Transcript ME 575 Hydrodynamics of Lubrication Fall 2001

ME 575
Hydrodynamics of Lubrication
By
Parviz Merati, Professor and Chair
Department of Mechanical and
Aeronautical Engineering
Western Michigan University
Kalamazoo, Michigan
ME 575
Hydrodynamics of Lubrication
Fall 2001

An overview of principles of lubrication
– Solid friction
– Lubrication
– Viscosity
– Hydrodynamic lubrication of sliding surfaces
– Bearing lubrication
– Fluid friction
– Bearing efficiency
– Boundary lubrication
– EHD lubrication
ME 575
Hydrodynamics of Lubrication
Movie on “Lubrication Mechanics, an Inside Look”
 General Reynolds equation
 Hydrostatic bearings
 Thrust bearings
 Homework #1
 Journal bearings
 Homework #2
 Hydrodynamic instability
 Thermal effects on bearings

– Viscosity
– Density
ME 575
Hydrodynamics of Lubrication


Viscosity-pressure relationship
Laminar flow between concentric cylinders
– Velocity profile
– Pressure
– Mechanical Seals
– Moment of the fluid on the outer cylinder

Homework #3
Solid Friction

Resistance force for sliding
– Static
– Kinetic

Causes
– Surface roughness (asperities)
– Adhesion (bonding between dissimilar materials)

Factors influencing friction
– Frictional drag lower when body is in motion
– Sliding friction depends on the normal force and frictional
coefficient, independent of the sliding speed and contact area
Solid Friction

Effect of Friction
– Frictional heat (burns out the bearings, ignites a match)
– Wear (loss of material due to cutting action of opposing

Engineers control friction
– Increase friction when needed (using rougher surfaces)
– Reduce friction when not needed (lubrication)
Lubrication

Lubrication
– Prevention of metal to metal contact by means of an intervening
layer of fluid or fluid like material

Lubricants
– Mercury, alcohol (not good lubricants)
– Gas (better lubricant)
– Petroleum lubricants or lubricating oil (best)

Viscosity
– Resistance to flow
– Lubricating oils have wide variety of viscosities
– Varies with temperature
Lubrication

Hydrodynamic lubrication (more common)
– A continuous fluid film exists between the surfaces

Boundary lubrication
– The oil film is not sufficient to prevent metal-to-metal contact
– Exists under extreme pressure

Hydrodynamic lubrication
– The leading edge of the sliding surface must not be sharp, but must be
beveled or rounded to prevent scraping of the oil from the fixed surface
– The block must have a small degree of free motion to allow it to tilt and to
lift slightly from the supporting surface
– The bottom of the block must have sufficient area and width to float on
the oil
Lubrication

Fluid Wedge
– The convergent flow of oil under the sliding block develops a pressure-
hydrodynamic pressure-that supports the block. The fluid film lubrication
involves the ‘floating” of a sliding load on a body of oil created by the
“pumping” action of the sliding motion.

Bearings
– Shoe-type thrust bearings (carry axial loads imposed by vertically
mounted hydro-electric generators)
– Journal bearings (carry radial load, plain-bearing railroad truck where the
journal is an extension of the axle, by means of the bearings, the journal
carries its share of the load)
– In both cases, a tapered channel is formed to provide hydrodynamic lift
for carrying the loads
Fluid Friction

Fluid friction is due to viscosity and shear rate of the fluid
–
–
–
–
Generates heat due to viscous dissipation
Generates drag, use of energy
Engineers should work towards reducing fluid friction
Flow in thin layers between the moving and stationary surfaces of the
bearings is dominantly laminar
 Z
 = shear stress
Z = viscosity
dU/dy = shear rate
dU
dy
Fluid Friction

Unlike solid friction which is independent of the sliding velocity and
the effective area of contact, fluid friction depends on both

Unlike solid friction, fluid friction is not affected by load

Partial Lubrication (combination of fluid and solid lubrication)
– Insufficient viscosity
– Journal speed too slow to provide the needed hydrodynamic pressure
– Insufficient lubricant supply
Overall Bearing Friction

A relationship can be developed between bearing friction and viscosity,
journal rotational speed and load-carrying area of the bearing
irrespective of the lubricating conditions
F  f ZNA
F = Frictional drag
N = Journal rotational speed (rpm)
A = Load-carrying area of the bearing
f = Proportionality coefficient
Overall Bearing Friction

