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1. INTRODUCTION

The parallel sliding smooth surfaces generate unstable hydrodynamic film by couette velocity

variations with zero pressure distribution which can collapse when the external force is applied. One

of the better methods to produce stable hydrodynamic film between the parallel sliding surfaces is by

providing the determined surface textures. Deterministic micro textures are the surface features that

have specific pattern in terms of shape, size, and orientation. Depending on the size, shape, orientation

and distribution of the textures, the hydrodynamic lubrication characteristics of the surface can vary

significantly. Surface textures are of two types, namely, protrusions (bumps, posts and recesses

(holes which are shown in the !ig. ".

!igure". #egative and $ositive %sperities

2. LITERATURE REVIEW

&n the modern technology, there are several techni'ues to mae the surface textures including

chemical etching )"*, laser ablation )+*, &-% process )*, $hotolithography )/*. 0tsion )1* explained

the different techni'ues to mae the micro textures on the surfaces. The idea of an increased pressure

generated by micro2textures under conditions of hydrodynamic lubrication was originated in the late

"345s. 6ligerman et al., )4* numerically analyzed the effect of surface textures on circumferential gasseal by !07 modelling. &t has been observed that the textures have significant effect on the

hydrodynamic performance of gas seal. % lot of research wor has been done for the reduction of

friction on the reciprocating components and on mechanical seal )823*.

9rizmer et al., )"5* analytically finds the optimum area density of the dimples for maximum

load carrying capacity on the parallel thrust bearings by analyzing the full width textured and partial

width textured surfaces. 0tsion et al., )""* experimentally analyze the model explained by 9rizmer et

al., )"5* and the results showed good correlation with the theoretical results. Siripuram et al., )"+*

utilized numerical modelling techni'ues to explore the effect of basic asperity properties comprised of

shape, size, concavity and orientation on lubrication characteristics for a simple thrust slider

application. The authors examined different regular shapes, all distributed in s'uare array. They found

that friction coefficient is largely independent of asperity shape and orientation but very sensitive to

asperity area fraction (size and the leaage is dependent on asperity shape, concavity, orientation andsize. 7athematically, 9uscaglia et al., )"* analyzed the effect of surface texture on the static

characteristics of thrust bearing. The result shows that the load carrying capacity increases and friction

coefficient decreases by including textures. %rghir et al., )"/* utilized the numerical techni'ue to

explore the lift effect due to the pressure generation in the different macro2roughness textured cell.

The results indicate that by increasing the convective inertia, the macro2roughness patterns produces

higher lift force on the flat surface. :ahmani et al., )"1* solved the "D :eynolds e'uation with

sommerfeld boundary condition for partially textured thrust bearing to find the optimum geometry of

s'uare shaped micro2dimples. !rom the study it has been stated, for partially textured surfaces

increasing the number of dimples would not help in improving the load capacity or friction coefficient

in a pure hydrodynamic mode.

;upillard et al., )"4* numerically analyzed the inertia effect on textured parallel sliders in one

dimensional with #avier2Stoes e'uation and Stoes e'uation. The two types of models have beensolved by using finite volume method with software pacage ;!< "".5. The result shows depending


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!ig. + % model of textured surface

!ig. ;ross2section of the textured surface !ig. /

&ndividual cell of single texture

The film thicness hbetween the parallel surfaces of positive textures is

above the protrusion

elsewhere

gC hhC

=

The film thicness hbetween the parallel surfaces of negative textures is

above the recess

elsewhere

gC hh

C

+=

The #on2dimensional 'uantities used to non2dimensionalize the film thicness is

,h

hC

=

gh

HC

=

The non2dimensional form of the film thicness for positive textures is

" above theprotrusion

" elsewhere

Hh

=

The non2dimensional form of the film thicness for positive textures is

" above the recess

" elsewhere

Hh

+=

!.2 Theor$ Of ()%i* I+erti&

&nertia effects will be significant in super2laminar flow, and=or when there is a rapid change in

the cross2section )+5*. The #avier2Stoes e'uation with fluid inertia effect by neglecting the bodyforce is given as


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u u u u p uu v w

t x y z x y y

+ + + = +

5 p

y

=

w w w w p wu v w

t x y z z y y

+ + + = +

#avier2Stoes e'uation contains four unnowns, but there are only three e'uations. &n order to solve

this mathematically, another one e'uation is needed which is involving the velocity components. This

e'uation is provided by the principle of conservation of mass. The resulting e'uation is nown as the

e'uation of continuity.

