Performance characteristics of a centrally loaded 120 ° partial journal bearing using pseudoplastic lubricants M. Malik*, B. Dasst and R. Sinhasan*

The additive mixed oils, which are classified as non-Newtonian fluids, are widely used in many lubrication systems. During the past decade, several analyses of plane journal bearings using non-Newtonian lubricants have been reported, although no work has been reported on partial bearings. This paper attempts to fill this void. The non-Newtonian lubricant is taken to be pseudo- plastic and two types of rheological model, based on a cubic shear stress law and a shear strain rate power law, have been considered. The computed resu Its reported here are dimensionless and include the static as well as the dynamic characteristics of the bearing

Keywords: journal bearings, non-Newtonian lubricants, apparent viscosity

Ever since its derivation almost a century back, the Rey- nolds equation has remained the backbone of all major researches and developments in fluid film lubrication. Two important assumptions involved in the derivation of this equation are that the lubricant is Newtonian and that the lubricant viscosity is constant across the film, although it can be a function of the coordinates along the film. Non- Newtonian lubricants are characterized by nonlinear shear stress-strain rate relationships. The viscosity field of a non- Newtonian hydrodynamic film varies along as well as across the film coordinates. Thus, as such, the usual form of Reynolds equation is not applicable to non-Newtonian hydrodynamic load supporting films. In fact, due to the dependence of the viscosity on the shear strain rate and hence the pressure field, the non-Newtonian hydrodynamic lubrication problem is basically nonlinear.

The earliest attempts in the theoretical analysis of non- Newtonian hydrodynamic lubrication were made with infinitely long bearing configurations ~-3 . The theoretical analyses of the finite length journal bearing are based on the various forms of modified nonlinear Reynolds equa- tions 4-9 , each derived for a specific rheological model of the lubricant. The more recent analysis of Tayal et al x°'n presents a general solution as it is based on the fundamental Navier-Stokes equation and is applicable to any form of constitutive law for the lubricant. However, considering the tedious algebraic efforts and the expensive computations due to the three dimensional finite element formulation, this analysis ~°'u does not appear to be very practicable from the engineering design viewpoint.

In recent works, Dass 12 and Malik et a113 also approached the non-Newtonian hydrodynamic lubrication problem with a view to presenting a general analytical method, vis-a- vis the constitutive laws, based on the usual form of Reynolds equation. In this approach, the lubricant viscosity function is defined on the basis of the shear strain rates averaged integrally across the film. The viscosity field is thus reduced to a function which varies across the film,

*Department of Mechanical and Industrial Engineering, University of Roorkee, Roorkee 247 672, India

tCentral Building Research Institute, Roorkee 24 7 6 72, India

making it possible to apply the usual Reynolds equation to the non-Newtonian hydrodynamic flow field. Interestingly, the results of this analysis 12'~3 compare extremely well with the already published results of the infinite slider bearing 2 , the journal bearing experimental results 14 , and, also, the theoretical static as well as dynamic characteristic results of the journal bearings 8'9 .

In this paper, the aforementioned analytical approach 12,13 has been used to analyse a centrally loaded 120 ° partial journal bearing lubricated with a pseudoplastic lubricant following a cubic shear stress law (a~ "3 + r - ~ = 0) and a shear strain rate power law (r = "~1 ~1 t~-l) model. The com- puted results are dimensionless and cover all the perfor- mance characteristics as required in the bearing design. Details of the averaging technique and the solution proce- dure are also included. The details of the solution of the Reynolds equation and the computation of the performance characteristics, which are based on the finite element method, may be found elsewhere 13 .

~y

Fig I Partial bearing geometry and coordinate axes

0301-679X/83/010009-08 $03.00 © 1983 Butterworth & Co (Publishers) Ltd 9

Malik, Dass and Sinhasan - Partial bearing performance

Integral averaging across the film The shear strain rate averaged integrally across the film is defined as

5'av = ~ ~dz (1) 0

The shear strain rate ~ in incompressible viscometric flow conditions is given by ~s

r 1 ?== L' aO l az I j

+ L ' R - " ~ + - - - + - - (2) R ~0 ~z R ~-

[ C a w a u ] 2

+ L R - ' - ~ +--az

For a thin film of the bearing clearance space, it can be seen easily on an order of magnitude basis, that only the terms au/az and av[az need to be retained in Eq (2). The expres- sion for averaged shear strain rate then becomes

