface texture and boundaryslip on tribological performances The schematic diagram ofthis slider bearing is presented in Figure 1 and its physicalphotograph was shown in Figure 9 of [34] The inner radiusof this slider bearing 119903

119894is 285mm and its outer radius 119903

119900is

45mm The width of oil supply channel at mean radius is2mm This slider bearing consists of 12 bearing pads Thedimplewidth is 200 120583mand its depth is 9 120583mThearea densityof dimples that is the ratio of total area of the dimples to totalarea of the textured surface is 025 that is (119897

119889119897119888)2= 025The

textured fraction is 05 in circumferential direction and is 09in radial direction that is 120579

119905120579119901= 05 and 119861

119905119861119901= 09 Thus

dimple number is 22 in circumferential direction and is 38 inradial direction

The corresponding slider bearing with slip surface ispresented in Figure 2 The only difference compared to thetextured bearing as shown in Figure 1 is in these regionswheredimples locate these regions are flat in the slip bearing andslip boundary condition is applied onto them Both the otherstationary wall and moving wall are applied with traditionalnoslip boundary condition Due to the periodicity of thebearing geometry only a single bearing pad is built up foranalysis in this study and the tribological performance ofthe whole slider bearing is obtained from multiplying thecorresponding performance of the single bearing pad by 12

In the experiments conducted by Marian et al therotational speeds 120596 of moving wall were from 500 rpm to800 rpm and the imposed external axial loads 119882 were 100Nand 200N [34] It should be pointed out that in ordernot to cumulate the error between the experimental andthe theoretical values of the fluid film thickness ℎ

119888 their

values in these slip slider bearings of this work are setequal to the experimental values presented in [34] under thesame rotational speed The experimental values of fluid filmthickness under different working conditions are presentedin Table 1 again for convenience And the axial loads are thecalculated values in this theoretical study but they were inputdata in the experimental test of [34] The other conditionsin this study keep the same as the experimental conditionsin [34] Both the inlet and outlet pressure are equal to theambient pressure The density of lubricant is 848Kgsdotmminus3 andits viscosity 120583 is 0022 Pasdots

To the geometry structure of the slider bearing in thisstudy only the first dimple column exists in small negativepressure and the cavitation phenomena can be neglected dueto the fact that the area of the first dimple column is insignif-icant compared to the total area of the bearing and this is

I

I

I-I view

Dimple

Dimple region

Stationary wall

Moving wall

Dimple

r120596

lc ld

hc

120579t 120579p

Bp

Bt

Figure 1 Schematic diagram of a bearing pad of the slider bearingwith square dimples

Table 1 Fluid film thickness ℎ119888

ℎ119888

119882

100N 200N120596

500 rpm 1708 120583m 1292 120583m600 rpm 1813 120583m 1375 120583m700 rpm 1875120583m 1458 120583m800 rpm 1917 120583m 1500 120583m

also confirmed by Marian et al [34] The result based on atesting analysis in [34] showed that the difference between thepressure considering cavitation and that without cavitationphenomena is extremely small that is the introduction ofcavitation phenomena had negligible effect on the resultsso the final analyses did not take the cavitation phenomenainto account in [34] Thus the cavitation is not considered aswell in this study and the thermal effect is also neglected fordecreasing computing time

For noslip boundary condition it is assumed that theliquid molecules adhere to solid surface strongly at the fluid-solid interface so there is not relative velocity at fluid-solidinterface for slip boundary condition it is assumed thatthe relative velocity between liquid and solid at the fluid-solid interface is not equal to 0 and this relative velocityis defined as slip velocity 119880

119904 as shown in Figure 3 There

are two main slip boundary conditions the Navier boundarycondition and the limiting shear stress boundary condition[2 26 27 29 31 32] The Navier boundary condition alsocalled as slip length boundary condition states that the slipvelocity 119880

119904is proportional to the surface shear rate (120597119906120597119910)

and slip length 119887 that is 119880119904= 119887 sdot (120597119906120597119910) The slip length 119887 is

4 Mathematical Problems in Engineering

I

I

Slip surface

Slip region

Stationary wall

Moving wall

Slip surface

I-I view r120596

hc

Figure 2 Schematic diagram of a bearing pad of the slider bearingwith slip surfaces

