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Hydrodynamic behaviour of two-phase (liquid-solid) lubricants
Wear, 66 (1981) 223 - 240 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
HYDRO~YN~IC LUBRICANTS
223
BEHAVIOUR OF TWO-PHASE (LIQUID-SOLID)
ALBERT E. YOUSIF and SOMER M. NACY Mechanical Engineering Deparfmenf, (Received
January 11,198O;
University of Baghdad, Baghdad (Iraq)
in revised form March 27,198O)
Summary An analytical hy~odyn~ic model was developed to predict the behaviour of a simple two-phase (liquid-solid) lubricant. An experimental study was also carried out with such lubricants using conventional full-film hydrodynamic bearings. Both the analytical and the experimental results showed #at the presence of suspended solid particles in a ne~oni~ lubricant enhanced the load-carrying capacity and reduced friction.
1. Introduction Oils in service become contaminated with solids in the form of wear debris, soot and sludge. These are usually held in suspension and may act as thickening agents. The thickening power of a suspension of rigid spherical particles can be expressed by the Einstein relation rf = ~~(1+ 2.5~) where n is the viscosity of the suspension, q. the viscosity of the carrier oil and c the volume concentration of the particles. It can thus be appreciated that ~on~inat~g solids act to enhance the viscosity of oils. However, the detailed rheological behaviour of such oils is not known, particularly when they contain added polymers. Recent work [l] has shosvn that non-newtonian viseoelastic effects commence even at very low percentage condensations of solid particles in a newtonian fluid. Particles in a dilute two-phase system of suspended solids in a newtonian lubricant contained in a wedge when subjected to a velocity gradient will undergo translational and rotational motion and, if they are deformable, their shape will change. This has been shown expe~men~ly by Milne [ 2 ] using white mineral oil containing aluminium flakes in a Perspex model of a slider bearing. The flakes rotated slowly during passage through the bearing, whereas particles dispersed in grease became oriented almost parallel to the bearing surface. Mason and Bartok [3] stated that deformation and rotation have an effect on viscosity. When the particles are asym-
224
metrical, rotation can cause them to assume preferred orientations and thus cause thixotropy. Translational movements cause particles to collide with one another and in certain cases to coalesce to form aggregates. In this paper the hydrodynamic effects of solids in oil are studied both analytically and experimentally. Owing to the mathematical complexities of the problem an approximate hydrodynamic solution was first obtained and then an analysis was made of a single particle moving in oil contained in a wedge-type bearing. The problem was reduced to five non-linear algebraic equations, the analytical solution of which was found to be extremely difficult, if not impossible. Thus a numerical solution was obtained using a digital computer. The results showed that there was an increase in the loadcarrying capacity and a decrease in the friction coefficient for a bearing containing a single particie compared with a bearing containing no particles. The experimental investigation of the hydrodyn~ic effects of solid particles in oil was carried out on a htichell pad thrust bearing and on journal bearings. The experimental results confirmed the theoretical predictions regarding both the load-carrying capacity and the friction coefficient. It is hoped that the work will aid the understanding of the behaviour of contaminated oils and worked greases in mechanical bearings [4]. It should also aid the future development of improved oils containing the optimum particle type, size, shape and distribution.
2. Analytical
hydrodynamic
model for a two-phase
lubricant
of the problem The formulation of an analytical hydrodynamic model for the system shown in Fig. l(a) is a formidable task because of the complexity of the problem. Five parameters, namely particle velocity, shape, size, distribution and elasticity, require consideration, A particle may have three velocity components (Fig. l(b)), a tr~~tion~ component u in the direction of the bearing, a vertical component u perpendicular to the axis of the bearing and a rotational component 52. From a mathematical point of view the particles can take the shape of spheres, double cones, ellipsoids or rods. They can be taken to be smaller than the minimum film thickness of the bearing, and their distribution can take any configuration provided that all particles 2.1. Nu ture
-U
(a) Fig. 1, (a) Slider bearing with small particles suspended components of a particle suspended in oil.
0) in the oil; (b) the three velocity
225
are separated by the liquid lubricant. Particles can deform and change shape as a result of pressure and velocity gradients. However, to a first approximation they can be assumed to be rigid. Such considerations indicate the complexity of the problem. Hence any attempt to include all five parameters of the problem in determining the characteristics of a bearing is extremely difficult. 2.2. Basic equations The Reynolds equation
[ 51 can be written
ur -u2
ax
2 Integrating
ah
(1)
+ v2 --vi
twice with respect to x gives
(2) where A and B are constants tance x is given by
of integration.
The film thickness
h at any dis-
h=Hi-aX
(3)
dh = -0~ dx
(4)
Then
where a=
Hi --Ho
Substituting
(5)
L in eqn. (2) and applying
h = Hi
p = Pi
atx=O
h=H,
p=P,
atx=L
the boundary
conditions
of Fig. 2(a)
gives
(a)
(b)
Fig. 2. (a) The velocity components of the two surfaces of a sliding bearing; (b) a slider bearing with M laminae.
