The document discusses computational methods for solving hydrodynamic lubrication problems. It begins by introducing Reynolds' equation, which relates the pressure field in a fluid to the geometry, velocities, and viscosity. The document then describes how the finite difference method can be used to computationally solve Reynolds' equation. Specifically, it discusses how a grid is placed over the domain, difference equations are used to approximate derivatives, and the resulting equations can be solved iteratively to find the pressure field. The document provides an example application to rectangular lubrication pads.

1. COMPUTATIONAL ASPECTS OF
HYDRODYNAMIC LUBRICATION
Whenever engines of this kind exist in the capitals and universities of the world it is obvious
that all enquirers who wish to put their theories to the test of number, will apply their efforts
to shape the analytical results at which they have arrived, so that they may be susceptible to
calculation by machinery in the shortest possible time, and the whole course of their analysis
will be directed towards this effort.
Memoirs of the Life and Labours of the Late Charles Babbage Esq. F.R.S. Chapter 8, H.W.
Buxton, republ. The MIT Press, 1988.

2. 2
CONTENTS
1. BACKGROUND
1.1 General Introduction
1.2 Reynolds Equation
1.3 Introduction to Finite Difference Method
1.4 Non-dimensionalisation
2. SOLUTION OF ISOTHERMAL REYNOLDS EQUATION
2.1 Pressure Fields for 2-D Pads
2.2 Pressure Fields for 2-D Journal Bearings
2.3 Integration to give W, F, Q
3. MORE COMPLEX HYDRODYNAMIC PROBLEMS
3.1 Thermal Hydrodynamics
3.2 Non-Steady State Problems
3.3 EHD Problems
3.4 Control Volume Approach

3. 3
1. BACKGROUND
1.1 General Introduction
In hydrodynamic lubrication, the motion of one or both of two opposing, solid surfaces drags
lubricant between them. As this lubricant is forced into the converging gap, a pressure field is
generated within the fluid. This pressure field pushes the surfaces apart against any applied
load. Since the pressure field decreases rapidly with increasing separation, the system should
stabilise at a particular separation when the integral of the pressure field matches the applied
load and the system then operates in the regime of hydrodynamic lubrication. Clearly, if the
combined surface roughness is greater than the minimum hydrodynamic separation, full
hydrodynamic lubrication cannot be reached.
The pressure generated within a fluid film is generally described by Reynolds' Equation
which relates the pressure field to the separation and geometry of the surfaces, the viscosity of
the lubricant and the velocities of the two surfaces. We need to be able to solve this equation
in order to determine the pressures in lubricated contacts and thus the film thicknesses and
friction in practical systems such as bearings and piston rings.
A complication is that the pressure generated in hydrodynamic lubrication depends upon the
viscosity of the oil, which, in turn depends upon the temperature of the lubricant. However
the temperature of the lubricant in a hydrodynamic contact may itself vary due to heat
generated by shear within the fluid film. Thus we need also to be able to solve for the
temperature rise due to shear in rubbing contacts. This is described by the Energy Equation.
In practice, both equations are 2-D or 3-D partial differential equations and can only be solved
analytically with drastic simplification. The main systems that can be solved analytically are
isothermal cases where the 2-D Reynolds equation is reduced to a 1-D approximation (either
the long or the short bearing) for a few, relatively simple, gap shapes (linear pad, exponential
pad etc.)
To obtain realistic solutions for most other cases, computational solutions are employed. In
the early days of computers these were produced by specialists on mainframe computers and
then summarised in graphical or regression form for use by designers. In the last few years
however, programs for the general user are becoming available on microcomputers. This
trend is likely to continue.
In practice, custom-written computer programs are rarely versatile enough to solve all
problems and it is also important for serious tribologists to understand the basis and

