Wear, 76 (1982) 299 - 319
299
EFFECTS OF CONSISTENCY
VARIATION
LUBRICANTS
IN SQUEEZE FILMS
OF POWER
LAW
J. B. SHUKLA and K. R. PRASAD
Department
(India)
of Mathematics,
Zndian Institute of Technology,
Kanpur, Kanpur 208016
PEEUYSH CHANDRA
The Mehta Research Institute of Mathematics and Mathematical Physics, 26 Dilkusha,
New Katra, Allahabad 211002 (India)
(Received December 15,198O;
in revised form September 26,1981)
summary
The characteristics
of squeeze film bearings with power law lubricants
have been investigated by considering the effects of consistency variation.
Various bearing geometries have been considered with rigid surfaces as well
as with compliant layers. With stiff solids, a high consistency layer adjacent
to the bearing surface increases the load capacity and time of squeezing and
this increase is enhanced by the pseudoplastic
behaviour of the lubricant.
For squeeze film bearings with compliant layers, the film thickness increases
with load, compliancy and conformity
of the surfaces even with power law
lubricants. It also increases with the consistency of the layer adjacent to the
bearing surface.
1. Introduction
Many lubricants such as silicone fluids behave as non-newtonian
power
law fluids at low shear rates [l -31. The characteristics
of these lubricants
have been studied in different systems, such as squeeze films [4 - 61,
externally pressurized bearings [ 7 - 91, journal bearings [ 10 - 141 and
elastohydrodynamic
rollers [ 31. Although the effects of viscosity variation
on the behaviour of lubricated systems using newtonian lubricants have been
investigated [ 15 - 191, little attention has been given to study the effects of
consistency
variation on bearing characteristics
using non-newtonian
lubricants [ 20,211.
The effects of consistency variation on the characteristics
of squeeze
film stiff bearings have thus been investigated using power law fluids as
lubricants. The squeeze film between two compliant surfaces has also been
discussed.
0043-1648/82/0000-0000/$02.75
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300
2. Basic equations
Consider the flow of a power law fluid in a thin clearance (slit) of film
thickness 2/z(x) as shown in Fig. 1. Under the usual assumptions of lubrication theory the equation governing the flow of the fluid is given by
;
[ml;
I”--‘!!!]
= dE&
(1)
where u(x, y) is the velocity, p(x, y) is the fluid pressure, m(x, y) is the
consistency of the fluid and n is the flow behaviour index of the fluid.
Since u decreases as y increases for y > 0, i3u/ay is negative and eqn. (1)
can be rewritten as
yao
Solving eqn. (2) and using the boundary
(2)
conditions
au
-=0
at
y=O
at
y=h
ay
Ll=O
(3)
gives
u
=
(- !!fi”)!!(;T”;
dy
(4)
The total volume flux Q across the width b of the slit is given by
Q = 2bju
dy
0
which on using eqn. (4) gives
(5)
Fig. 1. The flow of a power law fluid through a slit.
301
where
l/n
F=hyx
m
JO
(6)
dy
0
Equation (5) determines the flux of the fluid in the slit for any given
consistency function m(x, y).
Owing to various effects, e.g. consolidation,
reaction, absorption and
thermal effects, the consistency of the lubricant in the central layer may be
different from that of the peripheral layer and the consistency
function
m(x, y ) may be assumed to be given by
m(x, y) = ml
O<y<h-a
m(w)
h-a<y<h
= km1
(7)
where a is the thickness of the peripheral layer and 12is a constant. Using
eqns. (5) - (7) the final expression for flow flux can be obtained as
where
3. Squeezing
between
stiff solids
Equation (8) is used to study the lubrication
with stiff or compliant surfaces.
of squeeze film bearings
3.1. Parallel plates
Consider the squeeze film between two parallel plates with film thickness 2h; each plate approaches the other symmetrically
with velocity V as
shown in Fig. 2. The flow flux is given by eqn. (8) which can also be obtained
by using the equation of continuity
as [ 22, 231
Q = 2bVx
Fig. 2. Squeeze film behaviour between parallel plates.
(10)
302
Thus, from eqns. (8) and (lo),
g
= _+~
Integrating
the equation
governing
the pressure is
EJg””
eqn. (11) and usingp
p = ml(y
_!J;p”’
= 0 at x = 1 gives
zfl+;--;:::
(12)
where 21 is the length of the bearing and F. is given by eqn. (9).