( ) ( ) ( )5

u v w

t x y z

+ + + =

The #on2dimensional 'uantities used to non2dimensionalize the #avier2stoes and continuity

e'uations are+

X

pCp

UL= ,

X

xx

L= ,

yy

C= ,

Z

zz

L= ,

uu

U= , X

vLv

UC= ,

ww

U= , X

Z

Lk

L=

C

= , :e

UC

= , :e :e

X

C

L

=

The non2dimensional form of the #avier2Stoes e'uation including the fluid inertia effect is

+

+:e

u u u p uu v kw

x y z x y

+ + = +

+

+:e w w w p wu v kw k x y z z y + + = + ("

and the continuity e'uation is

( ) ( ) ( )5

u v wk

x y z

+ + =

!irst2order perturbation series in :e was used to determine the pressure generated in the

conformal contacting surfaces.

,ert%rb&tio+ -%&+titieA

5 ":ep p p= +

5 ":eu u u= +

5 ":ev v v= +

5 ":ew w w= +!irst2order perturbation method gives better results only at smaller values of the :educed2

:eynolds number. %ssume the flow is in the laminar region. Substitute the perturbation parameters in

the 0'. (" and separate the zeroth and first order terms of:e .

eroth or*er ter/ ( )5

:e A

+

5 5

+5

p u

x y

= +


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+

5 5

+5

p wk

z y

= +

( ) ( ) ( )5 5 5 5

u v wk

x y z

+ + =

(irt or*er ter/ ( )"

:e A

+

5 5 5 " "5 5 5 +

u u u p uu v kw

x y z x y

+ + = +

+

5 5 5 " "5 5 5 +

w w w p wu v kw k

x y z z y

+ + = +

( ) ( ) ( )" " "

5u v w

kx y z

+ + =

!rom the zeroth order term, :eynolds e'uation can be derived as follows

( ) + 5 5 4p p

h k h hx x z z x

+ =

!or the iso2viscous fluid, should be constant.

+ 5 5 4p p h

h k hx x z z x

+ = (+

!rom the first order term, the modified :eynolds e'uation can be derived for the solution of

perturb pressure

+ " "

5 5 5 5 5

5 5 5 5 5

"++

"++

y y yh h

x x

y y yh h

z z

p p hh k h K dydydy K dydy

x x z z x

hk K dydydy K dydy

z

+ =

+

(

Bhere,

5 5 55 5 5x

u u uK u v kw

x y z

= + +

5 5 5

5 5 5z

w w wK u v kw

x y z

= + +

The velocity components can be evaluated from the zeroth order terms

( )55"

+

p yu y y h

x h

= +

, ( )55

"

+

pw k y y h

z

=

( ) ( )5 5 55

" y

v u k w dyx z

= +

The boundary conditions and the periodicity condition in non2dimensional form is given as

5 5( , " 5, ( , 5 5p x z p x z= = = = (1

5 5( 5, ( ", p x z p x z= = = (4&n the present analysis, :eynolds cavitation condition is used. &t implies that, at the cavitation

boundary, the pressure gradient with respect to the direction normal to the boundary is zero. The 0's.

(+, (, and (/ is solved by finite difference method. % grid size of ( )< C ?"15 "15 "55 # # # and the convergence of 4"5 was chosen based on the accuracy, the graphs are shown in the next

chapter. The finite difference method leads to a set of algebraic e'uations which should be solved


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along with boundary condition 0'. (1 and periodicity condition 0'. (4. These e'uations are solved

with -auss2Siedel iterative method which is convenient for the evaluation of pressure distribution

with previously unnown cavitation region.