3 ' av=h o ~zz + a z / .J dz (3)

Like Newtonian lubricants, the rheological characteristic of non-Newtonian lubricants, usually referred to as apparent viscosity or viscosity function zs , is defined by the ratio of shear stress to shear strain rate. This characteristic, defined on the basis of'~av, may appropriately be referred to as averaged apparent viscosity (flay). Thus

r/av = r/')'av (4)

Since in a bearing clearance space the film thickness is only a function of coordinates along the f'dm (0,~'), it becomes obvious that qav = ~?av(0 ,~') and r~av = r/av(0,~" ). Also the Reynolds equation

- - + = 6 + 12 ( 5 ) ao ao

is now applicable to the non-Newtonian hydrodynamic flow field.

Solution procedure The finite element method with Galerkin's criterion 16 has been used for the solution of Eq (5). The computations have been made with eight noded rectangular isoparametric elements. Eq (5) is implicitly nonlinear as the viscosity ~Tav itself depends on the pressure field. An iterative method has been adopted for establishing the pressures in which the viscosity and pressure fields are refined in successive trials. The trials in this scheme begin with some initial guess of the pressure field. The steps of the scheme are as follows:

(1) Determine the averaged shear strain rate distribution using the available pressure field and Eq (3).

(2) With the known shear stress-shear strain rate relation- ship (constitutive law), determine the shear stress distribution using the shear strain rates obtained in Step (1).

(3) Using the shear stress and shear strain rate distributions obtained in the above steps and Eq (4), determine the averaged apparent viscosity.

(4) Solve Reynolds equation using the current viscosity field to obtain a new pressure distribution.

(5) Compare the current pressures with the pressures of the previous trial using some convergence criterion. If convergence has been achieved, proceed further to obtain bearing performance characteristics, otherwise go to Step (1).

The convergence criterion used in this work is based on the maximum pressure and may be stated as

I -/+1 ; I" ~+1 ,Umax -Pmax /Pmax <~ 0.1 (6)

where Pmax is the maximum pressure and i is the iteration count.

Although the convergence of this scheme is fast, the number of iterations depends on the initial guess of the pressure

Nomenclature All dimensional quantities are represented by symbols with a bar. The corresponding symbols without a bar refer to dimensionless quantities. The SI units of the dimensional quantities are given in brackets.

C

P

FL a

K

?-

U,I~, W

B1/¢(-•o/•4/•a), fluid film damping coefficients N s m -1 );/,k = x,y

Radial clearance (m)

he', film thickness (m)

M c ( ~ ~4/co C 3 ), critical mass of the journal above which journal bearing system becomes unstable (kg)

P ~ o ~ 2 / ~ - a ), fluid film pressure (N m-2 ) ~-av

PL~-o ~2~4/~, power loss (W) 0

Q(,~2C~o), side leakage from the bearing (m a s -1 )

Journal radius (m)

Slk (~o~'*/C'3) , fluid film stiffness coefficients v- (N m-l ) ; j ,k =x,y

t /~ , time (s) r-

(u,v,w)Rw, fluid velocity components in 0, ~" and z directions respectively (m s)

x,y

or,3

W o ( ~ o ~ 4 / C a) , steady state load capacity (N)

Cartesian coordinate axes (Fig 1)

zC, coordinate across the film (m)

Rheological constants

@(Rffa/C), shear strain rate (s -l )

Eccentricity ratio

~'/~, axial coordinate of the bearing (m)

Absolute viscosity of Newtonian lubricant, reference viscosity for non-Newtonian lubricant (N s m -2 )

r/argo, averaged apparent Viscosity (N s m -2)

Circumferential angular coordinate (Fig 1)

Bearing length to journal diameter ratio (aspect ratio)

Whirl frequency of the journal at the onset of film induced instability (s -1 )

r(~o~R/C), shear stress (N m-2)

Altitude angle

Angular rotational speed of the journal (s -1 )