4. 4
consequent limitations of such programs. These notes are intended to introduce how to set
about solving hydrodynamic lubrication problems using computational methods.
These notes begin with Reynolds' equation which is essentially only valid for isothermal
systems, i.e. those at quite slow speeds. They describes how finite difference methods can be
used to solve Reynolds' equation. The notes then point the reader towards how more complex
problems can be tackled.
(In these notes the terminology generally used are; x, vx are position and velocity in the main
sliding direction, y, vy are transverse to this, z,  are through the thickness of film).
1.2 Reynolds' Equation
A derivation of Reynolds' Equation was given in Tribology I and is only summarised here. It
is obtained by combining the reduced Navier-Stokes Equation and the mass continuity
equation;
Reduced Navier-Stokes:










=


z
v
z
x
p x
 









=


z
v
z
y
p y
 (1)
Mass Continuity Equation:
( ) ( ) ( ) 0
=


+


+


+


t
z
v
y
x
v
x
y
x



 (2)
Inherent in the reduced Navier-Stokes are the assumptions:
(i) body forces are negligible compared to shear forces
(ii) fluid flow is laminar, (inertial forces negligible compared to shear forces)
(iii) pressure is constant through thickness of film
(iv) fluid strain rates through the fluid thickness are dominant over all other strain rates,
i.e. vx/z and vy/z are the only significant strain rate components
(v) surface and film curvatures are negligible
By making the following further assumptions:
(vi) no slip at walls
(vii) lubricant density constant
(viii) lubricant viscosity constant through the thickness of the film
Reynolds' Equation is obtained:

5. 5
( )
1
2
3
3
12
12
12 



−
+


+


=














+














y
h
v
x
h
v
y
p
h
y
x
p
h
x
y
x
(3)
where the entrainment velocities, x
v = (vx1+ vx2)/2 and y
v = (vy1 + vy2)/2. vx1, vx2 etc. are the
tangential velocities of the bottom and top surface and  and  are their normal velocities.
In polar coordinates, assuming rotation but no translation the equivalent equation is (1):
( )
1
2
3
2
3
12
2
12
1
1








−
+








=














+













 h
p
h
r
r
p
rh
r
r
(4)
where  is the angular velocity.
These equations are fully valid only for isothermal systems since the assumption that the
lubricant viscosity is constant though the film thickness will break down if significant
amounts of heat are generated in the film due to shear. The Hydrodynamic Lubrication
handout of Tribology I discussed approximate ways of allowing for thermal effects. The
thermal case is briefly described later in section 3 of this handout.
1.3 Finite Difference Method
By far the commonest way to solve Reynolds' Equation is using the 2-D finite difference
approach. In this a grid of points is placed over the area or domain in which a property 
(typically pressure p) varies, (figure 1).
j=N
j=1
i=1 i=M
area of
interest
e.g. thrust pad
Figure 1
x
y

6. 6
The value of  varies from grid point to grid point and its first and second partial differential
at each point, (i,j) can be approximated in terms of  at neighbouring points by the following
difference equations:
x
x
j
i
j
i
ij
ij

−
=

 −
+
2
,
1
,
1 


y
y
j
i
j
i
ij
ij

−
=

 −
+
2
1
,
1
, 


(5)
( )2
,
1
,
1
2
2
2
x
x
ij
j
i
j
i
ij

−
+
=

 −
+ 



( )2
1
,
1
,
2
2
2
y
y
ij
j
i
j
i
ij

−
+
=

 −
+ 



The above expressions can then be
substituted into a partial differential
equation, e.g.
0
2
2
2
2
=
+


+


+


+


E
x
D
x
C
y
B
x
A



 (6)
and ij obtained in terms of neighbouring points. From the above equation we obtain, after
substitution and rearrangement:
ij
j
i
ij
j
i
ij
j
i
ij
j
i
ij
ij G
CW
CE
CS
CN +
+
+
+
= −
+
−
+ ,
1
,
1
1
,
1
, 