The load capacity W,, is given by
W,, = 2+1
dx
0
which on using eqn. (12) gives
(13)
The squeezing time tPz for the plates to approach from an initial film
thickness 2hr to a final thickness 2h2 is obtained by putting dh/dt = -V in
eqn. (13) and integrating:
(14)
For k = 1, the case of no consistency variation, F. = 1 and the expressions for load capacity W,, and time tpl of approach can be obtained from
eqns. (13) and (14) as
(15)
(16)
To assess the effects of the peripheral layer and consistency variation
on the characteristics
of a squeeze film, we assume that the peripheral layer
thickness a = 6h, where 6 < 1, is a constant. Then, from eqns. (13) - (16),
W
---P2
WPl
tP2
---
t Pl
1
-
(17)
Fo"
_
1
(18)
J'o
where F, is defined by eqn. (9) with a = 6h.
The ratios given by eqns. (17) and (18) are plotted for various values
of 6, k and n in Figs. 3 and 4. As k, which is greater than unity, increases, these
ratios increase showing that the effect of the high consistency peripheral
303
4
k-2
o-
.---‘-kc2
I’
I
/
/
/
/
3 o-
I
//
t
/’
‘pz
f
w
‘PI
I
I
I
WPl
Fig. 3. Variation in Wp2/Wpl
l.O;-.--,
n = 1.5.
with 6 for various values of k and n: - - -, n = 0.5; -,
Fig. 4. Variation in tpZ/tpl with 6 for various values of k and n: ---,
l.O;-.-,
n = 1.5.
n = 0.5; -,
n =
n =
layer adjacent to the surface is to increase the load capacity and squeezing
time. Also, the load capacity and squeezing time increase as 6 increases and
this increase is enhanced owing to the pseudoplastic
behaviour of the fluid.
3.2. Cylindrical surfaces
Consider the case of the squeeze film between two cylindrical surfaces
using a power law lubricant, as shown in Fig. 5. The film thickness is given by
h=h,,+;
(19)
0
where 2ho is the minimum film thickness and R. is the radius of curvature
of the approaching cylinders. The load capacity W,, and squeezing time tc2
are
Fig. 5. Squeezing between cylindrical surfaces.
(21)
where 21 is the length of the projected film and F, and h are given by eqns.
(9) and (19) respectively.
For h = 1, eqns. (20) and (21) take the following forms:
(22)
(23)
For a = Ah, the ratios W,,/W,, and tcz/t,. also take the same form as in
eqns. (17) and (18) and hence the existence of a high consistency peripheral
layer for k > 1 increases the load capacity and time of squeezing in this case
also.
To assess the infraction
of geometry of the surface and the nonnewtonian behaviour of the lubricant on load capacity and time of squeezing,
eqns. (20) and (21) are compared with the corresponding
eqns. (15) and (16)
for the case of parailel plates having the same projected area and the same
minimum film thickness. In such a case, from eqns. (15), (Xl), (20) and (Zl),
xn+l
we2
-
=(n
+
w Pl
2,5
o
“2
= (n+ l)(n
t
t Pl
I;;,“(1
+2)“”
-
dx
+ CILXs)sncl
H1(“‘1)‘”
Hl
j.J,‘” + u/n - 1
n
(24)
J/J
1
1
IIn
Xn+l
dx
61l11,
i
* F~2"(.?&) +alX2)2n+1
(25)
where in eqn. (25) WC2= W,, and
z&=1-(1-P’“)
(2n +1)/n
a
1 --x2
t
(
(2n + 1)/n
(1
F,, = 1 -
H
-/i-l’n)
a1
1l-
=
(1 -
x2
0
:‘;
a
-
hz
a=
1
1
-
I
l2
=oho
l2
a!--
I-
2Roh2
306
-Equations (24) and (25) are plotted for various values of 01,cxl, a, al, k
and n in Figs. 6 - 9. These ratios increase as h increases and this increase is
enhanced by the pseudoplastic
behaviour of the fluid for k > 1. From Figs.
k.2
--._
---.__
--_
-_
---_
illr--_
k.,
72
;k=O5
00 60
Fig. 6. Variation in W,./ W,, with k for various
0.5;-,
n = l.O;-.--,
n = 1.5.
Fig. 7. Variation
-,
ii = 0.05.
in Wc2/Wp1
0
0 03
007
0 10
014
a
0
values of n (a = 0.01; si = 0.1): ---,
with (Yfor various values of k and Z (n = 0.5): ---,
n =
ii = 0.1;
095
1
i
.