Stationary wall

Moving wall u

NoslipSlip

b

U0

Us

Figure 3 Comparison of the noslip and slip boundary phe-nomenon

defined as the fictive distance below the solid surface wherethe velocity extrapolates linearly to zero as shown in Figure 3The limiting shear stress boundary condition assumes thatthere is a limitingcritical shear stress 120591

119888at the fluid-solid

interface and the wall slip occurs only when the wall shearstress 120591

0reaches the critical value If slippage occurs the

surface shear stress 120591119904is equal to the critical value that is

120591119904

= 120591119888(if 1205910

ge 120591119888) and 120591

119904= 1205910(if 1205910

lt 120591119888) where

1205910= 120583 sdot (120597119906120597119910) However the exact value of the slip length

is difficult to determine for a certain structure and the sameproblem still exists for critical shear stress

Not only the Navier boundary condition but also thelimiting shear stress boundary condition indicates that theslip velocity 119880

119904is always related to fluid velocity 119880

119894at the

nearest region close to this fluid-solid interface For theconvenience to determine the slip intensity a slip-intensityfactor 119891 is used thus a slip boundary condition with a slip-intensity factor can be derived from the limiting shear stressboundary condition then and is expressed as

119880119904= 119891 sdot (119880

119894minus 119899 sdot ⟨119880

119894sdot 119899⟩) (2)

where 119899 is the surface normal vector and this part 119899 sdot ⟨119880119894sdot 119899⟩

represents the velocity component of119880119894in the surface normal

direction Equation (2) indicates that boundary slip occursin the tangential direction and the normal component ofvelocity is zero Assume that the three components of slipvelocity 119880

119904are (119906

119904 V119904 119908119904) and the three components of fluid

velocity119880119894are (119906

119894 V119894 119908119894) under this coordinate system 119909

119899-119910119899-

119911119899 where the direction of surface normal vector 119899 is minus119911

119899 as

shown in Figure 4 When (2) is applied we have

119906119904= 119891 sdot 119906

119894

V119904= 119891 sdot V

119894

119908119904= 0

(3)

For possessing physical meaning the slip-intensity factoris restricted from 0 to 1 that is 119891 isin [0 1] This condition119891 = 1 represents a shear-free condition that is a perfectslip condition and at this situation we have 120597119906120597119910 = 0thus this condition is also equivalent to the limiting shearstress slip boundary conditionwhen the critical shear stress iszero And this situation 119891 = 0 is equivalent to the stationarywall without slip that is noslip boundary condition withzero speedThis slip boundary condition with a slip-intensityfactor is applied onto these slip surfaces in the followingstudies of this work The numerical analyses are performedusing OpenFOAM

A careful check on mesh independence is conductedfirstly to ensure the accuracy of numerical solution Themesh independence analyses are based on this slip bearingmodel with 119891 = 1 when the rotational speed is 500 rpmand the fluid film thickness is 1708120583m (ie the imposedexternal axial load in experiment was 100N) the analysisresults indicate that the mesh density of 546 kilo is goodenough for the accuracy of numerical solution thus thesame mesh density is used for all the numerical models inthis work The tribological performances of the slip sliderbearing predicted theoretically in this study are comparedwith the corresponding experimental and theoretical valuesof the textured slider bearing demonstrated in [34]

3 Results

The effects of slip-intensity factor 119891 on the load-carryingcapacity and friction torque are illustrated in Figures 5 and 6

Mathematical Problems in Engineering 5

n

Fluid-solid interface

wi

Ui

i

ui

zn

ws

Us

s

us

yn

xn

Figure 4 Schematic diagram of velocity

Table 2 The values of slip-intensity factor 119891 when 119882 asymp 100N and200N