 (7)
where, for each point (i,j)
CNij = (B/(y)2 + D/2y)/U
CSij = (B/(y)2 - D/2y)/U
CEij = (A/(x)2 + C/2x)/U (8)
CWij = (A/(x)2 - C/2x)/U
Gij = -E/U
j+1
j
j-1
i-1 i i+1
x
x
y
y
 



i,j
i,j-1
i-1,j
i,j+1
i+1,j
Figure 2

7. 7
CN stands for "coefficient north" etc. (i.e. "north" in the layout in figure 1) and
U = 2(A/(x)2 +B/(y)2)
Equation 7 is the computing finite difference equation for .
The general approach to solving the differential equation is now as follows.
(i) Place a grid of points (M x N) over the area to be integrated.
(ii) Calculate A, B, C, D, E and hence CN(i,j), CS(i,j) etc. for each point (i,j) within the
area (not the perimeter points). This gives an expression for (i,j) at each such
point from equation 7.
(iii) Set the perimeter points (i,j) to suitable boundary values (and also any interior
points whose values of  are to be fixed).
(iv) Stages (ii) and (iii) give a set of M x N simultaneous equations for the values of  at
the M x N grid positions which can be solved algebraically or often more simply,
iteratively.
If iteratively:
(v) Guess the initial values of (i,j) within the area (often = 0).
(vi) For each value of (i,j) in turn (except the fixed perimeter values and any interior
fixed values) recalculate  using equation 7 until  converges for all points.
1.4 Non-dimensionalisation
It is common, although not essential, for equations to be made non-dimensional before putting
them into computing form. This reduces the number of variables (both dependent and
independent), increases the level of generality of the solution and can also reduce the absolute
size of values and so reduce round off errors. Non-dimensionalisation was more important in
the earlier days of computing when it was essential to produce general solutions to save
computing time.

8. 8
2 SOLUTION OF ISOTHERMAL REYNOLDS EQUATION
2.1 Rectangular Pads
For uniform viscosity and no squeeze effect, Reynolds' Equation can be reduced to:
x
h
v
y
p
h
y
x
p
h
x
x


=










+











12
3
3 (9)
Using the non-dimensional conversions:
p
B
v
h
p
h
h
h
B
y
y
B
x
x
x
o
o

12
/
/
/
2
*
*
*
*
=
=
=
=
equation 9 reduces to
*
*
*
*
3
*
*
*
*
3
*
*
x
h
y
p
h
y
x
p
h
x 

=














+














 (10)
where  = B2/L2. Differentiating by parts and dividing by h*3 yields
3
*
*
*
*
*
*
*
*
*
*
*
*
*
2
*
*
2
2
*
*
2
1
3
3
h
x
h
y
p
y
h
h
x
p
x
h
h
y
p
x
p


=




+




+


+

 
 (11)
This is a second order partial differential equation of the form (equation 6):
0
2
2
2
2
=
+


+


+


+


E
y
p
D
x
p
C
y
p
B
x
p
A (12)
where A =1, B = , C = 3/h*.h*/x*, D = .3/h*.h*/y*, E = -1/h*3. h*/x*. (12a)
Hence apply the finite difference method (see last section).
Procedure:
(i) For most pads, h*/y* = 0, i.e. D = 0 in equation 12.
(ii) For a linear pad h* = 1 + Kx* and dh*/dx* = K (13)

9. 9
(iii) Cover the pad with a rectangular grid of M x N points (M x N ~ 400 is suggested
initially). Note that iteration time is proportional to (no. of points)3. You need not
have the same number of points in the x as in the y direction.
(iv) Work out CN(I,J), CS(I,J), CE(I,J), CW(I,J), G(I,J) for all interior grid points by
determining the appropriate local A to E values in equation 12 at each point using
equations 12a, 13 and substituting these into equation 8.
(v) Set all pressure values, P(I,J) to zero or to non-dimensional atmospheric pressure
(including the perimeter points).
(vi) Sweep through all interior points, I=2 to M-1, J=2 to N-1 recalculating P(I,J) until
the solution converges. Note that you are holding all the perimeter pressures to
zero or atmospheric pressure throughout. These are the boundary conditions.
Use successive over-relaxation to speed solution. "Successive" means that you
change the value of P(I,J) as soon as you calculate it (Gauss Seidl iteration).
"Overrelaxation" means that you overcorrect the new value of P(I,J) by a set
factor, typically 1.5
Monitor the weighted absolute residual until the pressures reach a preset
convergence criterion, e.g. mean weighed absolute residual R <10-5.