Coefficient of friction (friction force divided by the load that presses
the two surfaces together)
F
ZN

 f
L
P
 is the coefficient of friction and is equal to F/L.
L is the force that presses the two surfaces together.
P is the pressure and is equal to L/A.
Overall Bearing Friction

ZN/P Curve
– The relationship between  and ZN/P depends on the lubrication
condition, i.e. region of partial lubrication or region of full fluid film
lubrication. Starting of a journal deals with partial lubrication where as
the ZN/P increases,  drops until we reach a full fluid film lubrication
region where there is a minimum for . Beyond this minimum if the
viscosity, journal speed, or the bearing area increases,  increases.
Analysis

Proper bearing size is needed for good lubrication.
– For a given load and speed, the bearing should be large enough to operate
in the full fluid lubricating region. The bearing should not be too large to
create excessive friction. An oil with the appropriate viscosity would
allow for the operation in the low friction region. If speed is increased, a
lighter oil may be used. If load is increased, a heavier oil is preferable.

Temperature-Viscosity Relationship
– If speed increases, the oil’s temperature increases and viscosity drops, thus
making it better suited for the new condition.
– An oil with high viscosity creates higher temperature and this in turn
reduces viscosity. This, however, generates an equilibrium condition that
is not optimum. Thus, selection of the correct viscosity oil for the
bearings is essential.
Boundary Lubrication
– Viscosity Index (V.I) is value representing the degree for which the oil
viscosity changes with temperature. If this variation is small with
temperature, the oil is said to have a high viscosity index. A good motor
oil has a high V.I.

Boundary Lubrication
– For mildly severe cases, additives known as oiliness agents or film-
strength additives is applicable
– For moderately severe cases, anti-wear agents or mild Extreme Pressure
(EP) additives are used
– For severe cases, EP agents will be used
Boundary Lubrication

Oiliness Agents
– Increase the oil film’s resistance to rupture, usually made from oils of
animals or vegetables
– The molecules of these oiliness agents have strong affinity for petroleum
oil and for metal surfaces that are not easily dislodged
– Oiliness and lubricity (another term for oiliness), not related to viscosity,
manifest itself under boundary lubrication, reduce friction by preventing
the oil film breakdown.

Anti-Wear Agents
– Mild EP additives protect against wear under moderate loads for boundary
lubrications
– Anti-wear agents react chemically with the metal to form a protective
coating that reduces friction, also called as anti-scuff additives.
Boundary Lubrication

Extreme-Pressure Agents
– Scoring and pitting of metal surfaces might occur as a result of this case,
seizure is the primarily concern
– Additives are derivatives of sulfur, phosphorous, or chlorine
– These additives prevent the welding of mating surfaces under extreme
loads and temperatures

Stick-Slip Lubrication
– A special case of boundary lubrication when a slow or reciprocating action
exists. This action is destructive to the full fluid film. Additives are added
to prevent this phenomenon causing more drag force when the part is in
motion relative to static friction. This prevents jumping ahead
phenomenon.
EHD Lubrication
In addition to full fluid film lubrication and boundary lubrication, there is
an intermediate mode of lubrication called elaso-hydrodynamic (EHD)
lubrication. This phenomenon primarily occurs on rolling-contact
bearings and in gears where NON-CONFORMING surfaces are subjected
to very high loads that must be borne by small areas.
-The surfaces of the materials in contact momentarily deform elastically
under extreme pressure to spread the load.
-The viscosity of the lubricant momentarily increases drastically at high
pressure, thus increasing the load-carrying ability of the film in the contact
area.
Reynolds Equation

In bearings, we like to support some kind of load. This load is taken
by the pressure force generated in a thin layer of lubricant. A
necessary condition for the pressure to develop in a thin film of fluid is
that the gradient of the velocity profile must vary across the thickness
of the film. Three methods are available.
– Hydrostatic Lubrication or an Externally Pressurized Lubrication- Fluid
from a pump is directed to a space at the center of bearing, developing
pressure and forcing fluid to flow outward.
– Squeeze Film Lubrication- One surface moves normal to the other, with
viscous resistance to the displacement of oil.
– Thrust and Journal Bearing- By positioning one surface so it is slightly
inclined to the other and then by relative sliding motion of the surfaces,
lubricant is dragged into the converging space between them.
Reynolds Equation