%fter solving the e'uations, the pressure can be found as

5 ":ep p p= +

Once the pressure distribution is evaluated in the film region, the non2dimensional loadcarrying capacity, non2dimensional end flow and friction parameter are calculated from the

expressions" "

5 5

W pdxdz = "

5 "

5 5 5 5 5 5

:e"+ "+ +

y y yh h

z z

p pkh kh hQ K dydydy K dydy dx

z z

= + +

( XF

L CW

=

Bhere,

( )" "

5 "

5 5 5 5 5

" " " "!riction !orce :e

+ +

yh h

x x

p pF h h K d y K d ydy dxdz

x h x h

= + + + +

0. E((ECT O( SUARE S"A,E ,ROTRUSION ON ,ARALLEL SLIDIN CONTACTS

WIT" INCLUDIN T"E (LUID INERTIA

The numerical analysis was performed to determine the effect of various non2dimensional

parameters lie aspect ratio, texture height and :eynolds number on the steady2state hydrodynamic

performance characteristics of the parallel surface with single s'uare2shaped protrusion by including

the inertia effect. The limits of these parameters areA%spect ratio (%A 5." % 5.3 Texture height ratio (HA 5." 5.1h :educed :eynolds number ( :e A 5.+ :e "./ !irst, the mesh size and convergence parameter for pressure calculation are considered from the

following !ig. 1. The load carrying capacity variation is decreasing when the mesh size increases for

the convergence value of 4"5 .

15 "55 "15 +555."+

5."/

5."4

5.">

5.+

5.++

5.+/

5.+4

5.+>

7esh Size

#on2d

imensional,oadcarryingcapacity

;onvD"e2/

;onvD"e21

;onvD"e24

5 +5 /5 45 >5 "55 "+5 "/5 "45 ">5 +555.+8

5.+>

5.+3

5.

5."

5.+

#y mesh size

#on2d

imensionalloadcarryingcapacity

!ig. 1 7esh size and convergence of XN (ongitudinal, ZN (Transverse and YN(across the film.

andX ZN N


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!rom the figures, it has been concluded, for further calculation mesh size of "15 < "15


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5." 5."1 5.+ 5.+1 5. 5.1 5./ . . . .5

5.1

"

".1

+

+.1

Texture height ratio( H

5." 5."1 5.+ 5.+1 5. 5.1 5./5

5.51

5."

5."1

5.+

5.+1

5.

5.1


5." 5."1 5.+ 5.+1 5. 5.1 5./5

"5

+5

5

/5

15

45


!ig. 8 Gariation of #on2dimensional oad carrying capacity, 0nd flow and !riction parameter with

Texture height ratio.

!ig. 8, shows the hydrodynamic performance characteristics gives better result as the non2

dimensional texture height increases. !ig. > shows the variation of W and Q with the aspect ratio for

5./H= . The W and Q increases at lower aspect ratio but decreases at higher aspect ratios. %s thetexture size increases, the area of constant film thicness increases due to which the pressure

distribution gets uniform over the region and thus the W is reduced. %s the texture size increases

there is increase in obstruction to the flow due to which the Q is reduced. The ( XL C is @ust the

inversion of the W . The W is maximum for the aspect ratio of 5./, the Q is maximum for the aspect

ratio in between 5.+25. and the ( XL C is minimum for the aspect ratio in between 5.25./.


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5." 5.+ 5. 5./ 5.1 5.4 5.8 5.> 5.35.5/

5.54

5.5>

5."

5."+

5."/

5."4

5.">

5.+

5.++

i

i

ll

i

i

.

%spect ratio ( A

5." 5.+ 5. 5./ 5.1 5.4 5.85

5.5"

5.5+

5.5

5.5/

5.51

5.54

5.58

5.5>

5.53

5."

i

i

l

.

%spect ratio ( A

5." 5.+ 5. 5./ 5.1 5.41

"5

"1

+5

+1

5

1

/5

%spect ratio ( A

!ig.> Gariation of load carrying capacity, end flow and friction parameter with aspect ratio by varying

:eynolds number

The variation of hydrodynamic performance characteristics with the aspect ratio is shown in the !ig. 3

for :e 5./= .

5." 5.+ 5. 5./ 5.1 5.4 5.8 5.> 5.35

5.51

5."

5."1

5.+

5.+1

5.

.

ll

.

%spect ratio ( A

5." 5.+ 5. 5./ 5.1 5.4 5.85

5.5"

5.5+

5.5

5.5/

5.51

5.54

5.58

5.5>

i

i

l

l .

%spect ratio ( A


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5." 5.+ 5. 5./ 5.1 5.4 5.8 5.> 5.35

+5

/5

45

>5

"55

"+5

"/5

"45

l

l .