226 A=--...-__
H,
HiHo
Hi2 Ho2
+H
1
t”l
-u2)
-6qLtH_
0
+ H
l(P,-po)
L
0
LHi --H7(V2 I
-
L2W2
-VI)
(7)
(Hi - Ho2)(Hi + Ho)
+
The load-carrying
w=
(6)
Vl)
0
capacity
of the bearing is given by
11‘pdx
0
Inserting the expressions for A and B, substituting eqns. (3), (4) and (5) in eqn. (2), integrating, collecting Hi/Ho by n gives w = yJ1
-
In n
u2w2
2
I (n-l)2-n2-1
K12
for h, dh and x from terms and replacing
I
2 ln n - 12n( V2 - V1)L3 _~ I (n - 1)3 - (n - l)2(n + 1) t + HCJ3 n -Pi i n+l
+L
1 + -Pp, n+l
(8)
1
The centre of pressure, 2, is obtained wz
=
from
jpxdx 0
=
6v(U,
- U2)L3
n(n + 2)lnn - 0.5(n - 1)(5n + 1) (n - l)3(n + 1)
Ho2 - 1277(V2 - V1)L4 Ho3
n(n + 2)In n + 0.5(n - l)( 5n + 1) + (n - l)(n + 1)
+ L2p, n21nn - n2(n - 1) + 0.5n2(n - 1)2 L
_L2P
n21nn-n2(n-l)+0.5(n-l)2 0
The friction h, are given by
(n - l)3(n + 1) (n - l)3(n + 1)
(9)
forces on both surfaces of the bearing, i.e. at y = 0 and y =
227 L
F y’o =
Ty=o dx
J
0
L
FYzh
s
=
dx
‘fyph
0
A newtonian fluid is assumed, i.e. du T=% where the term du/dy at both values of y is obtained from the velocity distribution
3. -dp
u=2q
h-Y
--&I +- h
Y(Y
&
Integrating for FY =o and FY=h Ul
FYzo=-
-
H
u2
v2
$z+, =
Ul
4
n--l
0
-677
rlL -
gh?S
-
lnn--
6 n+l *a
vr Inn
f&z
-
% + 6~(V2
t
2 7IL -Inn--t n-1
u2
-V11L2
H,2
+-
n+
l(Pi-po)
(11)
6 n+l 2
1-
n2-1
1
- (n-1)2
The oil flow through the bearing is given by Q,
=judy 0
Substituting for u from eqn. (10) and integrating with the assumption that the oil flow will be the same at all values of x if the side leakage is neglected gives Q,
=
H
O
Ul
~ l_
6qL
+ u2
2
n2H03 n+l
n-H,ul;U,n>;+
(131
2.3. Hypothetical modei 1 Let us suppose that a quantity of suspension is introduced between two infinite sliders and that the solids become redistributed by some mechanism
228
into M laminae as shown in Fig. 2(b). If h, = H,/(l mth lamina is given by
M+l-m ~=-----M+l
+ M), the velocity
of the
u
similar. Regions 1,2, . . . , M + 1 are assumed to be geometrically It can be concluded from eqns. (8) and (11) that for sliding motion only 1
1 Wa-$'aho2
ho
so for M + 1 regions the total load-carrying force F, are W
t
capacity
W, and the total friction
!!!
=K
l Ho2
where K1 and K2 are constants of proportionality that can be determined from eqns. (8) and (11) respectively. The coefficient of friction is given by Ft pt=W=-
K2 t
Ho -_=&
-
HO
M+l
KIM+1
where K3 = K2/K1 is a constant. For squeezing motion only, eqns. (8) and (11) give
Wt =K, y Ft =K, Ho2 pt
K5 =-_
Ho --=K6M+1
HO
K4M+l where K4 and K5 areconstants of proportionality and KG = K5/K4. This simplified solution was analysed by solving the following example: Wt =lON
V
U = 1000 cm s-l
= 100 cm s-l
77 =0.8P
n=Hi/Ho-= 2
L=5cm
The results are shown in Figs. 3(a) and 3(b) for sliding and squeezing respectively.
motion
229
co ;
lr~
ir
%
3 “I-
x
‘s
?
$
”
m 2 20.
2cI.
15
l( )_
00
2
4
h
Number
6 of
10 tamhat
12
11
16
10
20
o-12
0
H
I‘ Number
at iomime
16
16
H
lb) Fig, 3. The variations in the friction coefficient /A,the friction force Ft, the thickness H, of the outlet film and the thickness h, of the outlet film between the laminae for (a) sliding and (b) squeezing motion.
2.4.
Hypothetica
model 2
Let us consider the model of a weightless particle and a plane slider bearing shown in Fig. 4(a). Each oil film exerts a tangential (frictional) force and a normal (lift) force on the region of the particle surface that it contacts. These forces are resolved into two component in the x direction and the y direction. There are also two forces acting on the ends of the particle. If the acceleration of the particle is zero and there are no other forces acting on it, then equilibrium requires that (a) the sum of the forces in the x direction is zero, (b) the sum of the forces in the y direction is zero and (c) the sum of >/----
RcaPalntlOl
fs$=$+
TL (a)
PC
%___/
(b)
Fig. 4. (a) Geometrical representation of one particle suspended in the oil; (b) the forces acting on the particle.
230
the moments acting on the particle is zero. Continuity of flow requires that (a) the flow to the left of the particle equals the flow to the right of the particle and (b) the flow to the left of the particle is equal to the sum of the flow above and below the particle. The rotational velocity R of the particle is neglected. Five equations are obtained from these five requirements (Fig. 4(b)). The equilibrium equations are ZF,=O (P, -Pd)t
cos Cp+ (F, - Fb) cos # + (W, - IV,) sin Q1= 0
(14)
sin Cp+ (W, - W,) cos $ = 0
(15)
CF, =0 (Pd -P,)t
(PL?-Pd)
sin #I + (Fb -F,)
f + W,r7, - W,rl, -F,_,t=O
and the continuity
of flow equations
W-5)
are
8, = Qtj
(17)
Q, = 9, + Qt,
(13)
Hydrodynamic theory can be applied to each region in the bearing (i.e. regions A, B, C and D). Equations (14) - (18) can be simplified by using eqns. (8), (9) and (11) - ( 13) and assuming that (a) the thickness t of the particle is very small (i.e. t2 = 0), (b) the angle # of inc~nation of the particle is very small (i.e. 9 = sin I# = tan 4) and (3) the vertical velocity of the bearing is neglected. These equations contain five unknowns, the translational velocity u of the particle, the vertical velocity u of the particle, the height Ii of the centroid of the particle from the slider, the angle Cpof inclination of the particle and the length L, from the entrance of the bearing to the head of the particle. The length L of the bearing, the angle cy of inclination of the bearing, the inlet film thickness iYi, the outlet film thickness N,, the length B of the particle, the thickness t of the particle and the pressures Pa and Pd at opposite ends of the particle are known. All the known values except P, and Pd are constants and can be assigned numerical values. Pa and Pd are calculated in the program as they change in magnitude along the bearing. The five non-linear algebraic equations obtained cannot be solved analytically, so they were solved numerically by substituting numerical values for H, $J and L, . The unknown values u and u were found, and therefore the total load-carrying capacity Wt and the total friction force Ft could be calculated for small # and CYfrom w, = w, + w, + w, + Wd
(19)
Ft = F, + F, + Fd
(20)
The friction
coefficient
pt was calculated
from
231
The results of these calculations are presented in the following dimensionless forms: dimensionless load-carrying capacity W* = WJW; dimensionless friction force F* = Ft /F; dimensionless friction coefficient /.J* = pt /cc; dimensionless translational velocity U* = u/U; dimensionless vertical velocity V* = u/U. W, F and p are the load-carrying capacity, the friction force and the coefficient of friction respectively of the bearing in the absence of particles and U is the sliding velocity of the slider. Two special cases which could not be solved by the main computer program arise from the solution of this problem, so two special programs were written to allow a logical solution. The first special case is when the angle @Iof inclination of the particle is zero and hence both W, and F, are zero. Therefore wt = w, + w, + w,
(21)
Ft
(22)
= F, + F,,
The second special case is when the angle 4 of inclination of the particle is equal to the angle CYof inclination of the bearing. In this case Wb and Fb will equal zero and therefore wt = w, + w, + Wd
(23)
Ft = F, + F, + Fd
(24)
A numerical example computer program: L
B=4cm
=lOcm
Hi = 0.1 cm
including
= 1000 cm s-l
n
= Hi/H,
data was solved using the
t = 0.01 cm
H, = 0.05 cm
U
the following
7) = 0.8 P
CY= 0.005 rad
= 2.0
Some of the main results for the variations in W* and cc* with the tilt ratio @/CY for various positions LJL of the particle and various height ratios H/H, are presented in Figs. 5 and 6.