10 TR I BO LOGY international February 1983

Malik, Dass and Sinhasan - Partial bearing performance

Table 1 Comparison of 120 ° partial journal bearing data for Newtonian lubricant (X=I, e=0.0,/3=1,0)

e Wo Vs Sxx Sxy - Syx Svv Bxx - Bxy t ByK

1 * 0.2989 72.00 0.2724 0.4152 3.177 0.6098 0 . 8 5 1 1 0,2764 0.1

2 x 0,2990 7 2 , 3 1 0.2727 0.4145 3,173 0,6103 0.8516 0,2761

1 1.062 50.22 1.093 0.6124 4.987 2.490 1.448 1.206 0.3

2 1.061 50.19 1.093 0.6139 4.986 2.489 1.450 1.206

1 2.531 39.39 3.097 1.147 9.809 7.305 3.277 3.990 0.5

2 2.533 39.53 3.103 1.135 9.800 7.307 3.278 3.993

1 6.465 31.95 7.984 0.8606 23.84 29.10 6.523 10.46 0.7

2 6.441 32.31 8.754 1.897 24.95 27.38 8.276 13.08

1 30.50 21.30 41.05 -23.61 150.8 366.9 19.31 49.53 0.9

2 30.31 21.57 44.42 -16.28 159.5 347.7 25.68 64.96

6.375

6.364

10.21

10.21

20.81

20.76

51.67

55.12

313.6

347.6

*Present work (finite element method) X Reference 19 (series solution)

tBxy = Byx

I0

~.o

1.0

0.5

o . i I I I I I O. I 0 . 3 0 . 5

Fig 2 Bearing load capacity

Cubic 1ow OT 3 + T -~mu = 0 Curve a

I 0 . 0 2 0.1 3 1.0 4 I0.0 X = I

I I I 0.7

I 0 . 9

f 0.1

I I I 0 . 3 0 . 5

Power low

Curve

I 1.0 2 0 . 9 5 0 . 7 4 0 . 5

X=I I I I

0.7 I

0 . 9

TRIBOLOGY international February 1983 11

field. Choosing the initial pressures corresponding to the Newtonian viscosity (~av = 1), the iterations may sometimes be as many as six. However, when the initial pressures are chosen as equal to the already established pressures for approximate values of eccentricity and rheological constants, the number of iterations never exceeds four.

Results and discussion The performance characteristics reported here are for a centrally loaded 120 ° partial bearing (Fig 1) of aspect ratio equal to unity. The following two types of constitutive equation have been used for the pseudoplastic lubricant.

Cubic shear stress law:

otr 3 + "r - ~'av = 0 O~/> 0

0

Shear strain rate power law:

7" = "Yav I "Yav I t3-1 /3 ~ 1

where t~ and ~l are rheological constants (measures of the deviation from rheological linearity of the lubricant). Also, both ct = 0 and/3 = 1 refer to the Newtonian lubricant. The values of the theological constant for the additive mixed lubricants depend on the type as well as the contents of the additive. With the same type of additive, the theological nonlinearity of the lubricant increases, generally, with increase in the additive contents. The numerical computations have been made for three values of each of the theological constants of pseudoplastic lubricants (c~ = 0.1, 1.0 and 10.0;/3 - 0.9, 0.7 and 0.5). The ranges of a and ~ have been chosen on the basis of their

0

0.1 0.1

0.2

0.:5

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1.0 0 o I0 o 2 0 °

0 .2

0.3

0 . 4

0 . 5

0 . 6

0.7

O.E

0 . 9

Malik. Dass and Sinhasan - Partial bearing performance

1.0 0 o i0 °

q, 2 0 °

Fig 3 Journal centre loci

12 TR I BOLOGY international February 1983

500

Malik, Dass and Sinhasan - Partial bearing performance

B

I00 - -

50 - -

I0

/ /

i/I/l

/ /

/ I

/

// I I 13

I I I I I i 11//4

/ / / / / l l l l

/ / / / / j / i s

B

m

m

m

m

m

I I 2

//// / I

/ Cubic law / Power low =- I

Curve I 0 . 0 I 1 .0

0.5 2 0.1 2 0.9 3 I .0 3 0 .7 4 10.0 4 0 .5

X = I X = I

Sxx -- - - Sxx

YY YY

o.= I I I I I I I I I I I I I I I I 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9

E (

Fig 4 Direct stiffness coefficients

Table 2 Power loss and oil requi rements of the bear ing (~ = 1 .0)

Power loss coefficient, PL/Wo Bearing side/eakage, Q

e" Newtonian Cubic shear Shear strain Newtonian Cubic shear Shear strain stress law rate power law stress law rate power law