%spect ratio ( A

!ig.3 Gariation of load carrying capacity, end flow and friction parameter with aspect ratio by varying

texture height ratio

!rom the above results, it can be concluded that the aspect ratio should be low and the non2

dimensional texture height should be high to get the better performance characteristics results.

3. E((ECT O( T"E DI((ERENT S"A,E O( ,OSITIVE TE4TURES ON ,ARALLEL

SLIDIN SUR(ACES

The numerical analysis is performed for the effect of different shapes of positive textures

namely, S'uare, ;ircular, Hexagonal, Dome, Triangular (apex perpendicular to flow and 0llipsoidal

(ma@or axis is parallel to flow on the tribological performance characteristics of the parallel sliding

contacts including the fluid inertia. To validate the numerical results, comparison is made with the :ef

)"+* by maing :e 5= because the analysis done in the :ef )"+* is without including the fluidinertia. !igure "5 shows the comparison of different shapes of textures with the :ef )"+*, a good

correlation is obtained. The variation is due to the mesh size.

0.0 0.2 0.4 0.6 0.8 1.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

---------

Triangular

Hexagonal

CircularSquare

PresentReerence !21"

#il$t

%ic&ness'$(

)s*ect Ratio0.0 0.2 0.4 0.6 0.8 1.0

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Reerence !21"

Present++++++++++

Triangular

Hexagonal

Circular

Square

Coeicientoriction'(

)s*ect ratio

!ig. "5 Galidation results of different shape of textures

The maximum aspect ratios of the different shapes of textures in an imaginary cell are shown in the

Table. ".

Table. ". 7aximum aspect ratios of different shapes

Sl. #o Init ;ell Description 7aximum %spect ratio

" S'uare 5.3


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+ ;ircular 5.8>

Hexagonal 5.41

/ Dome 5.8>

1 0llipsoidal 5.>

4 Triangular 5.+

The effect of different shape of textures on parallel sliding contacts with the variation of aspect ratio

for a certain value of :e 5./= and 5./H= is shown in the !ig. "". %s the aspect ratio increases,

the non2dimensional load carrying capacity ( W first increases and then decreases for the cases of allshapes of textures except triangular and dome. This is because as the aspect ratio increases, the more

area has the constant film thicness which uniforms the pressure distribution over the surface and thus

the W is reduced. %s the aspect ratio increases, the fluid gets more obstruction to the flow and thus

the Q is reduced.

5

5.51

5."

5."1

5.+

5." 5.+ 5. 5./ 5.1 5.4 5.81

"5

"1

+5

+1

5

%spect ratio ( A

!ig. "" 0ffect of different shapes on non2dimensional load carrying capacity, non2dimensional end

flow and friction parameter with the aspect ratio.


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The triangular shape of texture shows better non2dimensional load carrying capacity( W than the

other shapes. The hexagonal shape shows higher preferential end flow than the other shapes.

!or initial values of aspect ratio, the dome shape have lower non2dimensional load carrying capacity,

non2dimensional end flow and higher friction parameter but after the aspect ratio of 5. elliptical have

lower non2dimensional load carrying capacity, non2dimensional end flow and higher friction

parameter. !or an aspect ratio of 5.+A= and :e 5./= the effect of different shape of textures withthe variation of texture height ratio is shown in the !ig. "+. The non2dimensional load carrying

capacity, non2dimensional end flow increases and friction parameter decreases with the texture height

ratio. This is because as the texture height increases the film thicness decreases which develops

higher pressure and thus the non2dimensional load carrying capacity increases and friction parameter

is @ust inversely proportional to the non2dimensional load carrying capacity. The triangular shape

shows better load carrying capacity at higher texture height ratio and the hexagonal shape gives higher

preferential end flow than the other shapes. 0xcept dome and elliptical, remaining shapes has

negligible difference in the friction parameter. !orm this it can be concluded that the friction

parameter is independent for some shape of textures not for any shape.

5

5.1

"

".1

5." 5."1 5.+ 5.+1 5. 5.1 5./ 5./15

+5

/5

45

>5


!ig. "+ 0ffect of different shapes on the non2dimensional load carrying capacity, non2dimensional end

flow and friction parameter with the texture height ratio.