3. Experimental
results
Experiments were carried out on three different full-film hydrodynamic test rigs, a Michell pad thrust bearing, a journal bearing for pressure distribution measurement and a journal bearing for friction measurement. The tests were performed using various volume concentrations of carbon black in a mineral oil. 3.1. Michell pad thrust bearing The apparatus consisted essentially of a plane aluminium square pad of dimensions 10 cm X 10 cm which could be positioned accurately relative to a
232
Fig. 5. The variation in the dimensionless load with the tilt ratio for different ratios:(a) La/L = 0.0;(b) La/L = O.l;(c)L,/L = 0.2;(d) La/L = 0.3.
height
moving plastic-coated belt. The main body of the apparatus consisted of an aluminium casting carrying two steel drums which in turn carried a plasticcoated belt. One drum was driven by a variable-speed a.c. motor and the other was carried in a slotted mounting to permit adjustment of belt tension. The drum surfaces were serrated to minimize belt slip and the drums were crowned to ensure correct tracking of the belt. The apparatus was located in a plastic tank containing oil to a level such that the lower part of the belt was submerged. The belt travelled along the machined top surface of the main casting which was slotted to drain off any excess oil. The pad was supported on two eccentric shafts fitted with hand wheels which carried inside members bolted to the main frame of the apparatus. The clearance between the pad and the moving belt was measured by two micrometers located in line with the leading and trailing edges of the pad respectively. Typical values of the clearance ranged from 0.5 to 2.0 mm. It was thus several hundred times as great as the clearance in a real Michell
233
00 Tilt
(cl
.OllO
0,
cl?
03
04 Tilt
bid
05 micJ
06
07
06
09
$I<
(d)
Fig. 6. The variation in the dimensionless friction coefficient with the tilt ratio for different height ratios: (a) La/L = 0.0;(b) La/L = 0.1; (c) La/L = 0.2; (d) LJL = 0.3.
pad thrust bearing and in consequence could easily be measured to the required accuracy. The pressure developed between the pad and the moving belt was indicated by 13 graduated tubes secured to the pad. Seven of these tubes were equally spaced along the axis of the slider in the direction of motion, while the remainder were located transversely in a plane which approximately coincided with the point at which maximum pressure was developed.
10
234
The ratio of the inlet to the outlet film thickness was kept constant at 2 in all the runs. Typical transverse and longitudinal pressure distribution curves for various solid concen~ations are shown in Fig. 7(a). The area under the longitudinal pressure distribution curves was calculated using Simpson’s rule. This area is used as a parameter defining the load-carrying capacity. Figure 7(b) shows a plot of this area against the percentage solid concentration for three different speeds. 3.2. Journal bearing for pressure d~tribution measurement The apparatus used consisted of a plain steel shaft (5 cm in diameter) encased in a clear Perspex bearing (5.5 cm in diameter) and directly driven by a small electric motor, The bearing was freely supported on the shaft and sealed at the motor end with a light rubber diaphragm. The clearance was large so that the oil could be seen clearly. The motor speed was precisely controlled and adjusted by an electronic speed control unit and could be run in both directions. The speed was read directly from a calibrated meter in the control unit. The bearing contained 12 equispaeed pressure tappings around its circumference and four along its top side on a vertical radial plane. All the tappings were connected by light flexible plastic tubes to a manometer panel so that the pressure head of oil at all 16 points could be observed clearly in all the tubes. Oil was supplied to a low pressure region at both ends of the bearing by an adjustable Perspex reservoir fitted to the side of the rear panel. The bearing could be loaded by attaching various weights to the arms attached beneath it. After selecting the direction of rotation, the bearing was allowed to warm up under a light load for about 30 min at about 1500 rev min-I. Subsequently, care was taken to ensure that the pressure heads were steady before noting readings for each test speed (stepless variation between 500 and 3000 rev min-I). Typical axial and cir~umferenti~ pressure dist~bution curves for various solid concentrations are shown in Fig. 8. 3.3. Journal bearing for friction measurement The apparatus was arranged for frictional torque measurement under a known applied normal load. It consisted of a motor equipped with a speed control system. The motor rotated a steel shaft 4 cm in diameter by means of a pulley and a V-belt; this acted as a journal rotating in a lubricated bronze bearing (4.2 cm in diameter) which was forced to move by the frictional torque. The frictional torque could be balanced by an opposite torque produced by the balancing .weight wb. A small oil reservoir containing an oil filter and a valve which allowed oil droplets to flow through the bearing was attached to the casing. The normal load W was applied by a lever connected by two strings attached to the casing of the bearing, A thermometer hole in the top of the casing normal to the bronze bearing allowed the bearing temperature, which was approximately equal to the mean temperature of the oil, to be measured.
235 200-
SO1
8
236567 Sfation
(4
0 (b>)
5 Percentage
9
10
6
lo solid
II
12
I3
NO,.
mFlcentration
15 by
volume
Fig. 7. (a) Prcssute distribAon in a Michell pad thrust bearing for various solid concentrationa(U= 11 ems-l;Hi = 1 mm;H, = 0.5 mm); (b) variation in the area under the pressure distributiy curve with the solid concentration (0, U = 11 cm s-l ; 0, U = 15 cm 6-l ; 0, U= 22cms ).