Use Navier-Stokes equation and make the following assumptions
– The height of the fluid film h is very small compared with the length and
the span (x and z directions). This permits to ignore the curvature of the
fluid film in the journal bearings and to replace the rotational with the
transnational velocities.
Reynolds Equation
– Since the fluid layer is thin, we can assume that the pressure gradient in
the y direction is negligible and the pressure gradients in the x and z
directions are independent of y
p
 0.0
y
p
p
and
 fn( y )
x
z
–
–
–
–
Fluid inertia is small compared to the viscous shear
No external forces act on the fluid film
No slip at the bearing surfaces
Compared with u/y and w/y, other velocity gradient terms are
negligible
Reynolds Equation
1 p
 2u

 x
y 2
1 p
2w

 z
y 2
B.C.
y = 0.0, u = U1 , v = V1 , w = W1
y = h, u = U2 , v = V2 , w = W2
Integrating the x component of the above equations would result in the
following equation.
Reynolds Equation
1 p
y

2 
u
(hy  y )  U1  (U 2  U1)
2 x
h


Integrating the z-component
1 p
y


2
w
(hy  y )  W1  (W2  W1)
2 z
h


Reynolds Equation

u and w have two portions;
– A linear portion
– A parabolic portion
Reynolds Equation

Using continuity principal for a fluid element of dx, dz, and h, and
using incompressible flow, we can write the following relationship
 qx
 qz
q x  q z  q1  q x 
dx  qz 
dz  q 2

x

z
Where,
h
q x   u dy dz
0
h
q z   w dy dx .
0
Reynolds Equation
h3 p
U1  U 2
qx  
dz 
h dz
12 x
2
h3 p
W1  W2
qz  
dx 
h dx
12 z
2
Fluid moving into the fluid element in the Y direction is q1
q1  V1 dx dz
Reynolds Equation
h
h
q 2  V2 dx dz  U 2
dx dz  W2 dx dz
x
z
1   h3 p
 h3 p 
h
h
(
)

(
)

(
U

U
)

2
(
V

V
)

(
W

W
)


1
2
1
2
1
2
6  x  x
z  z 
x
z
h


(U1  U 2 )  h (W1  W2 )
x
z
The last two terms are nearly always zero, since there is rarely a change
in the surface velocities U and W.
Reynolds Equation in Cylindrical Coordinate
System
3
1 1 
h3 p

h
p 
h
1 h
1
(
r
)

(
)

(
R

R
)

2
(
V

V
)

(
T

T
)


2
1
2
1
2
1
2
6  r r
 r r    
r
r 

h




r
(
R

R
)

(
T

T
)

1
2
1
2 
r  r


R1 and R2 are the radial velocity of the two surfaces
T1 and T2 are the tangential velocity of the two surfaces
V1 and V2 are the axial velocity of the two surfaces
Hydrostatic Bearings

Lubricant from a constant displacement pump is forced into a central
recess and then flows outward between bearing surfaces. The surfaces
may be cylindrical, spherical, or flat with circular or rectangular
boundaries.

If the pad is circular as shown in the following figure,
Hydrostatic Bearings

p
(r
)0
r r
Total Load P
D
2
P   p (2 r ) dr 
B.C.
D
2
d
p  p 0 at r 
2
p  0 at r 
D
p
 2r
p 0 ln D
d
ln
d
2
 d2
4
The hydrostatic pressure
required to carry this load is
p0.
8 P ln ( D / d )
p0 
 (D2  d 2 )
p0
Hydrostatic Bearings
What is the volumetric flow rate of the oil delivery system?
h
Q  2  r  ur dy
0
Using Reynolds Equation for rectangular system, and
substituting x with r, and considering that U1 and U2 are
zero, the following relationship can be obtained for radial
component of the flow velocity ur.
4 P(hy  y )
ur 
r ( D 2  d 2 )
2
Hydrostatic Bearings
4 P h3
Q
3 ( D 2  d 2 )
What is the power required for the bearing
operation?
Power Re quired 
( p0 A) V


p0 Q

A = Cross sectional area of the pump
delivery line
V = Average flow velocity in the line
 = Mechanical efficiency
Hydrostatic Bearings
What is the required torque T if the circular pad is rotated with speed n
about its axis ?
The tangential component of the velocity is represented by Wt and the
shear stress is shown by 
Wt  2 r n
D
2
y
h
T   r dF   r  dA   r
d
2
w 2  r n
  t 
y
h
T
 2 n
16 h
(D4  d 4 )
2  r n
2 r dr
h
Thrust Bearings

