Frictional behaviour of nonNewtonian lubricants Z. S. Safar*
This investigation is concerned with the prediction of the frictional behavior of nonNewtonian fluids in a statically loaded journal bearing, The pressure distribution, friction and bearing load capacity are obtained for various values of the flow behaviour index, n. For the pseudoplastic fluids (n 1) exhibited increases in the friction forces and load. The coefficient of friction was found to decrease as the value of n is increased, provided the dimensionless load exceeded a certain value Previous experimental studies comparing the behavior of mineral oils and nonNeWtonian lubricants in journal bearings 13 showed superior performance of the nonNewtonian fluid. That is, for the same eccentricity ratio and friction force, the nonNewtonian oil had the greater load capacity. Also, they concluded that the improvement is greater as the eccentricity is increased. They also confirmed lower wear and friction for polymercontaining lubricants. In steady shearing flow, nonNewtonian behavior has been studied by Horowitz and Steidler4 for shear thinning models. It was shown that these models give a reduced load carrying capacity. Recently, Safar s# has shown that for shear thickening fluids (n > 1), such as high molecular weight polymers in water and glycerin, the load capacity may be greatly increased, while for the pseudoplastics (n < 1), such as greases, the load carrying capacity is decreased. There have been many analytical studies of timeunsteady shearing Flow79, which concluded that the nonNewtonian, elasticoviscous lubricants cause an increase in load carrying capacity. Viscoelastic effects in squeeze films have also been studied by several authors. Recently, Tichy and Winer 1° concluded that bearing performance may be improved for a shear thickening lubricant, while the use of shear thinning oils causes a decrease in the load capacity of the bearing. This study is concerned with the friction force and the friction coefficient of nonNewtonian, inelastic pseudoplastic and shear thickening oils.
Analysis The problem considered is a journal bearing geometry with a nonNewtonian fluid within the gap. The purpose of the analysis is to predict the load carrying capacity, the friction force and the friction factor using the powerlaw relationship between the shear stress tensor and the rate of deformation tensor. It has been shown s'6 that the nondimensional pressure distribution is given by: 0 f (1) n  1
o +
H L N P
P(°2) =
P
u=rl+~~l (  1 )
=
0
(2)
1
\2rtK ~ /
n+l
\r/

(3)
The friction force at the journal is given by: L x2 F]= f f r dx dz 0 xl n=l eccentricity film thickness consistency coefficient
(4)
KUn1
dimensionless coefficient  ,hn_C'I
uUL (R/c)
76 TRIBOLOGY international April 1 9 7 9
o=o2
n1
e
R bearing radius U journal speed Wx, IVy components of the bearing load c radial clearance
aP
The dimensionless velocity profile is given by:
h K
dimensionless film thickness bearing length rotational speed lubricant pressure P dimensionless pressure/nV (R/c) 2
dO
where A is a constant to be determined by the boundary condition. Here the Reynolds boundary conditions are used at the trailing edge:
*Mechanical Engineering Department, University o f California, Berkeley,, California 94 720, USA
Nomenclature D bearing diameter Fi frictional forceat the journal F/ dimensionless friction force 
(1)
7
flow behaviour index velocity components in the x,y directions Cartesian coordinates viscosity of Newtonian fluid stress tensor
0,71
dimensionless coordinates,
n u, P x, y, g
11
01,02 A
x y R'h the beginning and the end of the film W dimensionless load = " IEV(R/c) 2 LD
0301679X/79/0207603 $02.00 © 1979 IPC BusinessPress
Using the nondimensional quantities, equation (4) may be written in the form:
I0 09
1~=
F/ pUL(R/c)
02 f01
=
(5)
~ dO
07
d
., d u/.
~
/
I
= (  1)n  1 .~
1
n
II
w
(1) n aP n +~ k2n~" ~'O/ .J
(6)
Substituting equation (1) into equation (6): O2
nt
= (1)
~
.~
+
(2n+l) n
c ]] n
[1 + ~
AV = 
~ly
pJV(R/c) 2 LD
= 1
2
~2 P(O)sin 01
ill~, 0
20
40
~

n =

I
n=05
I 14O
I 160
I 180
200
dO
//////jl~/"!
,
0.8
/ / / /
,,,/,,'/
.£
~6 0.6
,'///
/','1i/ ///// .~:°,5:2
The friction factor is given by:
,///
~
=~
(9)
04

ku
02
where:
+ ~y
'
1.0 0
Wx 1 ~2 P(O) cosOdO Ax = IJN(R/c)2LD = 2 01
a=~/~
/
Fig 2 Load criterion distribution with the eccentricity ratio for n = 0.5, 1, 1.5, and 2
(a)
f~
n = 2 n=15
[ _[ J_ 80 I00 120 Load criterion, A
60
(7)
The components of the load carrying capacity of the journal are :