119891119882

100N plusmn 1 200N plusmn 1120596

500 rpm 0775 079600 rpm 0755 0768700 rpm 0732 0758800 rpm 0705 0735

For the textured bearing models corresponding to the slipbearing models whose results are shown in Figure 5 theimposed load-carrying capacity 119882 in Marianrsquos experimentis 100N and it is 200N for that corresponding to the slipbearing models whose results are shown in Figure 6The the-oretical results demonstrate that the effects of slip-intensityfactor on load-carrying capacity and friction torque areboth monotone The load-carrying capacity monotonicallyincreases with the slip-intensity factor and the friction torquemonotonically decreases with the slip-intensity factor Onthe other hand there is one and only value of slip-intensityfactor 119891which makes the predicted theoretical load-carryingcapacity based on the slip boundary condition presenting in(2) equal to the imposed value in the experiment conductedby Marian et al and the corresponding values of this slip-intensity factor119891 are presented in Table 2 when the predictedtheoretical load-carrying capacity is equal to the experimentvalue that is 100N and 200N respectively

The theoretical friction torques based on these valuesof slip-intensity factor as shown in Table 2 are presentedin Figure 7 and they are also compared to the measuringvalues and theoretical prediction values reported by Marianet al [34] The theoretical friction torques in this work andMarianrsquos study are both smaller than the measured valuesreported in [34] But the errors between the measured valuesand the theoretical values predicted in this study are smallerthan that between the measured and theoretical values of[34] The physical structure of dimples was really built up

in the theoretical models of [34] As pointed out by Marianet al the differences between the measuring and theoreticalfriction torques may result from the measurement errors dueto vibrations the measuring values are oscillating over timeand their mean values are used to make comparison withtheoretical values

The 2D dimensionless pressure distribution is illustratedin Figure 8 over a slip bearing pad The dimensionlesspressure 119875 is defined as 119875 = 119901ℎ

119888

2(120583119903120596119871) where 119871 denotes

the circumferential length of bearing pad The fluid filmthickness of this slip bearing is 10 120583m the numbers of slipsurface in circumferential and radial directions are both 22and other bearing parameters keep the same as the aboveTheslip-intensity factor 119891 is 08

The shape of this pressure distribution is similar to thepressure distribution of a step bearing as shown in Figures8 and 9 There is a big pressure peak at the rear of the slipregion Figure 9(a) illustrates the 3D dimensionless pressuredistribution over a bearing padTheonly difference comparedto a step bearing is that there is an additional pressure peak(net pressure) at the rear of each slip surfaceThis can be seenmuch more clearly from Figure 9(b) which illustrates thepressure distribution over a section when 119909119903

119894asymp minus129 Each

slip surface strongly affects its neighboring slip surface andthe pressure contributed by each slip surface has a collectiveeffect This pressure distribution based on slip bearing agreeswell with that based on textured bearing as shown in Figure 6of [34]The dimple (texture) numbers in circumferential andradial directions are equal to the number of slip surface inthese two directions respectively There is also a big pressurepeak at the rear of the dimple region and an additional netpressure at the rear of each dimple as well as a collective effecton net pressure

4 Discussions

The influences of slip surface on bearing tribological perfor-mances are very similar to those of concave texture (dimple)as demonstrated above Each concave texture and slip surfacewill both induce a pressure peak (net pressure) locating at itsrearThe global or cumulative impact of concave textures andslip surfaces will both result in a big pressure peak which justlocates at the rear of texture region and slip region For thesame bearing models except these zones of texture and slipthe value of slip-intensity factor is one and only value tomakethe bearing tribological performances predicted by the slipboundary condition presenting in (2) equal to the tribologicalperformances of the corresponding textured bearing

The relationship between the slip velocity and the textureparameters will be deduced in the following based on two 2Dparallel slider bearingmodels with a texture and a slip surfacein the stationary wall respectively These two slider bearingmodels are illustrated in Figure 10

For the 2D parallel slider bearing model the governingequations (1) can be simplified and reduced to

1205972119906

1205971199102 =1120583

sdot120597119901

120597119909 (4)

6 Mathematical Problems in Engineering

350

300

250

200

150

100

50

0

Load

-car

ryin

g ca

paci

ty (N

)

02 04 06 08 10

f

500 rpm600 rpm

700 rpm800 rpm

(a) Load-carrying capacity

02 04 06 08 10

f

05

04

03

02

01

00

Fric

tion

torq

ue (N

middotm)

500 rpm600 rpm

700 rpm800 rpm

(b) Friction torque

Figure 5 Tribological performances versus slip-intensity factor based on these bearing models with 119882 = 100