There should be a converging gap between specially shaped pad or
tilted pad and a supporting flat surface of a collar. The relative sliding
motion forces oil between the surfaces and develop a load-supporting
pressure as shown in the following figure.
– Using the Reynolds Equation and using h/z = 0, for a constant viscosity
flow, the following equation is obtained
 3 p
 3 p
h
(h
)  (h
)  6 (U1  U 2 )
x
x
z
z
x
Thrust Bearings
This equation can be solved numerically. However if we assume that the
side leakage w is negligible, thus p/z is negligible, then the equation
can be solved analytically
 3 p
h
(h
)  6 (U1  U 2 )
x
x
x
B.C.
h  h1 at x  0
h  h2 at x  b
h1  h2
Defining  
b
Thrust Bearings
6  (U1  U 2 )  x (b  x )
p
2
( 2h2   h ) h2   (b  x )
Total load can be found by integrating over the surface area
of the bearing.
Flat Pivot
Flat pivot is the simplest form of the thrust bearing where
the fluid film thickness is constant and the pressure at any
given radius is constant. There is a pressure gradient in the
radial direction. The oil flows on spiral path as it leaves the
flat pivot.
Thrust Bearings
r1
What is the torque T required to rotate the shaft?
T    2 r dr
2
0

Shear stress is represented by 
 r
h
 A  r1
T
, where A   r 2
2h
2
Thrust Bearings
What is the pressure in the lubricant layer?
Pressure varies linearly from the center value of p0 to zero at the outer
edge of the flat pivot.
r
p  p0 (1  )
r1
If we define an average pressure as pav
r1
pav A   p 2  r dr
0
pav 
p0
3
Thrust Bearings
What is the viscous friction coefficient?
 A  r1
2
T   f p 2  r dr 
2h
0
r1
N  r1
f  2
pav h
2
where N is the shaft RPM
Thrust Bearings
Pressure Variation in the Direction of Motion
  Thickness at X  X , at a dis tan ce x from the center
of the breadth
2e x
  h
B
1
1 dp 3
Flow across X  X  V 
2
 dx 12
x ' corresponding to max imum pressure,
dp
0
dx
Thrust Bearings
Continuity
1
2ex' 1
2ex 1 dp 3
V (h 
)  V (h 
)
2
B
2
B
 dx 12
dp Ve 12

( x' x)
3
dx
B 
Integrating and using the following boundary condition
1
x B, p0
2
p
1 a2 1
1 1

 
3VB
2 (  )2  2
(
)
eh
h
h
e
where a  , attitude of the bearing or pad surface
h
Thrust Bearings
Maximum pressure pm
pm
a2

2
3VB
2
(
1

a
)
(
)
eh

As the attitude of the bearing surface a is reduced, pressure magnitude
decreases in the fluid film and the point of maximum pressure
approaches the middle of the bearing surface. For a = 0, the pressure
remains constant.
Thrust Bearings
What are the total load and frictional force on the slider?
Define P and F' as the load and drag force per unit length perpendicular to
the direction of motion.
B/2
P 
 p dx
B / 2
F 
'
B/2
 q dx
B / 2
q is the shear stress and is defined by the following equation
q
V 1 dp
 