  . _ _ . . . .
......
/~lf/,/'i"
o4 05
,r r/=l
/
06
~ 05 t
t  S
,//3J / I/////// />/
08 .
where :
//~"~'/
.......
(10)


I 10
0
n :0.5
I 20
I 30
Dimensionless friction, Fj 18
" Fig 3 Dimensionless friction force distribution with the eccentricity ratio for n = 0.5, ], 1.5, and 2 . . . .
14 
/
'
\
// ,.
'\ \\
\
\\
/
I0
........   .  
n=2 n=15
  n = l
~\k
....
n=05
2 .,..c
N c
n=2 n = I n=0.5 ~=02
//
2
8o
"\\
..\ ~x\
6
g =
I
0
40
80
120
160
200
240
Angular displacement, 0
Fig I Pressure distribution along the circumferential direction at e = 0.2 for n = 0.5, 1, and 2
280
..... .<...,,
I
0
O2
]
I
03
04
I
05
I
06
I
I
I
07
08
09
Eccentricity ratio,
Fig 4 Friction factor coefficient with the eccentricity ratio for n = 0.5, 1, 1.5, and 2
TRIBOLOGY
international April 1 9 7 9
77
that if A < 30, the pseudoplastic fluid gives a lower coefficient o f friction than either Newtonian or shear thickening fluids for the same load. For A > 30 the pseudoplastic fluid and the Newtonian lubricant exhibit the same coefficient of friction, while the shear thickening fluid gives a lower value.
n = 2 n
g
.........
=
n=
[
05
Thus for smaller load capacities the pseudoplastic lubricant gives lower values o f coefficient of friction, while for larger values of load capacity shear thickening fluids exhibit better friction criterion than the Newtonian fluids.
References 0
I 40
I 80 Loed criterion
I 120
I 160
1.
Okrent E.H. The Effect of Lubricant Viscosity and Composition on Engine Friction and Bearing Wear, ASLE Trans.,
2.
Savage M.W. and Bowman L.D. Radioactive Tracer Measurements of Engine Bearing Wear, A SLE Trans., Vol. 4, 1961,
200
A
Fig 5 DistribUtion o r friction coefficient with the load criterion f o r n = 0.5, 1, and 2
Vol. 4, 1961, pp. 257262
p. 322
3.
Results Tha pressure distribution, load carrying capacity, friction and friction factor were calculated for eccentricity ratios of 0.2, 0.4, 0.6, 0.8, and 0.9 and for various values of the flow behaviour index ranging from 0.52. Fig 1 shows, by way o f example, the pressure distributions for an eccentricity of 0.2, for values of the flow behaviour index n o f 0.1, 1, and 2. Pressure increases as n increases, and the film ruptures at a larger angle as n decreases. The effect of the exponent n is more pronounced at higher eccentricity ratios. Fig 2 shows the distribution o f the load carrying capacity with the eccentricity ratios for various values of n. It is also noted that the load carrying capacity is increased with n. The effect is more pronounced at higher eccentricities. Fig 3 shows the normalized friction force at the shaft, which also increases with n, but the effect of n on the friction force is not as high as on the load carrying capacity. As a result, the coefficient o f friction is decreased as n is increased (Fig 4). Fig 5 gives the distribution of the dimensionles load with the coefficient o f friction. From the figure it is shown
78 T R I B O L O G Y
international April 1979
4.
Dubois G.B., Ocvirk E.W. and Wehe R.L. Study of Effect of a NonNewtonian Oil on Friction and Eccentricity Ratio of a Plain Journal Bearing, NASA TND 427, May 1960 Horowitz H.H. and Steidler F.E. The Calculated Journal Bearing Performance of Polymer Thickened Lubricants, ASLE Trans., Vol. 3, 1960, pp. 124133
5.
Safar Z.S. and Shawki G. Performance of Thrust Bearings Operating with NonNewtonian Lubricating Films, to be published in Tribology International
6.
7.
Safar Z.S. Journal Bearings Operating with NonNewtonian Lubricating Films, to be published in Wear Tanner R.1. Increase of Bearing Loads Due to Large Normal Stress Differences in Viscoelastic Lubricants, Journal o f Applied Mechanics, Trans. ASME, 1969, pp. 634635
8.
Harnoy A. and Hanin M. Second Order, Elasticoviscous Lubricants in Dynamically Loaded Bearings, ASLE Trans., Vol. 17, No. 3, 1974, pp. 166171
9.
Harnoy A. An Analysis of Stress Relaxation in Elasticoviscous Fluid Lubrication of Journal Bearings, Journal of Lubrication Technology, Trans. A SME, Vol. 100, 1978, pp. 287294
10.
Tiehy J.A. and Winer W.O. An Investigation into the Influence of Fluid Viscoelasticity in a Squeeze Film Bearing, Journal o f Lubricating Technology, Trans. ASME, Vol. 100, 1978, pp. 5664