236
Oi, 1
I.-
2
3
4
5 StatIon
No
Fig. 8. Pressure distribution in a journal 1000revmin-1:~,0%;~,2%,~,4%;X,6%;~,8%.
bearing
for various
solid concentrations
at
After a steady state condition had been attained, a balancing load was applied to the lever of the casing in order to make it approximately horizontal. The rotational speed, balancing load and temperature were recorded for each test. The speed of the journal was measured by a stroboscopic method. The viscosity of the oil was measured with an Engler viscometer and plotted as a function of temperature, so that the viscosity at each temperature was known accurately. Figure 9(a) shows the variation in the friction coefficient p against the dimensionless parameter ZN/F where 2 is the viscosity of the oil in centipoises, N is the rotational speed of the journal in revolutions per minute and p is the normal load per unit area in newtons per square centimetre. Figure 9(b) shows the variation in the dimensionless torque T* = T/pN2D4L) with the Reynolds number Re = 2nNRE/v. The particular case of concentric cylinders is plotted by using the Petroff equation T* = (n3/2)ReV1. The variation in the percentage friction reduction with the solid concentration by volume is shown in Fig. 10.
4. Discussion
of results
4.1. Theoretical results The results from hypothetical model 1 shown in Fig. 3 indicate that for both sliding and squeeze actions under constant load, viscosity, speed and
231
20 01
30
ZNtf’ (a)
50
40 Factof
60
70
’ OoMA
X103
12 Reynolds’
number
Re
14
16
i
18
X10
PI
Fig. 9.(a) Coefficient of friction VS.ZN/p for a journal bearing containing various solid concentrations (W = 11.573 N); (b) d imensionless friction torque us. Reynolds number for a journal bearing containing various solid concentrations (W = 7.03 N): 0,O vol.%; 0, 5 vol.%;A, 10 vol.%; v, 15 vol.%.
bearing configuration (a) the outlet film thickness H, increases with increasing number of laminae, (b) the outlet film thickness h, between any two successive laminae decreases with increasing number of laminae and (c) the friction force, and hence the friction coefficient, decreases with increasing number of laminae. Thus, if the wedge has a fixed geometry, then the presence of the laminae will increase the load-carrying capacity. However, if the bearing is allowed to adjust itself under constant load, the friction force will be reduced and there will be a reduction in the dissipated energy. The main results of hypothetical model 2 are shown in Figs. 5 and 6. All the graphs of the variation in the dimensionless load-carrying capacity W* with the tilt ratio #/ol show that W* is greater than unity which implies that the presence of particles in the oil enhances the load-carrying capacity of the bearing. The graphs of the variation in the dimensionless friction coefficient p* with the tilt ratio show that /J* is less than unity which implies that the presence of particles in the oil reduces the coefficient of friction. Experimental results The pressure distribution results given in Figs. 7 and 8 for both the Michell pad thrust bearing and the journal bearing indicate an increase in loadcarrying capacity with increasing solid concentration in the oil. The results 4.2.
Fig. 10. The variation in friction reduction with the percentage solid concentration journal bearing for various normal loads: 0, 7.03 N; 0, 9.3 N; 0, 11.57 N.
in the
obtained from the journal bearing indicate that the coefficient of friction decreases with increasing solid concentration up to a specific value and then increases. This is assumed to be due to the increase in lubricant viscosity, which changes its overall behaviour to that of a non-newtonian viscoelastic material. This change in friction behaviour occurred at a solid concentration of about 10 vol.%. When the solid concentration was kept constant, the coefficient of friction increased with increasing ZIV/puntil it reached the hydrodynamic region and then remained almost constant as shown in Fig. 9(a). The curves of dimensionless torque T* against the Reynolds number Re (Fig. 9(b)) lay above the theoretical Petroff line. As the solid concentration increased to 10 vol.% the curves became closer to the Petroff line but then moved away again as it was increased further. This implies that the value of the frictional torque decreased owing to the presence of solid particles. These curves also show that T* decreases as the Reynolds number increases. When the speed of rotation N increases, T* decreases. Figure 10 shows #at the percentage friction reduction increases with increasing solid concentration up to 10 vol.% and then decreases. This decrease can be attributed to the non-newtonian viscoelastic behaviour of the oil.
239
5. Conclusions In view of the qualitative nature of the experimental and analytical work, the following conclusions can be drawn. (1) When the configuration of the oil film is kept constant, the loadcarrying capacity increases when solid particles are added to the oil. This was confirmed by analytical hypo~eti~~ models. (2) The coefficient of friction was reduced by increasing the solid content of the lubricating oil up to an optimum volume percentage. (3) The experimental results obtained using the various conventional hydrodynamic test rigs confirm the validity of the theoretical results obtained from the qualitative models.
Nomenclature B c
c D E
F F* h
H
Hi HO L
M ; P
P Pi PO
Q
Re t T T* zi u lJ* u V V* W w w* R 2 cl
length of the particle percentage of solid content by volume mean radial clearance bearing diameter modulus of elasticity friction force dimensionless friction force oil film thickness height of the centroid of the particle from the slider inlet film thickness outlet film thickness axial length of the bearing number of laminae ratio of inlet to outlet film thickness rotational speed pressure W/LD, mean pressure of the bearing inlet pressure outlet pressure volumetric flow Reynolds number thickness of the particle torque dimensionless torque velocity in the x direction translational velocity dimensionless translational velocity velocity in the y direction vertical velocity dimensionle~ vertical velocity velocity in the z direction load-carrying capacity (normal load) dimensionless load-carrying capacity distance between the leading edge of the bearing and the oentre of pressure dynamic viscosity (cP) angle of inclination of the pad in the slider bearing
dynamic viscosity coefficient of friction dimensionless friction coefficient kinematic viscosity density shear stress angle of inclination of the particle rotational velocity of the particle
References 1 A. E. Yousif and K. D. Bogie, The rheological behaviour of a new high temperature synthetic grease, Proc., Inst. Mech. Eng., London, (1970). 2 A. A. Milne, A theory of grease lubrication of slider bearings, Proc. 2nd Int. Congr. Rheology, 1958. 3 S. G. Mason and W. Bartok,
on
in C. C. Mills (ed.), Rheology of Disperse Systems, Pergamon, Oxford, 1959, p. 16. 4 A. E. Yousif and J. Hailing, The frictional traction characteristics of greases in heavily loaded contacts, ASLE-ASME Lubr. Conf., Atlanta, October 16 - 18, 1973, Society of Mechanical Engineers, New York. 5 0. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication, McGraw-Hill, New York, 1961.