 −
=
s
new
s
old
new
p
p
p
R
value
p
all
value
p
all
Note. To save computing time, most pads are symmetrical about one or more position, e.g.
for rectangular pads along the centre line in the sliding direction, y*=0.5. Thus you need
compute only half the domain, J=1 to N/2 and have a dummy set of points along J = N/2+1
which you give the same values as the corresponding points along J = N/2-1 at the end of
each iteration. The use of symmetry to reduce computing time or memory used to be very
important but is becoming less so as computers become faster and cheaper.
The technique for sector shaped pads is identical, except that you start with the polar
coordinate form of Reynolds' Equation and a sector-shaped grid (2).
2.2 Holes and Pockets
It is easy to incorporate pockets, ports or grooves for oil feeding or jacking purposes. These
are simply held constant at the appropriate value of pressure during the iteration.

10. 10
p
ps1
s2
pressurising
ring
pivot line
p
s
Figure 3
2.3 Pressure Fields for 2-D Journal Bearings
2.3.1 Standard Method
Starting with the 2-D Reynolds Equation given in equation 9, for journal bearings we
substitute the bearing angle,  for the linear position x.
h = c(1+cos), x = R, dx = Rd
Using the non-dimensional terms,
p
R
v
c
p
c
h
h
L
y
y
x
12
/
/
2
*
*
*
=
=
=
equation 10 then yields
*
*
*
*
3
*
*
*
*
3
*
*



 

=














+













 h
y
p
h
y
p
h (14)
where  = R2/L2
Differentiation by parts gives:
3
*
*
*
*
*
*
*
*
*
*
*
*
*
2
*
*
2
2
*
*
2
1
3
3
h
h
y
p
y
h
h
p
h
h
y
p
p





 

=




+




+


+


(15)

11. 11
which is solved just as for the analogous pad equation in the previous section. The only
difference is in the boundary conditions. It is common to use Reynolds' boundary condition
at the exit, which can be obtained by setting P(I,J)=0 if P(I,J) < 0 during a sweep. This allows
the cavitation position to "float". Pockets can be used, just as for thrust pads.
2.3.2 Substituting M = ph3/2
When trying to solve equation 14 at high eccentricity values, difficulty arises in convergence
and in obtaining accurate pressure calculation. This is because the pressure field changes
very rapidly around the minimum film thickness, requiring an impractically fine grid. A
similar problem can arise in thrust pads of high convergence, K.
A common solution is to substitute M = ph3/2 in Reynolds' Equation. This is called the
Vogenpohl substitution (1). M does not tend to infinity as h tends to zero so solution by finite
difference is more accurate. If M is substituted into equation 9 we obtain:
















+


=


+


x
h
h
x
M
x
h
v
h
y
M
x
M
x
2
/
1
2
/
3
2
2
2
2
8
12
 (16)
2.4 Integration to give W, F, Q
Once the pressure field has been obtained, integration will provide the load support, friction
and flow.
2.4.1 Pads
For rectangular pads, load and friction are given by:
 
=
LB
pdxdy
W
00
 
=
LB
dxdy
F
00
 (17a)(17b)
Defining non-dimensional values:
W
LB
v
h
W
x
o
2
2
*
12 
= F
LB
v
h
F
x
o

12
*
=

12. 12
gives (assuming pure sliding with x
v = (vx1+vx2)/2 = vx1/2)
 
=
11
0
*
*
*
*
o
dy
dx
p
W (18a)
*
1
0
*
*
*
*
6
1
4
* dx
h
dx
dp
h
F 








+
−
= (18b)
For flow in the x direction,
dy
q
Q
L
x
x 
=
0
x
o
x
x Q
Lh
v
Q
1
*
= *
0
*
*
3
*
*
*
dy
x
p
h
h
Q
L
x 










−
= (19)
Total flow in the y direction transverse to sliding can be found by integration or from the
difference between the Qx values at the inlet and at the exit.
These integrations are best performed by Simpson's Rule, i.e.
( )
( )
*
*
5
*
4
*
3
*
2
*
1
*
*
...
2
4
2
4
1
3
N
x q
q
q
q
q
q
N
y
Q +
+
+
+
+
−