 2 dx
Thrust Bearings
P
3
1 a

(ln
 2a )
2
2
3VB
2a
1 a
(
)
h2
F'
2 1  a 3a
 (ln
 )
VB a 1  a 2
(
)
h
Coefficient of friction f is defined by the following relationship.
F'  f P
Thrust Bearings
If  is the angle in radians between the slider and the bearing
pad surface, then the following equations based on the
equilibrium conditions of the film layer exist.
P  Q cos   Fr sin 
F '  Q sin   Fr cos 
Since  is very small, film layer thickness h and e are small
relative to the bearing length B, sin    , and cos   1. It
is also safe to assume that Fr is small compared with Q.
Thrust Bearings
Critical value of  occurs when Fr =0. This will result in
F'
 
 tan   
P
 is the angle of friction for the slider. When  > , Fr becomes negative. This is
caused by reversal in the direction of flow of the oil film . The critical value of a
is thus obtained by using the following relationship.
2e
 f 
B
e
a   0.86
h
Thus the range of acceptable variation for a is 0 < a <0.86
Homework 1
For a thrust bearing, plot non-dimensionalized pressure along the breath of
the bearing for several values of the bearing attitude defined by a=e/h,
( 0  a  0.86). In addition, plot non-dimensionalized maximum pressure,
load per unit length measured perpendicular to the direction of motion,
tangential pulling force, and virtual friction coefficient versus the bearing
attitude. For each plot, please discuss your findings and provide
conclusions.
Note:
Please refer to figure 5.11 and sections 5.4.2, 5.4.3, and 5.4.4 of your
notes for additional information.
Journal Bearings
In a plain journal bearing, the position of the journal is directly related to
the external load. When the bearing is sufficiently supplied with oil and
external load is zero, the journal will rotate concentrically within the
bearing. However, when the load is applied, the journal moves to an
increasingly eccentric position, thus forming a wedge-shaped oil film
where load-supporting pressure is generated.
Journal Bearings
Oj = Journal or the shaft center
Ob = Bearing center
e = Eccentricity
0 e c
The radial clearance or half of the initial difference in diameters is
represented by c which is in the order of 1/1000 of the journal diameter.
 = e/c, and is defined as eccentricity ratio
0   1
If  = 0, then there is no load, if  = 1, then the shaft touches the bearing
surface under externally large loads.
Journal Bearings
What is the lubricant’s film thickness h?
Using the above figure, the following relationship can be obtained for h
h  c (1   cos  )
The maximum and minimum values for h are
hmax  c  e  c(1   )
hmin  c  e  c(1   )
r = Journal radius
r+c = Bearing radius
Journal Bearings
Using Reynolds equation and assuming an infinite length for the bearing,
i.e., p/ z = 0, and U = U1+U2 , the following differential equation is
obtained.
 3 p
h
(h
)  6 U
x
x
x
Reynolds found a series solution in 1886 and Sommerfeld found a closed
form solution in 1904 which is widely used.
Ur  6  sin  (2   cos ) 
p 2 
2
2
c  (2   ) (1   cos ) 
Journal Bearings
Modern bearings are usually shorter, the length to diameter ratio is often
shorter than 1. Thus, the z component cannot be neglected. Ocvirk in
1952 showed that he could safely neglect the parabolic pressure induced
part of the U component of the velocity and take into account the z
variation of pressure. Thus, the following simplified equation can be
obtained.
 3 p
h
(h
)  6 U
z
z
x
If there is no misalignment of the shaft and bearing, h and  h/  x are
independent of z, then the above equation can be easily integrated with the
following boundary conditions for a journal of length l.
Journal Bearings
B.C.
At z  0,
p
0
z
l
At z   , p  0
2
Ocvirk Solution of the Short Bearing Approximation
p
U l 2
rc 2
(
4
 z2)
3  sin 
(1   cos )3
Thus, axial pressure distribution is parabolic.
Journal Bearings
At which angle the maximum pressure occur? m=?
To find m’  p/  =0.
1  1  24 2
 m  cos (
)
4
1
What is the total load that is developed within the bearing?
The oil film experiences two forces, one from the bearing, the other from
the journal. The bearing force P passes through the center point of the
bearing, the journal force P passes through the journal center.
Journal Bearings
The hydrodynamic pressure force is always normal to the bearing and
journal surfaces. In order to find the total load, the pressure force over the
bearing surface must be integrated. Since the oil film is stationary, the
resultant of the external forces and moments, i.e. bearing and journal
forces and moments exerted on the oil film, must be zero.
The total load P carried by the bearing is calculated by the following
equation.
P  ( P cos ) 2  ( P sin  ) 2
Where  is defined as the attitude angle and is the angle between the line
of force and the line of centers. The two components of the load normal
and parallel to the line of centers are represented by P sin  and P cos .
Journal Bearings
Journal Load and the Attitude Angle
l
P cos  2   ( p r d dz) cos
00
l
P sin   2   ( p r d dz) sin 
00