HYDRO~YN~IC LUBRICANTS
223
BEHAVIOUR OF TWO-PHASE (LIQUID-SOLID)
ALBERT E. YOUSIF and SOMER M. NACY Mechanical Engineering Deparfmenf, (Received
January 11,198O;
University of Baghdad, Baghdad (Iraq)
in revised form March 27,198O)
Summary An analytical hy~odyn~ic model was developed to predict the behaviour of a simple two-phase (liquid-solid) lubricant. An experimental study was also carried out with such lubricants using conventional full-film hydrodynamic bearings. Both the analytical and the experimental results showed #at the presence of suspended solid particles in a ne~oni~ lubricant enhanced the load-carrying capacity and reduced friction.
1. Introduction Oils in service become contaminated with solids in the form of wear debris, soot and sludge. These are usually held in suspension and may act as thickening agents. The thickening power of a suspension of rigid spherical particles can be expressed by the Einstein relation rf = ~~(1+ 2.5~) where n is the viscosity of the suspension, q. the viscosity of the carrier oil and c the volume concentration of the particles. It can thus be appreciated that ~on~inat~g solids act to enhance the viscosity of oils. However, the detailed rheological behaviour of such oils is not known, particularly when they contain added polymers. Recent work [l] has shosvn that non-newtonian viseoelastic effects commence even at very low percentage condensations of solid particles in a newtonian fluid. Particles in a dilute two-phase system of suspended solids in a newtonian lubricant contained in a wedge when subjected to a velocity gradient will undergo translational and rotational motion and, if they are deformable, their shape will change. This has been shown expe~men~ly by Milne [ 2 ] using white mineral oil containing aluminium flakes in a Perspex model of a slider bearing. The flakes rotated slowly during passage through the bearing, whereas particles dispersed in grease became oriented almost parallel to the bearing surface. Mason and Bartok [3] stated that deformation and rotation have an effect on viscosity. When the particles are asym-
224
metrical, rotation can cause them to assume preferred orientations and thus cause thixotropy. Translational movements cause particles to collide with one another and in certain cases to coalesce to form aggregates. In this paper the hydrodynamic effects of solids in oil are studied both analytically and experimentally. Owing to the mathematical complexities of the problem an approximate hydrodynamic solution was first obtained and then an analysis was made of a single particle moving in oil contained in a wedge-type bearing. The problem was reduced to five non-linear algebraic equations, the analytical solution of which was found to be extremely difficult, if not impossible. Thus a numerical solution was obtained using a digital computer. The results showed that there was an increase in the loadcarrying capacity and a decrease in the friction coefficient for a bearing containing a single particie compared with a bearing containing no particles. The experimental investigation of the hydrodyn~ic effects of solid particles in oil was carried out on a htichell pad thrust bearing and on journal bearings. The experimental results confirmed the theoretical predictions regarding both the load-carrying capacity and the friction coefficient. It is hoped that the work will aid the understanding of the behaviour of contaminated oils and worked greases in mechanical bearings [4]. It should also aid the future development of improved oils containing the optimum particle type, size, shape and distribution.
2. Analytical
hydrodynamic
model for a two-phase
lubricant
of the problem The formulation of an analytical hydrodynamic model for the system shown in Fig. l(a) is a formidable task because of the complexity of the problem. Five parameters, namely particle velocity, shape, size, distribution and elasticity, require consideration, A particle may have three velocity components (Fig. l(b)), a tr~~tion~ component u in the direction of the bearing, a vertical component u perpendicular to the axis of the bearing and a rotational component 52. From a mathematical point of view the particles can take the shape of spheres, double cones, ellipsoids or rods. They can be taken to be smaller than the minimum film thickness of the bearing, and their distribution can take any configuration provided that all particles 2.1. Nu ture
-U
(a) Fig. 1, (a) Slider bearing with small particles suspended components of a particle suspended in oil.
0) in the oil; (b) the three velocity
225
are separated by the liquid lubricant. Particles can deform and change shape as a result of pressure and velocity gradients. However, to a first approximation they can be assumed to be rigid. Such considerations indicate the complexity of the problem. Hence any attempt to include all five parameters of the problem in determining the characteristics of a bearing is extremely difficult. 2.2. Basic equations The Reynolds equation
[ 51 can be written
ur -u2
ax
2 Integrating
ah
(1)
+ v2 --vi
twice with respect to x gives
(2) where A and B are constants tance x is given by
of integration.
The film thickness
h at any dis-
h=Hi-aX
(3)
dh = -0~ dx
(4)
Then
where a=
Hi --Ho
Substituting
(5)
L in eqn. (2) and applying
h = Hi
p = Pi
atx=O
h=H,
p=P,
atx=L
the boundary
conditions
of Fig. 2(a)
gives
(a)
(b)
Fig. 2. (a) The velocity components of the two surfaces of a sliding bearing; (b) a slider bearing with M laminae.
