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IMAC-U course 6th semester
Tribology 7. Hydrodynamic Lubrication: Reynolds Equation
Assoc. Prof. Takeshi YAMAGUCHI [email protected]
Stribeck curve Fr
icti
on
co
effi
cie
nt
(Viscosity×velocity)/load (equivalent to fluid film thickness)
Dry friction
Boundary lubrication
Mixed lubrication
Hydrodynamic lubrication
Bearing
Shaft
Lubricating oil
Upper fixed surface
Lower moving surface
U
x
y
dy
dx
t
t+dt
p+dp p
Microelement in fluid
h
dx
dhU
dx
dph
dx
d6
3
h(7.1)
If h=h(x) is known, a distribution of fluid pressure can be obtained.
Reynolds equation
Assumptions
1. Fluid is incompressible Newtonian fluid. 2. Flow in the gap is a laminar flow and viscosity of the
fluid is constant. 3. Inertia force of fluid is negligibly small compared to its
viscous force. 4. Pressure change in the gap direction can be neglected
because the thickness of fluid film is very small.
Derivation of Reynolds Equation (1)
0
dxdy
dy
ddxdydx
dx
dpppdy
ttt (7.2)
(7.3)
dy
d
dx
dp t (7.3)
dy
duht (7.4)
Where h is viscosity of fluid, and u is the flow velocity in the x direction.
(7.5)
Derivation of Reynolds Equation (2)
Derivation of Reynolds Equation (3)
2
2
dy
ud
dx
dph (7.5) (7.6)
2nd integration
Boundary condition: y=0, u= U; y=h, u=0
(7.7)
Couette flow and Piseuille flow
dx
dpyhy
h
yhUu
h2
1
（ ） （ ）
（ ） （ ）
(7.7)
Derivation of Reynolds Equation (4)
Flow rate per unit depth length Q:
h
dx
dphUhudyQ
0
2
122 h(7.8) (7.9)
Mass conservation raw
(7.1)
Generalization of Reynolds Equation
Upper moving surface
Lower moving surface U1
x
y
h z
U2
V
W2
W1
u
v
w h
2
2
2
2
y
w
z
p
y
u
x
p
h
h (7.10)
(7.11)
Motion equation of fluid Boundary conditions
y = 0: u = U1, w = W1 = 0
y = h: u = U2, w = W2 = 0 (7.12)
t
hh
x
UUh
x
hUU
z
ph
zx
ph
x
1266 2121
33
hh
Generalization of Reynolds Equation
Upper moving surface
Lower moving surface U1
x
y
h z
U2
V
W2
W1
u
v
w h
(7.13)
Wedge film action Stretch film action Squeeze film action
t
hh
x
UUh
x
hUU
z
ph
zx
ph
x
1266 2121
33
hh
Wedge film action
Wedge film action
U
0
x
hIt requires inclined surfaces to generate a fluid film wedge action that results in a pressure wave. Positive pressure is generated if the film thickness reduces in the x direction.
t
hh
x
UUh
x
hUU
z
ph
zx
ph
x
1266 2121
33
hh
Wedge film action
When the wall velocity reduces in the x direction, negative fluid pressure is generated. This is not usually occurred in normal sliding surfaces.
Stretch film action
0
x
U
t
hh
x
UUh
x
hUU
z
ph
zx
ph
x
1266 2121
33
hh
Wedge film action
C
A positive pressure is generated if the surfaces are approaching each other.
Squeeze film action
-V
0
t
h