 U l 3   2 (1   2 )  16 2
P
c2
4 (1   2 ) 2

1/ 2
2
P sin 
1  1  
  tan (
)  tan
P cos
4
1
Journal Bearings
With an increasing load,  will vary from 0 to 1 and the attitude angle 
vary from 90 degrees to zero. The path of the journal center Oj as the load
and eccentricity are increased is shown in the following figure.
Homework 2
Non-dimensionalize the hydrodynamic pressure and load of equations
5.48 and 5.51 of your notes, respectively. These are the Ocvirk equations
for short journal bearings. Plot this non-dimensionalized pressure versus
 at z = 0.0 for eccentricity ratios  = 0.1, 0.3, 0.5, 0.7, and 0.9. Plot the
location and magnitude of the maximum pressure with respect to  at
z = 0.0. Plot the non-dimensionalized load P and the attitude angle 
versus . For each plot, please discuss your findings and provide
conclusions.
Hydrodynamic Instability

Synchronous whirl
– Caused by periodic disturbances outside the bearing such that the bearing
system is excited into resonance. Shaft inertia and flexibility, stiffness and
damping characteristics of the bearing films, and other factors affect this
instability. The locus of the shaft center called the whirl orbit increases at
the critical shaft speed where there is resonance. It is usual procedure to
make the bearings such that the critical speeds do not coincide with the
most commonly used running speeds. This may be done either by
increasing the bearing stiffness so that the critical speeds are very high, or
reducing the stiffness so that the critical speeds are quickly passed through
and normal operation takes place where the attenuation is large. Stiffness
can be increased by reducing the bearing clearance. Introduction of extra
damping by mounting the bearing housings in rubber “O” rings or metal
diaphragms are other methods to suppress the synchronous whirl.
Hydrodynamic Instability

Half-Speed whirl
– This is induced in the lubricant film itself and is called “half-speed whirl”.
This is because due to existence of the attitude angle , the reaction force
from the lubricant on the shaft has a component normal to the line
connecting the centers of the shaft and the bearing. This component causes
the shaft to move in a circumferential direction, i.e., at the same time as
the shaft moves around its center, the shaft center rotates about the bearing
center. If the whirl takes place at the half the rotational speed of the shaft,
this will coincide with the mean rotational speed of the lubricant. Because,
the lubricant, on the average, does not have a relative velocity with respect
to the shaft, the hydrodynamic lubrication fails. Extra damping, axial
groves on the bearing housing, partial bearing are some of the techniques
to get rid of this instability.
Hydrodynamic Instability
Thermal Effects on Bearings
We have assumed that fluid viscosity and density remains constant in
deriving the Reynolds equations. In reality due to viscous dissipation
because of the large existing shear stress, the lubricant’s temperature rises
and thus the fluid density and viscosity change. Since the fluid is unable
to expand due to restriction, fluid pressure increases as the temperature
increases. This is called Thermal Wedge. Consider the General R.E. with
the viscosity and density variation in the sliding direction.
d  h 3 dp
d U h
(
)
(
)
dx 12  dx
dx
2
Thermal Effects on Bearings
After integrating the above equation,
6U
12 A 
p  2   dx  3  dx  B
h
h 
In this equation, A and B are constants. The variation of density with
temperature can be approximated by the following relationships.
  i   t (T  Ti )
x
(To  Ti )
L
x
  i 
(  o  i )
L
T  Ti 
Thermal Effects on Bearings
Contribution of viscosity variation for liquids compared with density
variation is negligible since viscosity increases with pressure and
decreases with temperature. Thus, we can assume that viscosity remains
constant in the sliding direction. Using the following boundary
conditions, the pressure variation due to temperature variation for a
parallel bearing can be obtained.
p  0 at x  0
p  0 at x  L

x


ln 1  (  '  1)  

6 UL  x
L


p



h2  L
ln  '