= (20)
where y is the non-dimensional grid spacing and N the number of points in the y direction.
Note that Simpson's integration only works easily if the number of points is odd.
For W* and F* use Simpson's rule twice, integrating all the points in one direction the first
time and then all the resulting values in the other direction. For F* and Q* the values
p*/x* and p*/y* are needed. For the inner points use the central difference formula
x
p
p
x
p j
i
j
i
ij
ij

−
=

 −
+
2
,
1
,
1
For the edge points fit an approaching parabola to the three edgemost values:

13. 13
c
bx
ax
pij +
+
= 2
b
ax
x
p
ij
ij
+
=


2
At x = 0, i = 1, so b
x
p
ij
ij
=


Solving for the parabola gives
( ) x
p
p
p
b
x
p
j
j
j
ij
ij

−
+
−
=
=


2
/
4
3 3
2
1 (21)
2.4.2 Journal Bearings
For journal bearings, the bearing angle  and the total load support W are found by
integrating for the two load components Wp and Wq.
 
=
L
p dy
d
pR
W
0
2
0
cos



 
=
L
dy
d
pR
Wq
0
2
0
sin



p
q
W
W
−
=

tan
( ) 2
/
1
2
2
q
p W
W
W +
= (22)
p
p
p
1j
2j
3j
x x
Figure 4
x

Wp
Wq


W
P
Figure 5

14. 14
3. MORE COMPLEX HYDRODYNAMIC PROBLEMS
The above notes considered steadily-loaded, isothermal bearings. The remainder of this
handout looks briefly at how this can be extended to more complex systems.
3.1 Thermal Hydrodynamics
The full solution of thermal hydrodynamic problems is quite complex and these notes are
only intended to give an introduction. A thorough review is given in reference (3). To allow
for thermal effects, an Energy Equation must be used. These are based on the energy balance
of an element in steady state, i.e.
Net rate of energy leaving element = Rate of work done on element
In general the main terms are:
Convection + Conduction = Shear dissipation + Compressive work
3.1.1 2-D Systems (2-D Energy Equation + Reynolds' Equation)
A proper treatment of thermal effects requires the temperature to vary through the thickness
of the films and thus involves a 3-D solution. This will be discussed in the next section. For
approximate solutions, however, 2-D energy equations have been developed and used to
solve 2-D hydrodynamic problems. These consider the energy flow into a column of fluid in
the contact and assume temperature is constant through the film thickness. They usually
neglect compression heating and often neglect conduction heat transfer, assuming, for
hydrodynamics, that convection is much greater than conduction. This question was
discussed in the Hydrodynamic Lubrication handout of Tribology I. A typical such energy
equation is (4):


















+








+
=








+


2
2
3
2
12
4
y
p
x
p
c
h
h
c
v
y
T
q
x
T
q
v
v
x
y
x



(23)
(convection) (shear dissipation)
This rearranges to;

15. 15


























+








+
+


−
=


2
2
3
2
12
4
1
y
p
x
p
c
h
h
c
v
y
T
q
q
x
T
v
v
x
y
x 

 (24)
To solve for the temperature field in a bearing, the procedure is then as follows. Put equation
24 in finite difference form. Then begin at some axial line (say an oil groove) where the
temperature and hence T/y is known and use equation 24 to march out the solution of T
downstream. (The approach will fail if qx becomes zero or locally negative at some point.
This limitation results from the assumption that T/z = 0).
In practice both the energy equation and Reynolds' equation must be solved iteratively until
the two converge at some particular solution of p(x,y), T(x,y), (x,y) over the domain.
3.1.2 3-D Systems (3-D Energy Equation + Generalised Reynolds' Equation)
The full energy equation considers heat flux balance in an element of fluid. A clear and full
derivation is given in reference (1). If all the shear dissipation is assumed to result from shear
along the main strain directions, u/x and v/y, if convection and velocity gradient in the z
direction are neglected then we obtain (5)
( ) ( )
)
n
compressio
(
)
shear
(
)
conduction
(
)
convection
(
2
2