,I6
k:2
077k=,
r
k.05
k-
Fig. 8. Variation in t,z/t,l
---,n=0.5;-,n=l.O;-.-,n=15
Fig. 9. Variation
---,ol
=O.l;-,a1
with k for various values of n (ii1 = 0.1; q
= O.Ol;H1
= 2.0):
. .
in tca/tpl
with 01~ for various values of k and ii, (n = 0.5; IfI = 2.0):
=0.05.
306
7 and 9 the effect of the curvature of the bearing surface is to decrease the
load capacity and the squeezing time.
3.3. Journal bearing *
Consider the case of a squeeze film in a journal bearing when the
journal and bearing are approaching symmetrically
with a velocity V as
shown in Fig. 10. The film thickness 2h is given by
2h = c(1 -e
cos0)
(26)
where c = R, - rl is the clearance and E = e/c is the eccentricity
ratio.
The flow flux Q can be obtained from eqn. (8) by replacing x by r,B .
This flux can also be obtained by using the equation of continuity as [ 231
Q = 2bVrl sin0
(27)
which on using eqn. (8) gives
-dp = -r 1n+lmlV” sin”0
d0
where F0 and h are defined in eqns. (9) and (26) respectively.
Integrating eqn. (28) and using the boundary condition
(28)
p = 0 at e = 7712
gives the pressure
p(B) = fi2[rln’Iml(
e
The load capacity
Wj2 = 2br,
Fig. 10. Squeezing
(29)
distribution
as
‘g
Wjz is given by
n/2
p cos6’ de
$
in a journal
bearing.
V sinB)qc(l
_ ~cosO)~~il--$]
d0
(30)
307
which on using eqn. (30) gives
Wj, = BlV”II
(31)
where
n/2
II =
F21
sin” + 9
de
oI F21”(1-E COSe)2n+l
1-
=
1i
(1- A?-l’n) 1 -
2
11-e
a2
1 I
2 2n+1 2n+l”
2a
=
case
(Zn+l)/n
-
B1 = 2bm1r,
n )
c
n+li;l
(
The squeezing time tja for the surfaces to approach from the initial
concentric position (e = 0) to a final eccentric position (E = cl) is given by
Ill”’ de
For k = 1 the expressions
are written as
7712
Wjl = BIV”
(32)
for load capacity
Wjr and squeezing
time tjl
sin”+l0
de
s
o (1 -E C0Se)2n+1
(33)
(34)
When a = 6h, the ratios Wj2/Wj, and tj2/tjl dre Ofthe same form as in
eqns. (17) and (18) and give the same results.
To assess the interaction of the geometry of the surface and the nonnewtonian behaviour of the lubricant on load capacity and time of squeezing,
the expressions given by eqns. (31) and (32) are compared with the corresponding eqns. (15) and (16) for parallel plates having the same projected
area and the same minimum film thickness. Thus taking
h = i(l-E)
l=r
h2 = +l)
in eqns. (15) and (16) and using eqns. (31) and (32),
3. =(1 -#n+l(n
+ 2)11
(35)
Pl
tj2
-
t Pl
=
(n +
2)lln
!$i'r,""
where Wj2 = Wpl in eqn. (36).
de/j($--r+l’n
- 11
(36)
308
Equations (35) and (36) are plotted for various values of E, Ed, iii, h
and n in Figs. 11 - 14. From Figs. 11 and 13 these ratios increase as k
increases and the increase is enhanced by the pseudoplastic
behaviour of
k.2
\
/ /
08 0 50
071
I 13
0 92
I34
, 50
ii-+
01
1
1
0 17
0 24
1
0 31
<
Fig. 11. Variation in Wjz/Wpl with k for various
II = 0.5;-,
n = l.O;-.-,
n = 1.5.
Fig. 12. Variation in Wjz/Wpl
0.1; -,
a = 0.05.
0 72
-. .