~=0.1 <x=l ~=10 /~=0.9 /~=0.7 /~=0.5 <x=0.1 ,',=1 ,',=10 / J=0 .9 /~=0.7 ~=0.5

0.1 14.37 0.2 7.20 O,3 4.68 0.4 3.32 0.5 2.43 0 . 6 1 .80 0.7 1.30 0.8 0.88 0.9 0.48

14.32 14.24 14.20 14.34 14.27 14.21 0.0897 7.12 7.01 6.95 7.15 7.05 6.97 0.1580 4.59 4.47 4.41 4.63 4.53 4 A3 0.2056 3.22 3.10 3.04 3.27 3.17 3.07 0.2381 2.33 2.22 2.16 2.38 2.28 2.19 0.2596 1,68 1.58 1.53 1.74 1.65 1.56 0.2739 1.16 1.08 1.05 1.25 1.16 1.08 0,2767 0.73 0.67 0.65 033 0.75 0.68 0.2643 0,34 0.31 0.30 0A4 0.38 0.33 0.2303

0.0901 0.0909 0.0914 0.0900 0.0907 0.0912 0.1600 0.1637 0.1660 0.1593 0.1622 0.1652 0.2099 0.2172 0.2216 0.2082 0.2139 0.2199 0.2450 0.2555 0.2619 0.2419 0.2501 0.2590 0.2691 0.2824 0.2903 0.2644 0.2747 0.2862 0.2859 0.3006 0.3092 0.2788 0.2906 0.3038 0.2940 0.3113 0.3209 0.2833 0.2983 0.3139 0.2874 0.3061 0.3162 0.2716 0.2981 0.3065 0.2596 0.2763 0.2849 0.2376 0.2539 0.2729

TR I BOLOGY internat ional February 1983 13

Malik, Dass and Sinhasan - Partial bearing performance

values for some typical non-Newtonian fluids and lubricants as available in the literature 14']7'1s . The journal centre loci (e, 4) for vertical load support have been determined accurately through a trial scheme 19 . Also, the Reynolds condition has been used at the interface of full and cavitated films in the bearing clearance space.

It has already been mentioned that the non-Newtonian characteristics obtained with the averaged apparent viscosity formulation compare very well with the available slider and plane journal bearing results ~2'13 . However, a comparison of the non-Newtonian characteristics for partial bearings could not be made due to the nonavailability of such results in the published literature. In fact, the computer software used in this study is the same as used for the plane journal bearings 13 ; only the algorithmic accuracy of the program for partial bearings needed to be ascertained by the com- parison of its Newtonian characteristics results with those available in the literature 19 . Thus, Table 1, which gives the comparison for the steady state load capacity, the altitude angle and the dynamic coefficients for a = 0.0 and/~ = 1.0, authenticates the accuracy of this analysis and its computer software.

2 0 0

The performance characteristics of the partial bearing using the pseudoplastic lubricants are given in Figs 2 to 7 and Table 2. To illustrate the effects of rheological nonlinearity, the performance characteristics with Newtonian lubricant are also included in these results. The results are presented in a manner to provide compaisons between the bearing characteristics with the two types of pseudoplastic lubricant.

The pseudoplastic lubricants are shear thinning, and on a Newtonian reference the viscosity of these lubricants is less than the Newtonian viscosity. Also, the increase in a for the cubic law and decrease in/3 for the power law both imply an increase in the shear thinning effects. Thus, as seen from Fig 2, the load capacities of the bearing with the non- Newtonian lubricants are always less than those with Newtonian lubricant. For the same load support, a bearing with pseudoplastic lubricant operates at a higher eccen- tricity (ie lower minimum film thickness) in comparison to a bearing with Newtonian oil. Thus, when using a pseudo- plastic lubricant, the bearing would have to be designed with a larger clearance than usually recommended for Newtonian oils.

!

I00

50

I0

1.0

0.5

Cubic low I a . T 3 ..I - T--~',=u =0 /

_ C u . , o / - - I 0 . 0 / 12 : z o.J / /

_ 3 I.o "//I I __ 4 I 0 .0 / 3 __ ~,=1 / / / / /

sxy / /

- _ _ _ _ - s x / I / / , 4

- - / / / i // /

/

_ / y / ; / / _-- / ' / /

- / / i , / / - - / / / /

/ / / /

I / / /

i j f

.....--"

o . i I I I I I I 0.1 0 . 3 0 .5 0 . 7

E

Fig 5 Cross coupled stiffness coefficients

I I 0 . 9

Power low l I T = '~Qvl ~"= lu I~ - I / , 2

_ Curvo /~ / ! - - I I . 0 / ' / i 3 - - 2 0 . 9 , / / / , 4 _- 3 0 .7 / / / , '