226 A=--...-__
H,
HiHo
Hi2 Ho2
+H
1
t”l
-u2)
-6qLtH_
0
+ H
l(P,-po)
L
0
LHi --H7(V2 I
-
L2W2
-VI)
(7)
(Hi - Ho2)(Hi + Ho)
+
The load-carrying
w=
(6)
Vl)
0
capacity
of the bearing is given by
11‘pdx
0
Inserting the expressions for A and B, substituting eqns. (3), (4) and (5) in eqn. (2), integrating, collecting Hi/Ho by n gives w = yJ1
-
In n
u2w2
2
I (n-l)2-n2-1
K12
for h, dh and x from terms and replacing
I
2 ln n - 12n( V2 - V1)L3 _~ I (n - 1)3 - (n - l)2(n + 1) t + HCJ3 n -Pi i n+l
+L
1 + -Pp, n+l
(8)
1
The centre of pressure, 2, is obtained wz
=
from
jpxdx 0
=
6v(U,
- U2)L3
n(n + 2)lnn - 0.5(n - 1)(5n + 1) (n - l)3(n + 1)
Ho2 - 1277(V2 - V1)L4 Ho3
n(n + 2)In n + 0.5(n - l)( 5n + 1) + (n - l)(n + 1)
+ L2p, n21nn - n2(n - 1) + 0.5n2(n - 1)2 L
_L2P
n21nn-n2(n-l)+0.5(n-l)2 0
The friction h, are given by
(n - l)3(n + 1) (n - l)3(n + 1)
(9)
forces on both surfaces of the bearing, i.e. at y = 0 and y =
227 L
F y’o =
Ty=o dx
J
0
L
FYzh
s
=
dx
‘fyph
0
A newtonian fluid is assumed, i.e. du T=% where the term du/dy at both values of y is obtained from the velocity distribution
3. -dp
u=2q
h-Y
--&I +- h
Y(Y
&
Integrating for FY =o and FY=h Ul
FYzo=-
-
H
u2
v2
$z+, =
Ul
4
n--l
0
-677
rlL -
gh?S
-
lnn--
6 n+l *a
vr Inn
f&z
-
% + 6~(V2
t
2 7IL -Inn--t n-1
u2
-V11L2
H,2
+-
n+
l(Pi-po)
(11)
6 n+l 2
1-
n2-1
1
- (n-1)2
The oil flow through the bearing is given by Q,
=judy 0
Substituting for u from eqn. (10) and integrating with the assumption that the oil flow will be the same at all values of x if the side leakage is neglected gives Q,
=
H
O
Ul
~ l_
6qL
+ u2
2
n2H03 n+l
n-H,ul;U,n>;+
(131
2.3. Hypothetical modei 1 Let us suppose that a quantity of suspension is introduced between two infinite sliders and that the solids become redistributed by some mechanism
228
into M laminae as shown in Fig. 2(b). If h, = H,/(l mth lamina is given by
M+l-m ~=-----M+l
+ M), the velocity
of the
u
similar. Regions 1,2, . . . , M + 1 are assumed to be geometrically It can be concluded from eqns. (8) and (11) that for sliding motion only 1
1 Wa-$'aho2
ho
so for M + 1 regions the total load-carrying force F, are W
t
capacity
W, and the total friction
!!!
=K
l Ho2
where K1 and K2 are constants of proportionality that can be determined from eqns. (8) and (11) respectively. The coefficient of friction is given by Ft pt=W=-
K2 t
Ho -_=&
-
HO
M+l
KIM+1
where K3 = K2/K1 is a constant. For squeezing motion only, eqns. (8) and (11) give
Wt =K, y Ft =K, Ho2 pt
K5 =-_
Ho --=K6M+1
HO
K4M+l where K4 and K5 areconstants of proportionality and KG = K5/K4. This simplified solution was analysed by solving the following example: Wt =lON
V
U = 1000 cm s-l
= 100 cm s-l
77 =0.8P
n=Hi/Ho-= 2
L=5cm
The results are shown in Figs. 3(a) and 3(b) for sliding and squeezing respectively.
motion
229
co ;
lr~
ir
%
3 “I-
x
‘s
?
$
”
m 2 20.
2cI.
15
l( )_
00
2
4
h
Number
6 of
10 tamhat
12
11
16
10
20
o-12
0
H
I‘ Number
at iomime
16
16
H
lb) Fig, 3. The variations in the friction coefficient /A,the friction force Ft, the thickness H, of the outlet film and the thickness h, of the outlet film between the laminae for (a) sliding and (b) squeezing motion.
2.4.
Hypothetica
model 2
Let us consider the model of a weightless particle and a plane slider bearing shown in Fig. 4(a). Each oil film exerts a tangential (frictional) force and a normal (lift) force on the region of the particle surface that it contacts. These forces are resolved into two component in the x direction and the y direction. There are also two forces acting on the ends of the particle. If the acceleration of the particle is zero and there are no other forces acting on it, then equilibrium requires that (a) the sum of the forces in the x direction is zero, (b) the sum of the forces in the y direction is zero and (c) the sum of >/----
RcaPalntlOl
fs$=$+
TL (a)
PC
%___/
(b)
Fig. 4. (a) Geometrical representation of one particle suspended in the oil; (b) the forces acting on the particle.
230
the moments acting on the particle is zero. Continuity of flow requires that (a) the flow to the left of the particle equals the flow to the right of the particle and (b) the flow to the left of the particle is equal to the sum of the flow above and below the particle. The rotational velocity R of the particle is neglected. Five equations are obtained from these five requirements (Fig. 4(b)). The equilibrium equations are ZF,=O (P, -Pd)t
cos Cp+ (F, - Fb) cos # + (W, - IV,) sin Q1= 0
(14)
sin Cp+ (W, - W,) cos $ = 0
(15)
CF, =0 (Pd -P,)t
(PL?-Pd)
sin #I + (Fb -F,)
f + W,r7, - W,rl, -F,_,t=O
and the continuity
of flow equations
W-5)
are
8, = Qtj
(17)
Q, = 9, + Qt,
(13)
Hydrodynamic theory can be applied to each region in the bearing (i.e. regions A, B, C and D). Equations (14) - (18) can be simplified by using eqns. (8), (9) and (11) - ( 13) and assuming that (a) the thickness t of the particle is very small (i.e. t2 = 0), (b) the angle # of inc~nation of the particle is very small (i.e. 9 = sin I# = tan 4) and (3) the vertical velocity of the bearing is neglected. These equations contain five unknowns, the translational velocity u of the particle, the vertical velocity u of the particle, the height Ii of the centroid of the particle from the slider, the angle Cpof inclination of the particle and the length L, from the entrance of the bearing to the head of the particle. The length L of the bearing, the angle cy of inclination of the bearing, the inlet film thickness iYi, the outlet film thickness N,, the length B of the particle, the thickness t of the particle and the pressures Pa and Pd at opposite ends of the particle are known. All the known values except P, and Pd are constants and can be assigned numerical values. Pa and Pd are calculated in the program as they change in magnitude along the bearing. The five non-linear algebraic equations obtained cannot be solved analytically, so they were solved numerically by substituting numerical values for H, $J and L, . The unknown values u and u were found, and therefore the total load-carrying capacity Wt and the total friction force Ft could be calculated for small # and CYfrom w, = w, + w, + w, + Wd
(19)
Ft = F, + F, + Fd
(20)
The friction
coefficient
pt was calculated
from
231
The results of these calculations are presented in the following dimensionless forms: dimensionless load-carrying capacity W* = WJW; dimensionless friction force F* = Ft /F; dimensionless friction coefficient /.J* = pt /cc; dimensionless translational velocity U* = u/U; dimensionless vertical velocity V* = u/U. W, F and p are the load-carrying capacity, the friction force and the coefficient of friction respectively of the bearing in the absence of particles and U is the sliding velocity of the slider. Two special cases which could not be solved by the main computer program arise from the solution of this problem, so two special programs were written to allow a logical solution. The first special case is when the angle @Iof inclination of the particle is zero and hence both W, and F, are zero. Therefore wt = w, + w, + w,
(21)
Ft
(22)
= F, + F,,
The second special case is when the angle 4 of inclination of the particle is equal to the angle CYof inclination of the bearing. In this case Wb and Fb will equal zero and therefore wt = w, + w, + Wd
(23)
Ft = F, + F, + Fd
(24)
A numerical example computer program: L
B=4cm
=lOcm
Hi = 0.1 cm
including
= 1000 cm s-l
n
= Hi/H,
data was solved using the
t = 0.01 cm
H, = 0.05 cm
U
the following
7) = 0.8 P
CY= 0.005 rad
= 2.0
Some of the main results for the variations in W* and cc* with the tilt ratio @/CY for various positions LJL of the particle and various height ratios H/H, are presented in Figs. 5 and 6.