0 38
k=I
,k=OS
0 45
-i,
values of n (is = 0.05; E = 0.3): ---,
with c for various values of k and 0 (n = 0.5): ---,
0 =
bk.2
k;l
>kZO
5
Fig. 13. Variation in tjz/tpl
n = 0.5;---,
n = l.O;-.--,
Fig. 14. Variation in tj2/tpl
0.1; -)
a, = 0.05.
with k for various values of n (61 = 0.05; ~1 = 0.3): - - -,
n = 1.5.
with ~1 for various
values of k and aI (n = 0.5): --
-, al =
5
309
the fluid for k > 1. From Figs. 12 and 14 the effect of the eccentricity
is to decrease the load capacity and the squeezing time.
ratio
3.4. Parallel circular plates
Consider squeezing between two parallel circular plates of film thickness 2h, approaching each other symmetrically
with a velocity V, as shown
in Fig. 15. The flow flux is given by eqn. (8) by replacing b by 2m. This
flux can also be obtained by using the equation of continuity
as [ 221
Q = 4nr2V
(37)
which on using eqn. (8) gives
vr n 1
dr,
/2n+l
-z-_-m
1
dr
2n+l
(38)
-
I
2n
F,
h 1
Ii
where F, is defined by eqn. (9).
Integrating eqn. (38) and usingp
the approaching surfaces, gives
= 0 at r = R, where R is the radius of
(3%
The load capacity
WPc2 is defined
by
R
WPC2
=
J
2nrp dr
0
which on using eqn. (39) gives
(40)
The elapsed time tpc2 to reduce the film from an initial film thickness
to a final thickness 2h2 is
t PC2
2n
= 7
+ 1
(
+3)/n
R PI
<
(2 $--~[FQh(2:+l),tl
dh
(41)
R-
”
km
)
__----__-
__---
--_-_-z
--_---_-_-_-z--_-_7--_
--:
2h
ml
BY
Cl
km,
---I-I---:--cc_-_-----
2h,
_
__-.
_-_-z----.7
”
Fig. 15. Squeeze film between parallel circular plates.
310
For k = 1, the expressions
tPCl are written as
WPC1
t PC1
=
=
for load capacity
WPcl and squeezing
7rm1
2n + ’ R’n+3”n(~
2(n + 1)
time
(42)
+--,“‘(
&(,?+I),” -h,cn:l,,nj
(43)
For a = 6h, the ratios Wp,JWpCl and tpc8/tpcl are of the same form as
in eqns. (17) and (18) and therefore the same results are also valid.
3.5. Spherical surfaces
Consider the case of a symmetrical squeeze film between two spherical
surfaces as shown in Fig. 16. The film thickness 2h is given by
h=h,,+
-
r2
(44)
m0
where 2ho is the minimum film thickness and R. is the radius of curvature
of the approaching surfaces.
Following the same procedure as in Section 3.4, we can write the load
capacity W,, and the squeezing time ts2 as
(45)
(46)
where R is the radius of the projected circular area and F, and h are defined
in eqns. (9) and (44) respectively.
Fork = 1, eqns. (45) and (46) take the form
(47)
(43)
Fig. 16. Squeezing between spherical surfaces.
311
For a = 6h the ratios W,,/W,, and tst/tsl take the same form as in eqns.
(17) and (18) and hence the same results are obtained.
To assess the effects of the interaction of the geometry of the surface
and the non-newtonian
behaviour of the lubricant on load capacity and time
of squeezing, eqns. (45) and (46) are compared with the corresponding
eqns.
(40) and (41) for parallel circular plates having the same projected area and
the same minimum film thickness. From eqns. (40), (41), (45) and (46),
ws2
-
WPC1
X
=(n
t
+ 3)j
o
3)l’n
Hr(” + l)‘”
n
t PC1
(49)
dx
Fs1”(1+ pX2)2n+1
(n + l)(n +
-=s2
n+2
x
HI’” +1)/n - 1
x s"' ?,~(~~~~~~2)2~+l~~1'n
1 1
cwJ
(50)
where W,, = WpClin eqn. (50) and
Cl
l-
Fs2 = 1 - (1- k-l’“)
Ho+
x=
I
H1 =
2
a1
-
R
a=
a
RZ
moho
Ho=
2
a
h2
ho
p=
=
01x2
pl=
RZ
mob
Equations (49) and (50) are of the same form as eqns. (24) and (25)
and hence the same results are obtained as with cylindrical surfaces.
3.6. Spherical bearing
Consider squeezing between two eccentric spherical surfaces of radii
rl and R1 which are approaching each other with velocity V, as shown in
Fig. 17. The film thickness 2h is given by
2h = ~(1 -E
cos0)
(51)
rl
is
the
clearance
and
E
=
e/c
is
the
eccentricity
ratio.
where c = R1 The flux in this case can be obtained by substituting b = 2nrl sin 0 in
eqn. (8) and also by using the equation of continuity
as [ 231
Q = 2nVr12 sin28
(52)
312
Fig. 17. Squeeze film in a spherical bearing.
which on using eqn. (8) gives
-dp = -r
de
1‘+lmlVn
1
sin”0 F nhan+l
0
2n+l”
i
2n
(53)
!
where F. and h are defined in eqns, (9) and (51) respectively.
