7. Hydrodynamic Lubrication: Reynolds Equation...Boundary lubrication Mixed lubrication Hydrodynamic...

IMAC-U course 6th semester

Tribology 7. Hydrodynamic Lubrication: Reynolds Equation

Assoc. Prof. Takeshi YAMAGUCHI yamatake@gdl.mech.tohoku.ac.jp

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