600

500

400

300

200

100

0

LLoa

d-ca

rryi

ng ca

paci

ty (N

)

02 04 06 08 10

f

500 rpm600 rpm

700 rpm800 rpm

(a) Load-carrying capacity

02 04 06 08 10

f

06

05

04

03

02

01

00

Fric

tion

torq

ue (N

middotm)

500 rpm600 rpm

700 rpm800 rpm

(b) Friction torque

Figure 6 Tribological performances versus slip-intensity factor based on these bearing models with 119882 = 200

The speed of moving plate is 1198800and the slip velocity at

the slip surface of stationary plate is 119880119904 The film thickness

at dimple zone is ℎ119889and it is ℎ

0at the other noslip and slip

zonesWith the boundary conditions 119906(ℎ) = 1198800and 119906(0) = 0

at noslip-surface zone or 119906(0) = 119880119904at slip-surface zone the

fluid velocity at noslip-surface zone can be given as

119906noslip =12120583

sdot120597119901

1205971199091199102+(

1198800ℎ

minusℎ

2120583sdot120597119901

120597119909)119910 (5)

And the fluid velocity at slip-surface zone is given as

119906slip =12120583

sdot120597119901

1205971199091199102+(

1198800ℎ

minusℎ

2120583sdot120597119901

120597119909)119910+

ℎ minus 119910

ℎ119880119904 (6)

where ℎ represents the film thickness The last term of theright hand side of (6) represents the contribution of boundaryslip to fluid velocityThus from the velocity equation the flowrate 119876 at a cross section can be obtained by the following

119876 = int

ℎ0

0119906 sdot 119889119910 (7)

So the flow rates at noslip- and slip-surface zones aregiven respectively by

119876noslip =1198800ℎ

2minus

ℎ3

12120583sdot120597119901

120597119909 (8)

119876slip =1198800ℎ

2minus

ℎ3

12120583sdot120597119901

120597119909+

119880119904ℎ

2 (9)

Mathematical Problems in Engineering 7

05

04

03

02

01

00

Fric

tion

torq

ue (N

middotm)

500 550 600 650 700 750 800

Rotational speed (rpm)

Present workMarianrsquos theoryMarianrsquos experiment

(a) When119882= 100N

06

05

04

03

02

01

00

Fric

tion

torq

ue (N

middotm)

500 550 600 650 700 750 800

Rotational speed (rpm)

Present workMarianrsquos theoryMarianrsquos experiment

(b) When119882= 200N

Figure 7 Comparison on friction torque

P

0033

003

002

001

0

Figure 8 2D pressure distribution over a bearing pad

The last term of the right hand side of (9) represents thecontribution of boundary slip to flow rate

For these parallel slider bearings the film thickness isconstant so this part 120597119901120597119909 is also constant that is thepressure variation is linear For the slider bearing with adimple the flow rate for each region can thus be given by

119876AB =1198800ℎ02

minusℎ0

3

12120583sdot (

1199012 minus 1199011119886

)

119876BC =1198800ℎ1198892

minusℎ119889

3

12120583sdot (

1199013 minus 1199012119887

)

119876CD =1198800ℎ02

minusℎ0

3

12120583sdot (

1199014 minus 1199013119888

)

(10)

The ambient pressure is applied onto the inlet and outletthat is 119901

1= 1199014

= 119901atm Because the flow is the continuitythe flow rate at every cross section is the equivalence that is119876AB = 119876BC = 119876CD From 119876AB = 119876CD and 119876BC = 119876CDrespectively we have

1199012 minus 119901atm119886

=119901atm minus 1199013

119888 (11)

1198800ℎ119889

2minus

ℎ119889

3

12120583sdot (

1199013 minus 1199012119887

)

=1198800ℎ02

minusℎ0

3

12120583sdot (

119901atm minus 1199013119888

)

(12)

Combining (11) and (12) to eliminate 1199012 we have

1199013 = 119901atm +61205831198800119887119888 (ℎ119889 minus ℎ0)

(119886 + 119888) ℎ1198893+ 119887ℎ0

3 (13)