Integrating eqn. (53) and using the boundary condition
p=O
at
.9=x/2
gives the pressure distribution
as
2n+l
1
F,”
t
The load capacity
(54)
W,,, is given by
7712
WSP2
de
1
P sin 0 cos
= 2c2$
e de
0
which on using eqn. (54) gives
Wsp2
(55)
= B,V”I2
where
and F,, is defined
in eqn. (31).
313
time tspZ for the surfaces to approach from the initial
(e = 0) to a final eccentric position (E = el) is given by
The squeezing
concentric position
c
tSP2
= 2
l/n
E,
i-1 J
B2
121’n de
WSP2
For k = 1 the expressions
t spl are written as
r/2
W SPl =B2V”J
o
(56)
0
~
(1 -e
for load capacity
sin” + 28
COSe)z”+l
Wspland squeezing time
d9
(57)
(53)
When u = Sh, the ratios Wsp2/Wsp1 and tsp2/tsP1 are of the same form
as in eqns. (17) and (18) and give the same results.
To assess the interaction of the geometry of the surface and the nonnewtonian behaviour of the lubricant on load capacity and time of squeezing
the expressions given by eqns. (55) and (56) are compared with the corresponding eqns. (40) and (41) for circular plates of the same projected area
and minimum film thickness. Thus, taking
h=
:(1--e)
hl = ;
h,=;(l-el)
.
in eqns. (42) and (43) and using eqns. (55) and (56),
W SP2
W PC1
-
t SP2
-
t PC1
= (n
= (n
+
+
3)(1--,)2”+112
3)“”
yd’1;‘”
(5%
de/~(-&-~+l’n
- 11
(60)
where Wsp2 = Wpclineqn. (60).
Equations (59) and (60) are of the same form as eqns. (35) and (36)
and hence give the same results as for journal bearings.
4. Squeeze film between compliant solids
The analysis of squeeze films presented above is applicable to stiff solids.
When the surfaces have low elastic moduli, as in the case of rubber, the
lubrication mechanism is modified owing to the large elastic deformation
caused by pressures which are low enough to leave the viscosity unaltered
[ 241. Though some investigations have been carried out using newtonian
lubricants [ 24 - 321, little attention has been given to non-newtonian
314
lubricants [33]. The effects of consistency variation and non-newtonian
behaviour on the characteristics
of squeeze films between compliant solids
can be studied following the procedure suggested by Fein [ 311.
In contact between compliant cylinders the elastic.effects
can be taken
into account by assuming the length Id of the deformed contact zone under
the applied load W to be the same as in dry contact [34], i.e.
R2
1
-=
=
!$
1
-
(61)
--v2
El
E
where r2 is the radius of each cylinder, v the Poisson ratio and E, the Young’s
modulus of the compliant surfaces. Similarly, in elastic contact between
two spherical surfaces, the radius rd of the deformed contact zone under the
applied load W can be written as [31, 321
where
R2=
$
and r2 is the radius of each sphere.
4.1. Ret tangular plates model (cylinders)
If we consider squeezing between two compliant cylinders, the
deformed zone may be approximated
in the form of rectangular plates, as
shown in Fig. 18. The load capacity W,, can be written from eqns. (13) and
(61) as
Fig. 18. Squeezing between compliant cylinders.
(63)
where F. is defined in eqn. (9).
The time tE2 of squeezing for the surfaces to approach from an initial
position 2hI to a final position 2h2 is given by eqns. (14) and (61) as
To assess the interaction
of the non-newtonian
behaviour of the
lubricant and the elasticity of the compliant surfaces the ratio tE2/tp2 can
be written from eqns. (14) and (64) as
(65)
From eqn. (65), the squeezing time increases as the elastic modulus E
decreases and as the radius R2 of curvature increases. From this equation this
ratio is greater for n = 0.5 than for n = 1.5. Hence, the above-mentioned
increase is enhanced owing to the pseudoplastic
behaviour of the fluid.