4 o. 5 / / / / - x : , / / / ,

- - - - yx l~/ / / / ]1

- /I: Iil,

/ . / / / IIII - / . / / / IIII

: 711

0.1 1 I L I I I I I

0 . 5 0 . 5 0 . 7 0 . 9

14 TR IBOLOGY international February 1983

5 0 0

Malik, Dass and Sinhasan - Partial bearing performance

o

o

=. o o

¢m

c

o . E o

m

I 0 0 - -

50

IO

I ) i

0/' • / , / /

I' / / • , / / '

. J . / I / t 7 /

n

I

I

m

I

n

/ 7 ,~I Cul)Ic low / =~ + r-?=~= o

/ Curve (i

0 . 5 I 0 . 0

/ z o.l ' I 3 I.o

I 4 io.o I' / X = l _ / Oxx

I - - - -a ,y : -sy= I e

O . I Y Y - -

0 .1 0.3 0 .5 0.7 0 . 9 O. I

Fig 6 Fluid film damping coefficients

Shear thinning effectively reduces the resistance to flow of the lubricant in the bearing clearance space. Table 2 gives the power loss and the oil requirements of the bearing. It may be seen that increase in shear thinning effects results in reduction in the power loss and increase in oil flow out of the bearing. This can be seen from the data in Table 1 ; for the same load support, the bearing temperature rise with pseudoplastic lubricant would be less because of higher operating eccentricity, than in a bearing operating with Newtonian oil.

For the same eccentricity ratio, the shear thinning effects increase the altitude angle (Fig 3) indicating an impairment in the dynamic performance. Thus, as seen from Figs 4 to 6, the fluid film stiffness and damping coefficients decrease continually with increasing shear thinning effects. However, the elastic and damping properties always improve with increase in eccentricity. In fact, for the same load support, a bearing with a pseudoplastic lubricant may exhibit better

I

/,' ' ;,':

Power low • . 8 - I

r = ~aul ~aul Curve

I 1.0

2 0.9 3 0 .7 4 0 . 5 h = l

Bxx . . . . B xy = - By x

" B . . . .

0 .3 0 . 5 0 . 7 0 .9

dynamic performance, due to a higher operating eccen- tricity, than a bearing with Newtonian oil. For the same eccentricity, it can be seen (Fig 7) that shear thinning results in an increase in the critical mass per unit load capacity. Also, the whirling frequencies, which decrease with increase in eccentricity ratio, are smaller for the non-Newtonian lubricants. Both of these effects are conducive to an improve. ment in the dynamic stability of a journal bearing system using a pseudoplastic lubricant. In comparison to the pseudoplastic lubricant following a power law, the shear thinning effects are more pronounced with the lubricant following a cubic shear stress law. The performance characteristics with the power law lubricant are quite close, up to an eccentricity ratio of approximately 0.3, to the characteristics with Newtonian lubricant. S u m m a r y

This paper has presented detailed performance characteris- tics of a centrally loaded 120* partial bearing using two

TR I B O L O G Y in ternat iona l February 1983 15

Malik, Dass and Sinhasan - Partial bearing performance

£ \

I00

50

I0

k\\ \ ,4 \\\ I c. ,o \ /

I_ =o \ \ \ J Curve a "~ \ \ \ 1 , 3

I O 0 \~I 2 2 o11 .WI

_ 3 , . o ) ' f i l l

- ~ Me/We 2 - - - -

I - 4

l 1 I I I 0.1 0 . 3 0 . 5 0 7

E

I I

Fig 7Journal bearing stability margins

types of pseudoplastic lubricant. Using the concept of an averaged apparent viscosity, the usual Reynolds equation has been applied to the analysis of the bearing. In com- parison to the earlier analysis of non-Newtonian hydrody- namic films, this analytical approach offers not only a major mathematical simplification but, also, a complete generality vis-a-vis the rheological models of the various lubricants. Since the performance analysis techniques with the usual Reynolds equation are well known to lubrication engineers, the present formulation can be used effectively for the development of design data of bearings lubricated with oils which obey different theological models.

For the same load capacity, a partial bearing with a pseudo- plastic lubricant operates, in comparison to a bearing with Newtonian oil, at higher eccentricities and shows improved static and dynamic performance.