Thus combining (11) and (13) we then have

1199012 = 119901atm minus61205831198800119886119887 (ℎ

119889minus ℎ0)

(119886 + 119888) ℎ1198893+ 119887ℎ0

3 (14)

For the slider bearing with a slip region the flow rate foreach region can thus be given by

1198761015840

AB =1198800ℎ02

minusℎ0

3

12120583sdot (

1199011015840

2 minus 1199011015840

1119886

)

1198761015840

BC =1198800ℎ02

minusℎ0

3

12120583sdot (

1199011015840

3 minus 1199011015840

2119887

)+119880119904ℎ02

1198761015840

CD =1198800ℎ02

minusℎ0

3

12120583sdot (

1199011015840

4 minus 1199011015840

3119888

)

(15)

8 Mathematical Problems in Engineering

003002001

0

08

06

04

020 minus16

minus14minus12

minus1minus08

P

yri xri

(a) Over a slip bearing pad

004

003

002

001

000

0 5 10 15 20 25

Circumferential angle (∘)30

Dim

ensio

nles

s pre

ssur

e

(b) At this section when 119909119903119894 asymp minus129

Figure 9 Pressure distribution

Texture bearing

Moving wall Moving wall

a

Stationary wallb c a b c

A B C D A B C DStationary wall

Slip bearing

Texture

Slip surface

x

y

p1 p2 p3 p4

U0 U0

h0 h0hd p9984004p998400

3p9984002p998400

1

Figure 10 2D slider bearing model

We also have 1199011015840

1 = 1199011015840

4 = 119901atm and 1198761015840

AB = 1198761015840

BC = 1198761015840

CDthus

1199011015840

2 = 119901atm minus6120583119880119904119886119887

(119886 + 119887 + 119888) ℎ02 (16)

1199011015840

3 = 119901atm +6120583119880119904119887119888

(119886 + 119887 + 119888) ℎ02 (17)

Under the condition that there are the same pressuredistributions for these two cases with a dimple and with a slipregion respectively that is 1199012 = 119901

1015840

2 and 1199013 = 1199011015840

3 and bycombining (14) and (16) or (13) and (17) we thus obtain

119880119904=

(119886 + 119887 + 119888) (ℎ119889 minus ℎ0) ℎ02

(119886 + 119888) ℎ1198893+ 119887ℎ0

3 1198800 (18)

This expression (18) indicates that the slip velocity has arelation to the geometry parameters of surface texture andthemovement condition For a givenmodel with determinatesurface texture and movement condition the correspondingslip velocity induced by surface texture is also determinateand its value is one and only value

5 Conclusions

Comparative analyses about the influences of surface textureand boundary slip on the tribological performances of sliderbearings are conducted The analysis results indicate thatthe tribological characteristics of slip and textured surfaceshave strong similarity For a given texture configuration

there is one and only slip velocity to make their tribologicalperformances equivalent A corresponding relation betweenslip velocity and texture parameters is also deduced based on2D slider bearing with one dimple texture The size of slipvelocity is directly related to the texture geometry parametersincluding its distribution position These results in this workare useful to further understand the slip phenomenon as wellas the relationship between boundary slip and surface texture

Nomenclature

119906V119908 Velocity components119901 Pressure119875 Dimensionless pressure120588 Density120583 Viscosity119903119894 Inner radius

119903119900 Outer radius

119903 Radius119897119889 Dimple length

119897119888 Dimple cell length

119880119904 Slip velocity

119887 Slip length1198800 Moving velocity

119880119894 Inner-fluid velocity

119891 Slip-intensity factor119899 Surface normal vector120596 Rotational speed119901atm Ambient pressure119909119910119911 Cartesian coordinates

Mathematical Problems in Engineering 9

120579119905 Circumferential angle of dimple region

120579119901 Circumferential angle of bearing pad

119861119905 Radial width of dimple region

119861119901 Radial width of bearing pad

ℎℎ119888ℎ119889ℎ0 Fluid film thickness

120591119888 Critical shear stress

1205910 Wall shear stress

120591119904 Shear stress of slip surface

119906119904V119904119908119904 Velocity components of 119880

119904

119906119894V119894119908119894 Velocity components of 119880

119894

119909119899119910119899119911119899 Cartesian coordinates where 119899 = minus119911

119899

119871 Circumferential length of bearing pad119882 Load-carrying capacity119886119887119888 Length of different bearing regionsABCD Position marks119876 Flow rate