+


−


















+








=
















+












+










−








+


y
p
v
x
p
v
T
z
v
z
v
z
T
K
z
y
T
K
y
x
T
K
x
T
c
y
v
T
c
x
v
y
x
y
x
p
y
p
x


(25)
where cv is the lubricant specific heat, K the lubricant thermal conductivity and  the
lubricant thermal expansivity. This is often further simplified by assuming the specific heat
and thermal conductivity do not vary with temperature/pressure and by neglecting conduction
in the x direction:
)
n
compressio
(
)
shear
(
)
conduction
(
)
convection
(
2
2
2
2
2
2








+


−


















+








=










+


−








+


y
p
v
x
p
v
T
z
v
z
v
z
T
y
T
K
y
T
v
x
T
v
c y
x
y
x
y
x
p 


(26)

16. 16
This equation allows the temperature to vary in the z direction. However the conventional
Reynolds' equation does not permit this. A form of Reynolds' equation known as the
"generalised Reynolds' equation" has been produced which holds pressure constant through
the film thickness but allows the viscosity and density to vary (6).
( ) ( ) ( ) ( )
( )( ) ( )( )






−
+


+






+
+
−


−






+
+
−


−








+


=








+


+








+



h
o
y
y
o
x
x
y
x
dz
t
G
V
F
G
F
v
v
y
G
U
F
G
F
v
v
x
y
v
x
v
h
y
p
G
F
y
x
p
G
F
x
0
1
1
2
2
3
1
2
3
1
2
3
1
2
3
1
2
2
2
2
2
1
2
1
2







(27)
where:
( ) 


 =
−
=
=
=
=
h
h
o
h
h
o dz
z
F
dz
z
z
z
F
F
F
z
zdz
F
dz
F
0
3
0
2
1
0
1
0
/






dz
z
z
G
dz
dz
z
z
G
dz
dz
z
dz
z
z
z
G
h
h z
h z
z

 
 



=










=














−


=
0
3
0 0
2
0 0
0
1






For steady state systems where there is no transverse sliding, the last two terms in equation 27
can be neglected. This version of Reynolds' equation can be solved using just the same 2-D
finite difference technique as used in section 2.
The solution to the thermal hydrodynamic problem involves the iterative solution of the
generalised Reynolds' equation and the energy equation until pressure convergence is
reached. Once conduction through the film is allowed, it is also necessary to take into
account the effect of heat transfer on the temperatures of the bounding solid surfaces. This
entails using a surface temperature rise/heat flux model. A typical example is given in
reference (7).
3.2 Non-Steady State Problems
In a number of hydrodynamic systems the load or the entrainment speed vary cyclically. Two
obvious examples are piston ring/liner contacts and dynamically-loaded crankshaft bearings.
These are usually solved as quasi-steady state problems. The applied load is assumed to be in
equilibrium with the fluid pressure, i.e.

17. 17

=
W
pdxdy
W (28)
The pressure is calculated from Reynolds' equation
dt
dh
x
h
v
y
p
h
y
x
p
h
x
x 12
12
3
3
+


=














+
















(29)
The squeeze rate dh/dt will be constant throughout the contact, and is thus equal to dho/dt
where h0 is the minimum film thickness. dho/dt is unknown but can be found by solving
equations 28 and 29 in combination.
The procedure for solving say a piston contact, (e.g. reference (8)) is now:
(i) Start at a particular position in the system cycle, e.g. for a piston, at mid-stroke.
Guess an initial minimum film thickness, ho at an initial time t = 0.
(ii) Solve equations 28 and 29 in combination to find dho/dt. This can be done
iteratively or by a numerical method of solving simultaneous non-linear integral
equations (7).
(iii) Work out the new minimum film thickness a small time step further round the
cycle (e.g. one crank angle) using a simple marching method; for time step i,
( )
i
i
o
i
i
o t
t
dt
dh
h
h −
+
= +
+ 1
,
0
1
,
(30)
(This is a very simple marching method; more complex ones, e.g Runga Kutta can
be used).
(iv) Repeat stages (ii) and (iii) round the system cycle.
(v) When you get all the way round the cycle (e.g. after four strokes for a piston),
start to compare your predicted minimum film thicknesses with the values on the
last cycle. Continue until the values converge throughout a cycle. (This usually
only takes about 2 cycles).
3.3 EHD Problems
The Reynolds' and energy equations outlined above are also used in solving EHD problems.
The main differences are:

18. 18
(a) The pressure are very much higher in EHD and the lubricant viscosity must be
allowed to rise with pressure.
(b) The solution of the hydrodynamic pressure is embedded as an inner loop in two
outer cycles, one to determine the effect of film pressure on geometry of the
contact due to surface deformation and an outer one to compare the minimum film
thickness with the applied load. The whole solution scheme was discussed in the
Elastohydrodynamic Lubrication handout of Tribology I.
There are two major problems:
(i) The viscosity varies in an exponential fashion with pressure, e.g.  = 0ep. This
makes Reynolds equation very non-linear. A substitution which helps to linearise
the equation is the "reduced pressure", q defined as:

p
e
q
−
−
=
1
This has the derivative dq = e-p.dp so that the differential in Reynolds' equation












x
p
h
x 
3
becomes












x
q
h
x o

3
(ii) The amount of elastic deformation produced in high pressure EHD contacts is
usually far greater than the final film thickness. This means that deformations
have to be calculated with great accuracy so as not to overwhelm film thicknesses.
The numerical solution of the EHD problem is not to be undertaken lightly. References 5,
and 9 describe some of the techniques.
3.4 Control Volume Approach
In all of this handout so far, the solution for fluid pressure in a film is based upon Reynolds'
equation. Recently, work has started to apply control volume methods to solve for fluid flow
in a 3-D arrangement of elements. The flow into each face of an element is related to the
bounding pressures using the reduced Navier-Stokes equations, equation 1. The net flow is
then conserved using the mass continuity equation 2. The bounding pressures and flows are
then equated for all elements to solve over the whole contact. This approach (10), which is
widely used in conventional fluid dynamics is more versatile than Reynolds' and in principle

19. 19
allows many of the underlying assumptions inherent in the Reynolds' approach to be
removed. It may also be more accurate.
REFERENCES
1. A Cameron, Principles of Lubrication , Chapter 3, publ. Longmans 1966.
2. Pinkus, O. and Lynn, W. "Solution of the Tapered-Land Sector Thrust Bearing", Trans.
ASME, 80, pp. 1510-1616, (1958).
3. Fillon, M., Frene, J. and, Boncompain, R. "Historical Aspects and Present Development
on Thermal Effects in Hydrodynamic Bearings", Proc. Leeds Lyon Symposium, Fluid
Film Lubrication - Osborne Reynolds Centenary, pp 27-47, ed. D Dowson et. al., publ.
Elsevier, 1987.
4. Stokes, M.J.and Ettles, C.M.M. "A General Evaluation Method for the Diabatic Journal
Bearing", Proc. Roy. Soc. Lond. A336, pp. 307-325, (1974).
5. R Gohar, Elastohydrodynamics, publ. Ellis Horwood, Chichester, 1988.
6. Dowson, D. "A Generalised Reynolds Equation for Fluid-Film Lubrication", Int. J.
Mech. Sci. 4, pp. 159-170, (1962).
7. Dong, Z. and Wen Shi-Zhu, "A Full Numerical Solution for the Thermoelastic Problem
in Elliptical Contacts." Trans. ASME J .of Trib. 106, pp 246-254, (1984).
8. Jeng Y-R "Theoretical Analysis of Piston Ring Lubrication. Part I - Fully Flooded
Lubrication", Tribology Trans. 35, pp 696-706, (1992).
9. C.H. Venner "Multilevel Solution of the EHL Line and Point Contact Problems" PhD
Thesis, Twente University, January 1991.
10. Blahey, A.G. "The Elastohydrodynamic Lubrication of Elliptical Contacts with
Thermal Effects", PhD Thesis, University of Waterloo, Ontario, 1985.