For (I = 6/z, eqn. (64) can be integrated and the squeezing time tE2 and film
thickness h2 written as follows for h2/hl Q 1:
(66)
t
h2 =
X
(67)
where
FE = 1 -
(1
-k-l’“){1
-
(1
+j)(2n+l)/n}
(68)
From eqns. (66) and (67) the time of approach and film thickness
increase as the compliant surface becomes more elastic (i.e. E decreases).
These quantities also increase owing to the radius of curvature and the load
as the elastic deformation
gives a larger load-bearing area. From eqn. (68),
FE decreases as k and6 increase. Thus from eqns. (66) and (67) the squeezing
time and film thickness increase as the thickness and consistency
of the
peripheral layer fluid increase for k > 1.
For no consistency variation, i.e. k = 1, the corresponding
expressions
for the load capacity WEI, the squeezing time tEl and the film thickness h2
can be written from eqns. (63), (66) and (67) respectively
as follows for
h2/hl < 1:
(69)
X
(71)
which are of the same form as obtained by Chandra [33]. From these
equations, for n = 1, the same results as discussed by Fein [31] are obtained.
4.2. Parallel cirqlar plates model (spheres)
We consider squeezing between two compliant spheres, the deformed
zone of which may be approximated
in the form of two parallel circular
plates as shown in Fig. 19. The load capacity W,, can be written from eqns.
(40) and (62) as
w,, = ;;;
--
(2% !J(;)z”1rd”+3
(72)
where F. is defined in eqn. (9).
The time tE2 of squeezing for the surfaces to approach from an initial
position 2h, to the final position 2h2 is given from eqns. (66) and (62) as
t,, = y
(73)
(~~~w~~/3(~~+3)‘“(l.15)“+s)~.iii/n~~h~~+”,”
t
To assess the interaction
of the non-newtonian
behaviour of the
lubricant and the compliance of the surfaces, the ratio tE2/tpc2 can be
written from eqns. (73) and (41) as
(74)
Fig. 19. Squeezing between compliant spheres.
317
Equation (‘74) is of the same form as eqn. (65) and hence the same
conclusions are obtained. For u = 61a,eqn. (73) can be integrated and the
squeezing time tE2 and film thickness h2 can be written as follows for
h2/ftl Q I:
From eqns. (75) and (76), the squeezing time and the film thickness
increase with iz, S and W,,.
For no consistency variation, ie, R = 1, the load capacity W& , squeezing
time tsl and film thickness la2 can be written from eqns. (72) and (73) as
follows for h2/hl 4 1:
which are of the same form as obtained by Ch~dra [33J. Frum these equations, for n = 1, the same results as discussed by Fein [31] are obtained.
5. Conclusion
The characteristics of various squeeze film b~~~~s with a power law
lubricant were investigated by considering the effects of consistency
variation.
With stiff solids the effect of a high consistency layer adjacent to the
bearing surface is to increase the load capacity and squeezing time, These
parameters also increase as the thickness of the high ~onsis~~cy peripheral
layer increases, This increase is enhanced by the pseudoplastic behaviour of
the fluid. The effect of bearing curvature with variable film thickness is to
decrease the load capacity and time of squeezing compared with the case of
constant film thickness with the same projected area and the same rn~n~um
film thickness.
318
For cylindrical and spherical surfaces with a compliant layer, the film
thickness increases owing to the compliance of the surface and the load and
conformity
of the surfaces. It also increases because of the high consistency
layer present at the bearing surface and this increase is enhanced by the
pseudoplastic
behaviour of the fluid.
Nomenclature
a
b
;
2h
2ho
2hl
k
‘d
ml
n
P
Q
r
ri
r’2
rd
R
Ro
Rl
u, u
V
w
x, Y
V
thickness of the peripheral layer
width of the bearing or length of the journal
clearance width in journal bearing or spherical bearing
Young’s moduli of the materials for compliant solids
film thickness
minimum film thickness between cylindrical surfaces
initial film thickness
consistency ratio of the peripheral layer to the middle layer
length of the deformed zone for compliant cylinders
consistency of the central layer
flow behaviour index
hydrodynamic pressure
flow flux
radial coordinate
radius of the journal or radius of the spheres
radius of the cylinders or spheres for compliant solids
radius of the deformed contact zone of spherical surfaces
radius of the circular plates
radius of the cylindrical surface or spherical surface
radius of the bearing
velocity components in the x, y directions
normal velocities of the approaching surfaces
load capacity
coordinate system
Poisson’s ratio of the compliant surfaces
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