References 1. Ng C.W. and Salbel E. Nonlinear Viscodty Effects in Slider

Bea~ag Lubrication. Trans. ASME, J. Basic Engng., 1962, 84 (1), 192-196

2. Hsu Y.C. and Saible E, Slider Bearing Performance with a Non- Newtonian Lubricant. ASLE Trans., 1965, 8 (2), 191-194

3. Hsu Y.C. Non-Newtonian Flow in Infinite Length Full Journal Bearing. Trans. ASME, J. Lub. Tech., 1967, 89 {3}, 329-333

4. Wada S. and Hayaehi H. Hydrodynamic Lubrication of Journal Beaxing by Pseudoplastic Lubricant (Part I, Theoretical Studies). Bull. JSME, 1971, 14 (69), 268-278

5. Hayaslii H. and Wsd= S. Hydrodynamic Lubrication of Journal Bearings by Psoudoplastic Lubricants (Part 3, Theoretical Analysis Couside~ag Effect of Correlation). Bull../SME, 1974, 17 (109), 967-974

6. Hay~dli H., Wada S. and Nak~ai N. Hydrodynamic Lubrication of Journal Beating by Non-Newtonian Lubricants. Bull. JSME, 1977, 20 (140), 224-231

7. Swamy S.T.N., Ptabhu B.S. amd Rao B.V.A. Stiffness and Damping Characteristics of Finite Width Journal Bearings With

Power lOW ~ \ \ I

T : -?,J ~,o~ I B-' ~ i Curve ~ 2

I 1 ,0 ~ 3 2 0 .9 4

_ 5 0 . 7 - - 4 0 . 5

k = l 3

[ I I I [ I

0 .90 . I 0 .3 0.5 0.7

E

0 .6

- - 0 . 5

- - 0 . 4 -

- - 0 . 3

- - 0 . 2

0.9

13

a Non-Newtonian Film and Their Application to Instability Prediction. Wear. 1975, 32 (3), 379-390

8. Swanly S.T.N., Prabhu B.S. and Rue B.V.A. Steady State and Stability Characteristics of a Hydrodynamic Journal Bearing With a No n-Newtonian Lubricant. Wear, 1977, 42 (2), 229-244

9. Rajalingham C., P~abhu B.S. and Rue B.V.A. Steady State Performance of a Hydrodynamic Journal Bearing With a Pseudoplastic Lubricant. Trans. ASME, J. Lub. Tech., 1979, 101 (4), 497-502

10. Tl~,al S.P., Sinliasan R. and Sil~h D.V, Analysis of Hydro- dynamic Journal Beatings Having Non-Newtonian Lubricant (Prandtl Model) by a Finite Element Method. Z Mech. Engng. Sci., 1981, 23 {2}, 63-68

11. Tayal S.P., Sinhemn R. and Singh D.V. Analysis of Hydrody- namic Journal Bearings Having Non-Newtonian Lubricants. Tribology International, 1982, 15 {1), 17-21

12. 1)ass B. Finite Element Analysis of Porous Sliders With Non- Newtonian Lubricants. M.E. Thesis, University o f Roorkee, India, 1981

13. Malik M., Da~ B. and Sinhl~m R. The Analysis of Hydrody- namic Journal Bearings Using Non-Newtonian Lubricants by Viscosity Averaging Across the Film. ASLE Trans., 1983, 26//)

14. WKia S. and Hayl~hi EL Hydrodynamic Lubrication of Journal Beatings by Pseudoplastic Lubricants (part 2, Experimental Studies). Bull.'JSME, 1971, 14 {69), 279-286

15. D~rby R. Viscoelastic Flnids.Marcel Dekker, Inc., New York, 1976

16. Heulmer K.H. The Finite Element Method for Engineers. John Wiley and Sons, New York, 1976

17. Bird R.B., Stewart W.E. and IdShtwood E.N. Transport Phenomena. John Wiley and Sons, New York, 1960

18. Sinha P. and Sil~i h C. Lubrication of a Cylinder on a Plane with a Non-Newtonian Fluid Considering Cavitation. Trans. ASME, Y. Lub. Tech., 1982, 104{2), 168-172

19. Malik M., Chlmdra M. and Sinhaem R. Analysis of a Centrally Loaded 120 ° Partial Porous Journal Bearing Taking Tangential Velocity Slip into Account. J. Mech. Engng. Sci., 1981, 23 (4), 171-178

16 T R I B O L O G Y in ternat iona l February 1983