Thermal Effects on Bearings
Where,
' 
o
i
 '  1
t
(To  Ti )
i
For mineral oil,
 t   0.00065
 i  0.9
gr
cm 3
gr
cm 3 C 
Thermal Effects on Bearings
Thus, for a rise in temperature of 100 C, ' =0.93. The dimensionless
pressure p' is
2
p
h
x

p' 
  13.78 ln 1  0.07
6 U L L

x
L 
Thermal Effects on Bearings
p´max is about 0.011 and for a plane-inclined slider, p´max is about 0.042.
The parallel surface bearing has a load capacity approximately 1/3.5 that
of the corresponding inclined slider. It is rare that the temperature rise is
100 C, usually the temperature rise due to viscous dissipation is in the
order of 2-20 C and under these conditions, it is safe to assume that the
effect of temperature is negligible.
Viscosity-Pressure Relationship
In some situations where extreme pressures can occur such as in the
restricted contacts between gear teeth and between rolling elements and
their tracks, viscosity relationship with pressure is represented by the
following equation.
   0 ep
Where 0 and  are reference viscosity and the pressure exponent of the
viscosity, respectively. In order to integrate R.E., we have to introduce
parameter q defined as
q
1

(1  e p )
Viscosity-Pressure Relationship
The differential equation that is obtained as the result of this substitution,
looks like a normal R.E. with viscosity term being 0. This equation can
then be integrated and pressure can be obtained from the following
relationship.
1
p   ln (1   q)

Although load remains finite, pressure is tending to approach an infinite
value between two disks rolling with some degree of sliding as shown in
the following figure. This does not happen in reality. In reality, large
pressures produce deformation of the bodies which distribute the pressure
over a finite area. This is called “Elasto-hydrodynamic” lubrication or
EHD.
Laminar Flow Between Concentric Cylinders
Using Navier-Stokes equations in cylindrical systems and the following
simplifications,
Vr =0
Vz = 0
v = u
The r-component is
dp
u 2

dr
r
The  component is
d
dr
1 d

(
ru
)
 r dr
0


Laminar Flow Between Concentric Cylinders
B.C. for velocity
r  R1 , u  R11
r  R2 , u  R2 2
2
2


1
R1 R2
2
2
u  2 2  ( R2  2  R1 1 ) r 
(1   2 )
r
R2  R1 

B.C. for pressure
r  R1 , p  p1
Laminar Flow Between Concentric Cylinders
p  p1 
2
2
 2
r 1 4 4
1
1 
2
2
2
2
2
2 ( r  R1 )
(
R


R

)
 2 R1 R2 (1   2 )( R2  2  R1 1 ) ln  R1 R2 (1   2 ) 2 ( 2  2 )
2
2
1
1
2
2 2 
2
R1 2
( R2  R1 ) 
R1 r 

For the case of mechanical seals where the inner cylinder is rotating and
the outer cylinder is stationary, i.e. 2 = 0
p
 1
p1
2
2
 1 r 2 R12
r 1  R2
R2 


(

)

2
ln



2
 R 2 r 2 
2 2 2 R 2
R
2
R



1
2
2
 1

p1
R

1 12 

R
12 R12 ( 1 ) 2  R2 
R2
1
Laminar Flow Between Concentric Cylinders
 R1
u
1
r R1 




R11 1  ( R1 ) 2  r
R2 R2 
R2
If the inner cylinder is at rest , 1 = 0, the moment of the fluid on a length
L of the outer cylinder is described by
M 2  2  R2 L  2 R2
 u
 2   r ( ) r R
2
r r
Laminar Flow Between Concentric Cylinders
 R12 R2 2
M2  4 L 2
2
2
R1  R2
Viscosity can be calculated from this equation if the moment on the outer
cylinder is measured.
Homework 3
Calculate and plot pressure ratio p/p1, and velocity ratio u/(R11) versus
the radial location (r-R1)/(R2-R1) for the flow between concentric
cylinders for water, oil, and sodium iodide solution. The radii of the inner
and outer cylinders are R1 = 0.031 m and R2 = 0.046 m, respectively. p1 is
the pressure at the inner cylinder surface and r is the radial location. The
outer cylinder is stationary and the inner cylinder is rotating at 1,200 rpm.
Density of water, oil and sodium iodide solution (67% by volume) are
1,000, 880, and 1,840 Kg/m3, respectively. Assume that p1 is atmospheric
pressure. Although the flow at this rotational speed is turbulent, the time
average of the flow velocity and pressure are close to the laminar flow
values. For each plot, please discuss your findings and provide
conclusions.
Thank You