0ffect of the shape of textures on the performance parameters with the fluid inertia i.e., by varying

reduced :eynolds number is shown in the !ig. " for a certain value of

5.+A= and

5./H=. The

figure clearly shows that the fluid inertia has significant effect on the performance parameters of


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different shapes. The non2dimensional load carrying capacity, non2dimensional end flow increases

and friction parameter decreases with the variation of reduced :eynolds number. The effect of fluid

inertia in dome shape is very small when compared with the other textures. The triangular shows

higher non2dimensional load carrying capacity and lower friction parameter whereas hexagonal shows

higher preferential non2dimensional end flow.

5.51

5."

5."1

5.+

5.+1

5 5.+ 5./ 5.4 5.> "4

>

"5

"+

"/

"4

:educed:eynolds number (:e

!ig. " 0ffect of different shapes on non2dimensional load carrying capacity, non2dimensional end

flow and friction parameter with reduced :eynolds number.

!rom the !igs. "", "+ and ", it can be concluded that the friction parameter is independent of some

shape of the texture but not for any shape. The triangular shape gives better non2dimensional load

carrying capacity and friction parameter when compared with the other shape of textures. !rom the

sealing point of view, the hexagonal shape gives better performance parameters than the other shapes.

5. E((ECT O( SUARE6S"A,ED 7ULTI6TE4TURES IN TRANSVERSE6DIRECTION ON

,ARALLEL SLIDIN SUR(ACES

The effect of the number of textures in the transverse2direction is analyzed for the

hydrodynamic performance on the s'uare2shaped texture parallel sliding surface. The effect of

number of textures on the hydrodynamic performance for a reduced :eynolds number and aspect ratio

of 5./ and 5.+ respectively is shown in !ig. "/.


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" + / 1 45

5."

5.+

5.

5./

5.1

5.4

5.8

5.>

5.3

" + / 1 45

5.5+

5.5/

5.54

5.5>

5."

5."+

5."/

5."4

" + / 1 45

"5

+5

5

/5

15

45

85

>5

!ig. "/ ;omparison of number of textures with load carrying capacity, end flow and frictionparameter for different texture height.

!ig. "5 shows that as the number of textures increases in the z2direction, the W and Q

decreases and ( XL C increases. The reduction of Q is favourable interms of leaage point of

view. !or a particular Hand A values of 5.+ and 5./ respectively, the steady2state performance

characteristics are shown in the !ig. "1.

" + / 1 45.54

5.5>

5."

5."+

5."/

5."4

5.">

" + / 1 45.5+>

5.5

5.5+

5.5/

5.54

5.5>

5.5/

5.5/+

5.5//

5.5/4


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" + / 1 44

8

>

3

"5

""

"+

"

"/

"1

!ig. "1 ;omparison of number of textures with load carrying capacity, end flow and friction

parameter for different :eynolds number.

The Q is reduced further with inertia effect incomparison with the Q without inertia when thenumber of textures increases more than five (See !ig. "1. !rom the above results, it is observed that

the number of textures should be less in the z2direction.

8. E4,ERI7ENTAL STUD# O( ,OSITIVE 7ICRO6TE4TURES ON T"RUST ,AD

9EARIN

0xperimental setup used for conducting the experiment is shown in the !ig. "4

!ig. "4 0xperimental set2up

". Thrust pad +. 7onitor . $roximity probe /. D%F card 1. oading arm

4. ;ontroller 8. Strain measuring system (S;%D 155 >. Tachometer


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The test specimens were made by using %luminium foil of 1Jm thicness. Different orientations of

texture are shown in Table. +

Table +. Different shape and orientation of textures

Orientation S'uare (%rea"55mm+ Triangle (%rea"55mm+

5 degree

5 degree

45 degree

35 degree

The results are obtained by varying speed for different load conditions, the !ig. "8 shows thevariations of film thicness with the speed. !rom the figure, it is observed that, for the case of low

load, the film thicness is decreasing as the speed increases.


17/20


18/20

:. CONCLUDIN RE7AR;S

&n the present wor, the numerical techni'ue is used to explore the effect of fluid inertia on the

different shaped protrusions of the parallel sliding contacts. !rom the results, it has been concluded

that

". There is a significant change in the non2dimensional load carrying capacity ( W , non2dimensional end flow (Q and friction parameter ( ( XL C when the fluid inertia effect is

considered.

+. The performance characteristics are very sensitive with the :eynolds number.