SuperscriptSubscript1015840 Slip bearingslip Slip-surface zonenoslip Noslip-surface zone1234 The position ABCD

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is financially supported by the National NaturalScience Foundation of China (no 51135004) and the TeacherResearch Support Program of Xirsquoan Jiaotong University (noDW013217K0000001)

References

[1] M Arghir N Roucou M Helene and I Frene ldquoTheoreticalanalysis of the incompressible laminar flow in a macro-rough-ness cellrdquo Journal of Tribology vol 125 no 2 pp 309ndash318 2003

[2] R Salant and A Fortier ldquoNumerical analysis of a sliderbearing with a heterogeneous slipno-slip surfacerdquo TribologyTransactions vol 47 no 3 pp 328ndash334 2004

[3] M Mahmoud and S Waheed ldquoEffects of slip and heatgenerationabsorption on MHD mixed convection flow of amicropolar fluid over a heated stretching surfacerdquoMathematicalProblems in Engineering vol 2010 Article ID 579162 20 pages2010

[4] Q Lin Z Wei and N Wang ldquoOptimum design of recessparameters for a high-speed hybrid journal bearing using fluid-structure interaction and improved orthogonal experimentmethodrdquo Journal of the Balkan Tribological Association vol 21no 2 pp 300ndash313 2015

[5] M Fesanghary and M M Khonsari ldquoTopological and shapeoptimization of thrust bearings for enhanced load-carryingcapacityrdquo Tribology International vol 53 pp 12ndash21 2012

[6] D Lu W Zhao B Lu and J Zhang ldquoCage speed of hydrody-namic rolling hybrid bearingsrdquo Tribology Letters vol 51 no 3pp 303ndash309 2013

[7] M E Salama ldquoThe effect of macro-roughness on the perfor-mance of parallel thrust bearingsrdquo Proceedings of the Institutionof Mechanical Engineers vol 163 no 1 pp 149ndash161 1950

[8] D B Hamilton J A Walowit and C M Allen ldquoA theory oflubrication bymicroirregularitiesrdquo Journal of Basic Engineeringvol 88 no 1 pp 177ndash185 1966

[9] J N Anno J A Walowit and C M Allen ldquoMicroasperitylubricationrdquo Journal of Lubrication Technology vol 90 no 2 pp351ndash355 1968

[10] K Toslashnder ldquoInlet roughness tribodevices dynamic coefficientsand leakagerdquo Tribology International vol 34 no 12 pp 847ndash852 2001

[11] V Brizmer Y Kligerman and I Etsion ldquoA laser surface texturedparallel thrust bearingrdquo Tribology Transactions vol 46 no 3pp 397ndash403 2003

[12] A V Olver M T Fowell H A Spikes and I G Pegg ldquolsquoInletsuctionrsquo a load support mechanism in non-convergent pock-eted hydrodynamic bearingsrdquo Proceedings of the Institution ofMechanical Engineers Part J Journal of Engineering Tribologyvol 220 no 2 pp 105ndash108 2006

[13] A Gherca A Fatu M Hajjam and P Maspeyrot ldquoInfluenceof surface geometry on the hydrodynamic performances ofparallel bearings in transient flow conditionsrdquo Tribology Trans-actions vol 56 no 6 pp 953ndash967 2013

[14] K Yagi and J Sugimura ldquoBalancing wedge action a contribu-tion of textured surface to hydrodynamic pressure generationrdquoTribology Letters vol 50 no 3 pp 349ndash364 2013

[15] M Fowell A V Olver A D Gosman H A Spikes andI Pegg ldquoEntrainment and inlet suction two mechanisms ofhydrodynamic lubrication in textured bearingsrdquo Journal ofTribology vol 129 no 2 pp 336ndash347 2007

[16] M B Dobrica M Fillon M D Pascovici and T CiconeldquoOptimizing surface texture for hydrodynamic lubricated con-tacts using amass-conserving numerical approachrdquoProceedingsof the Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 224 no 8 pp 737ndash750 2010

