Page 365 - Analysis and Design of Machine Elements

Figure 12.9 Velocity and pressure distributions in a converged wedge. Sliding Bearings 343

Figure 12.9 illustrates the superposition of these two terms to obtain the velocity at the
inlet, the maximum pressure point and the outlet. The parabolic term may be additive
or subtractive to the linear term, depending on the sign of pressure gradient at the inlet
and outlet. The figure also presents the variation of pressure gradient and pressure dis-
tribution. At the section where pressure reaches the maximum value, that is, p/ x = 0,
and the velocity gradient is linear.
Define Q as the volume of lubricant flowing across a section in the x direction per unit
of time. For unit width in the z direction,
h
Q = udy (12.7)
∫
0
Substituting the value of u from Eq. (12.6) and integrating, gives
3
vh h p
Q = − (12.8)
2 12 x
When p/ x = 0, the pressure reaches the maximum value p max ,and h = h , the volume
0
flow rate is
vh 0
Q = (12.9)
2
Becauseitisassumedthatthe lubricantisincompressible, theflow rate is constant for
all sections. Thus
h p vh vh 0
3
− + =
12 x 2 2
Convert it to
p 6 v
= (h − h ) (12.10)
0
x h 3
This is the classical one-dimensional Reynolds equation. It neglects side leakage, that
is, the flow in the z direction.