. The aspect ratio should be low to get the better hydrodynamic performance characteristics.

The W is maximum for the aspect ratio of 5./, the Q is maximum for the aspect ratio in

between 5.+25. and the CF is minimum for the aspect ratio in between 5.25./.

/. The number of textures in the transverse2direction should be less to get the high W and low

CF .

1. %s the number of textures increases in transverse2direction the Q decreases, which is

favorable from the application point of view to prevent the leaage.

4. The friction parameter is independent of some shape of the texture but not for any shapes.

8. The triangular shape of texture shows better performance than the other shape of textures.

>. !or sealing point of view, the hexagonal texture shows better result than the other shape of

textures.

The main limitation of the present wor is this method is applicable for smaller values of non2

dimensional :educed :eynolds number because the first order perturbation method is mainly

preferable for small perturbs.


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1=. NO7ENCLATURE

C maximum clearance between the surfaces

XL length of the imaginary cell inx2direction

ZL length of the imaginary cell iny2direction

, ,x y zN N N mesh size in x, y and z directions respectively:e :eynolds number

U maximum velocity inx2zplane

h film thicness of the lubricant

gh height of the protrusion

k ratio of the imaginary cell lengths ( X ZL L

l length of the s'uare protrusionp pressure of the lubricant film

u, v, w velocity components in thex,yandz directions respectively

5 5 5, ,u v w non2dimensional steady state velocity components

" " ", ,u v w non2dimensional first order perturb velocity components

A aspect ratio ( area of textured surface area of imaginary cell

F non2dimensional friction force

H texture height ratio (texture height=maximum clearance between the surfaces

Q non2dimensional end flow

:e :educed :eynolds number

( R C non2dimensional friction parameter

W non2dimensional load carrying capacity

h non2dimensional film thicness

H texture height ratio

p non2dimensional pressure of lubricant film

5p steady2state non2dimensional pressure

"p non2dimensional first order perturb pressure

, ,x y z non2dimensional co2ordinates (y non2dimension is across the film dynamic viscosity of the lubricant density of the lubricant

non2dimensional density of the lubricant

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7icroasperity2ubricated !ace SealsM, %S70 K.ubr. Technol., pp.8+4N8".

)+* 0tsion &., Halperin -., and :y -. >(+555, L&mproving Tribological $erformance of 7echanical

;omponents by aser Surface TexturingM, Kournal of the 9alan Tribological %ssociation, 5> +, pp.

8+N88.

)* 9ecer 0. B., 0hrfeld B., Hagmann $., 7aner % and 7unchmeyer D., ("3>4, L!abrication of

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-alvanoforming and $lastic 7oulding(&-%$rocessM, 7icroelectron 0ng., 0, pp.1N14.

)/* 7adou 7., ("338, L!undamentals of 7icrofabricationM,;:; $ress ;.

)1* 0tsion &., ?+551, LState of the art in laser surface texturingM, Transactions of %S70 Kournal ofTribology, 128, pp. +/>2+1.


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)4* 6ligerman and ?., 0tsion &., (+55", L%nalysis of the hydrodynamic effects in a surface textured

circumferential gas sealM, Tribology Transactions, 00, , pp. /8+2/8>.

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texturing for reciprocating automotive componentsM, Tribology Transactions, 03> /, pp. ///2//3.

)>* 0tsion &., and Halperin -.> ?+55+, L% laser surface textured hydrostatic mechanical sealM,

Tribology Transactions, 03, , pp. /52//.

)3*7ii #., %tsuo 6., %tsushi 6., 6o@i 7., Taashi 7., ?asuhisa %., Hatsuhio I and Shinya S.,

?+558, L%pplying micro2texture to cast iron surfaces to reduce the friction coefficient under

lubricated conditionsM, Tribology etters, 2:, pp. ""2"8.

)"5* 9rizmer G., 6ligerman ?., and 0tsion &., (+55,L% laser surface textured parallel thrust bearingM,

Tribology Transactions, 05, , pp. 382/5.

)""* 0tsion &., Halperin -., 9rizmer G., and 6ligerman ?.,?+55, L0xperimental investigation of laser

surface textured parallel thrust bearingsM, Tribology etters, 18, +, pp. +31255.

)"+* :avinder 9. Siripuram and yndon S. Stephens, (+55/, L0ffect of deterministic asperity

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