3. Experimental
results
Experiments were carried out on three different full-film hydrodynamic test rigs, a Michell pad thrust bearing, a journal bearing for pressure distribution measurement and a journal bearing for friction measurement. The tests were performed using various volume concentrations of carbon black in a mineral oil. 3.1. Michell pad thrust bearing The apparatus consisted essentially of a plane aluminium square pad of dimensions 10 cm X 10 cm which could be positioned accurately relative to a
232
Fig. 5. The variation in the dimensionless load with the tilt ratio for different ratios:(a) La/L = 0.0;(b) La/L = O.l;(c)L,/L = 0.2;(d) La/L = 0.3.
height
moving plastic-coated belt. The main body of the apparatus consisted of an aluminium casting carrying two steel drums which in turn carried a plasticcoated belt. One drum was driven by a variable-speed a.c. motor and the other was carried in a slotted mounting to permit adjustment of belt tension. The drum surfaces were serrated to minimize belt slip and the drums were crowned to ensure correct tracking of the belt. The apparatus was located in a plastic tank containing oil to a level such that the lower part of the belt was submerged. The belt travelled along the machined top surface of the main casting which was slotted to drain off any excess oil. The pad was supported on two eccentric shafts fitted with hand wheels which carried inside members bolted to the main frame of the apparatus. The clearance between the pad and the moving belt was measured by two micrometers located in line with the leading and trailing edges of the pad respectively. Typical values of the clearance ranged from 0.5 to 2.0 mm. It was thus several hundred times as great as the clearance in a real Michell
233
00 Tilt
(cl
.OllO
0,
cl?
03
04 Tilt
bid
05 micJ
06
07
06
09
$I<
(d)
Fig. 6. The variation in the dimensionless friction coefficient with the tilt ratio for different height ratios: (a) La/L = 0.0;(b) La/L = 0.1; (c) La/L = 0.2; (d) LJL = 0.3.
pad thrust bearing and in consequence could easily be measured to the required accuracy. The pressure developed between the pad and the moving belt was indicated by 13 graduated tubes secured to the pad. Seven of these tubes were equally spaced along the axis of the slider in the direction of motion, while the remainder were located transversely in a plane which approximately coincided with the point at which maximum pressure was developed.
10
234
The ratio of the inlet to the outlet film thickness was kept constant at 2 in all the runs. Typical transverse and longitudinal pressure distribution curves for various solid concen~ations are shown in Fig. 7(a). The area under the longitudinal pressure distribution curves was calculated using Simpson’s rule. This area is used as a parameter defining the load-carrying capacity. Figure 7(b) shows a plot of this area against the percentage solid concentration for three different speeds. 3.2. Journal bearing for pressure d~tribution measurement The apparatus used consisted of a plain steel shaft (5 cm in diameter) encased in a clear Perspex bearing (5.5 cm in diameter) and directly driven by a small electric motor, The bearing was freely supported on the shaft and sealed at the motor end with a light rubber diaphragm. The clearance was large so that the oil could be seen clearly. The motor speed was precisely controlled and adjusted by an electronic speed control unit and could be run in both directions. The speed was read directly from a calibrated meter in the control unit. The bearing contained 12 equispaeed pressure tappings around its circumference and four along its top side on a vertical radial plane. All the tappings were connected by light flexible plastic tubes to a manometer panel so that the pressure head of oil at all 16 points could be observed clearly in all the tubes. Oil was supplied to a low pressure region at both ends of the bearing by an adjustable Perspex reservoir fitted to the side of the rear panel. The bearing could be loaded by attaching various weights to the arms attached beneath it. After selecting the direction of rotation, the bearing was allowed to warm up under a light load for about 30 min at about 1500 rev min-I. Subsequently, care was taken to ensure that the pressure heads were steady before noting readings for each test speed (stepless variation between 500 and 3000 rev min-I). Typical axial and cir~umferenti~ pressure dist~bution curves for various solid concentrations are shown in Fig. 8. 3.3. Journal bearing for friction measurement The apparatus was arranged for frictional torque measurement under a known applied normal load. It consisted of a motor equipped with a speed control system. The motor rotated a steel shaft 4 cm in diameter by means of a pulley and a V-belt; this acted as a journal rotating in a lubricated bronze bearing (4.2 cm in diameter) which was forced to move by the frictional torque. The frictional torque could be balanced by an opposite torque produced by the balancing .weight wb. A small oil reservoir containing an oil filter and a valve which allowed oil droplets to flow through the bearing was attached to the casing. The normal load W was applied by a lever connected by two strings attached to the casing of the bearing, A thermometer hole in the top of the casing normal to the bronze bearing allowed the bearing temperature, which was approximately equal to the mean temperature of the oil, to be measured.
235 200-
SO1
8
236567 Sfation
(4
0 (b>)
5 Percentage
9
10
6
lo solid
II
12
I3
NO,.
mFlcentration
15 by
volume
Fig. 7. (a) Prcssute distribAon in a Michell pad thrust bearing for various solid concentrationa(U= 11 ems-l;Hi = 1 mm;H, = 0.5 mm); (b) variation in the area under the pressure distributiy curve with the solid concentration (0, U = 11 cm s-l ; 0, U = 15 cm 6-l ; 0, U= 22cms ).