[17] J Ji Y Fu and Q Bi ldquoThe influence of partially textured sliderwith orientation ellipse dimples on the behavior of hydrody-namic lubricationrdquo Industrial Lubrication andTribology vol 66no 2 pp 161ndash167 2014

[18] M Qiu B R Minson and B Raeymaekers ldquoThe effect oftexture shape on the friction coefficient and stiffness of gas-lubricated parallel slider bearingsrdquo Tribology International vol67 pp 278ndash288 2013

[19] L Hao Y Meng and C Chen ldquoExperimental investigationon effects of surface texturing on lubrication of initial linecontactsrdquo Lubrication Science vol 26 no 5 pp 363ndash373 2014

[20] J P Rothstein ldquoSlip on superhydrophobic surfacesrdquo AnnualReview of Fluid Mechanics vol 42 pp 89ndash109 2010

[21] B Bhushan Y Wang and A Maali ldquoBoundary slip study onhydrophilic hydrophobic and superhydrophobic surfaces withdynamic atomic forcemicroscopyrdquo Langmuir vol 25 no 14 pp8117ndash8121 2009

[22] F Feuillebois M Z Bazant and O I Vinogradova ldquoEffectiveslip over superhydrophobic surfaces in thin channelsrdquo PhysicalReview Letters vol 102 no 2 Article ID 026001 2009

[23] B Ono and Y Yamamoto ldquoPossibility of slip in hydrodynamicoil films under sliding contact conditionsrdquo Lubrication Sciencevol 14 no 3 pp 303ndash320 2002

[24] K Bhattacharyya G C Layek and R S R Gorla ldquoSlip effecton boundary layer flow on a moving flat plate in a parallel free

10 Mathematical Problems in Engineering

streamrdquo International Journal of Fluid Mechanics Research vol39 no 5 pp 438ndash447 2012

[25] Y Zhang ldquoReview of hydrodynamic lubrication with interfacialslippagerdquo Journal of the Balkan Tribological Association vol 20no 4 pp 522ndash538 2014

[26] H A Spikes ldquoThe half-wetted bearing Part 1 extendedReynolds equationrdquo Proceedings of the Institution of MechanicalEngineers Part J Journal of Engineering Tribology vol 217 no1 pp 1ndash14 2003

[27] A E Fortier and R F Salant ldquoNumerical analysis of a journalbearing with a heterogeneous slipno-slip surfacerdquo Journal ofTribology vol 127 no 4 pp 820ndash825 2005

[28] C-K Chen H-Y Lai and W-F Chen ldquoUnsteady unidirec-tional flow of second-grade fluid through a microtube withwall slip and different given volume flow raterdquo MathematicalProblems in Engineering vol 2010 Article ID 416837 17 pages2010

[29] C Wu ldquoPerformance of hydrodynamic lubrication journalbearing with a slippage surfacerdquo Industrial Lubrication andTribology vol 60 no 6 pp 293ndash298 2008

[30] Q Lin Z Wei N Wang and W Chen ldquoEffect of large-areatextureslip surface on journal bearing considering cavitationrdquoIndustrial Lubrication and Tribology vol 67 no 3 pp 216ndash2262015

[31] Q Lin Z Wei N Wang and W Chen ldquoEffects of large-areatexturedslip surface on slider bearingrdquo Journal of the BalkanTribological Association vol 21 no 1 pp 12ndash23 2015

[32] F Aurelian M Patrick and H Mohamed ldquoWall slip effectsin (elasto) hydrodynamic journal bearingsrdquo Tribology Interna-tional vol 44 no 7-8 pp 868ndash877 2011

[33] R Sharma A Ishak and I Pop ldquoPartial slip flow and heattransfer over a stretching sheet in a nanofluidrdquo MathematicalProblems in Engineering vol 2013 Article ID 724547 7 pages2013

[34] V G Marian M Kilian andW Scholz ldquoTheoretical and exper-imental analysis of a partially textured thrust bearing withsquare dimplesrdquo Proceedings of the Institution of MechanicalEngineers Part J Journal of Engineering Tribology vol 221 no7 pp 771ndash778 2007

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