236
Oi, 1
I.-
2
3
4
5 StatIon
No
Fig. 8. Pressure distribution in a journal 1000revmin-1:~,0%;~,2%,~,4%;X,6%;~,8%.
bearing
for various
solid concentrations
at
After a steady state condition had been attained, a balancing load was applied to the lever of the casing in order to make it approximately horizontal. The rotational speed, balancing load and temperature were recorded for each test. The speed of the journal was measured by a stroboscopic method. The viscosity of the oil was measured with an Engler viscometer and plotted as a function of temperature, so that the viscosity at each temperature was known accurately. Figure 9(a) shows the variation in the friction coefficient p against the dimensionless parameter ZN/F where 2 is the viscosity of the oil in centipoises, N is the rotational speed of the journal in revolutions per minute and p is the normal load per unit area in newtons per square centimetre. Figure 9(b) shows the variation in the dimensionless torque T* = T/pN2D4L) with the Reynolds number Re = 2nNRE/v. The particular case of concentric cylinders is plotted by using the Petroff equation T* = (n3/2)ReV1. The variation in the percentage friction reduction with the solid concentration by volume is shown in Fig. 10.
4. Discussion
of results
4.1. Theoretical results The results from hypothetical model 1 shown in Fig. 3 indicate that for both sliding and squeeze actions under constant load, viscosity, speed and
231
20 01
30
ZNtf’ (a)
50
40 Factof
60
70
’ OoMA
X103
12 Reynolds’
number
Re
14
16
i
18
X10
PI
Fig. 9.(a) Coefficient of friction VS.ZN/p for a journal bearing containing various solid concentrations (W = 11.573 N); (b) d imensionless friction torque us. Reynolds number for a journal bearing containing various solid concentrations (W = 7.03 N): 0,O vol.%; 0, 5 vol.%;A, 10 vol.%; v, 15 vol.%.
bearing configuration (a) the outlet film thickness H, increases with increasing number of laminae, (b) the outlet film thickness h, between any two successive laminae decreases with increasing number of laminae and (c) the friction force, and hence the friction coefficient, decreases with increasing number of laminae. Thus, if the wedge has a fixed geometry, then the presence of the laminae will increase the load-carrying capacity. However, if the bearing is allowed to adjust itself under constant load, the friction force will be reduced and there will be a reduction in the dissipated energy. The main results of hypothetical model 2 are shown in Figs. 5 and 6. All the graphs of the variation in the dimensionless load-carrying capacity W* with the tilt ratio #/ol show that W* is greater than unity which implies that the presence of particles in the oil enhances the load-carrying capacity of the bearing. The graphs of the variation in the dimensionless friction coefficient p* with the tilt ratio show that /J* is less than unity which implies that the presence of particles in the oil reduces the coefficient of friction. Experimental results The pressure distribution results given in Figs. 7 and 8 for both the Michell pad thrust bearing and the journal bearing indicate an increase in loadcarrying capacity with increasing solid concentration in the oil. The results 4.2.
Fig. 10. The variation in friction reduction with the percentage solid concentration journal bearing for various normal loads: 0, 7.03 N; 0, 9.3 N; 0, 11.57 N.
in the
obtained from the journal bearing indicate that the coefficient of friction decreases with increasing solid concentration up to a specific value and then increases. This is assumed to be due to the increase in lubricant viscosity, which changes its overall behaviour to that of a non-newtonian viscoelastic material. This change in friction behaviour occurred at a solid concentration of about 10 vol.%. When the solid concentration was kept constant, the coefficient of friction increased with increasing ZIV/puntil it reached the hydrodynamic region and then remained almost constant as shown in Fig. 9(a). The curves of dimensionless torque T* against the Reynolds number Re (Fig. 9(b)) lay above the theoretical Petroff line. As the solid concentration increased to 10 vol.% the curves became closer to the Petroff line but then moved away again as it was increased further. This implies that the value of the frictional torque decreased owing to the presence of solid particles. These curves also show that T* decreases as the Reynolds number increases. When the speed of rotation N increases, T* decreases. Figure 10 shows #at the percentage friction reduction increases with increasing solid concentration up to 10 vol.% and then decreases. This decrease can be attributed to the non-newtonian viscoelastic behaviour of the oil.
239
5. Conclusions In view of the qualitative nature of the experimental and analytical work, the following conclusions can be drawn. (1) When the configuration of the oil film is kept constant, the loadcarrying capacity increases when solid particles are added to the oil. This was confirmed by analytical hypo~eti~~ models. (2) The coefficient of friction was reduced by increasing the solid content of the lubricating oil up to an optimum volume percentage. (3) The experimental results obtained using the various conventional hydrodynamic test rigs confirm the validity of the theoretical results obtained from the qualitative models.
Nomenclature B c
c D E
F F* h
H
Hi HO L
M ; P
P Pi PO
Q
Re t T T* zi u lJ* u V V* W w w* R 2 cl
length of the particle percentage of solid content by volume mean radial clearance bearing diameter modulus of elasticity friction force dimensionless friction force oil film thickness height of the centroid of the particle from the slider inlet film thickness outlet film thickness axial length of the bearing number of laminae ratio of inlet to outlet film thickness rotational speed pressure W/LD, mean pressure of the bearing inlet pressure outlet pressure volumetric flow Reynolds number thickness of the particle torque dimensionless torque velocity in the x direction translational velocity dimensionless translational velocity velocity in the y direction vertical velocity dimensionle~ vertical velocity velocity in the z direction load-carrying capacity (normal load) dimensionless load-carrying capacity distance between the leading edge of the bearing and the oentre of pressure dynamic viscosity (cP) angle of inclination of the pad in the slider bearing
dynamic viscosity coefficient of friction dimensionless friction coefficient kinematic viscosity density shear stress angle of inclination of the particle rotational velocity of the particle
References 1 A. E. Yousif and K. D. Bogie, The rheological behaviour of a new high temperature synthetic grease, Proc., Inst. Mech. Eng., London, (1970). 2 A. A. Milne, A theory of grease lubrication of slider bearings, Proc. 2nd Int. Congr. Rheology, 1958. 3 S. G. Mason and W. Bartok,
on
in C. C. Mills (ed.), Rheology of Disperse Systems, Pergamon, Oxford, 1959, p. 16. 4 A. E. Yousif and J. Hailing, The frictional traction characteristics of greases in heavily loaded contacts, ASLE-ASME Lubr. Conf., Atlanta, October 16 - 18, 1973, Society of Mechanical Engineers, New York. 5 0. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication, McGraw-Hill, New York, 1961.