Mechanics of Fluids

Fundamentals of the theory of hydrodynamic lubrication 221
If the further assumption is made that the surfaces of the slipper and
bearing plate are parallel, the behaviour of the lubricant corresponds to
laminar flow between parallel boundaries, one of which is moving. Assuming
steady conditions and a constant viscosity, we may therefore use eqn 6.23:
Q
b =−δp∗
c
δx
3 Vc
+
12µ 2
In view of the negligible differences of elevation within the lubricant film, the
asterisk may be dropped from p ∗ , Then, with the volume flow rate divided
by width Q/b written as q, the equation may be rearranged to give (in the
limit as δx → 0)
dp 12µ
=
dx c3 � �
Vc
− q
2
(6.34)
We have assumed c constant, and from the principle of continuity and
the assumption of two-dimensional constant-density flow, q is constant.
So, from eqn 6.34, dp/dx = constant. But this constant must be zero because
the pressure has the same value at each end of the slipper, say p0 (usually
atmospheric). In other words, there is no variation of pressure throughout
the space between slipper and plate. The same pressure, p0, also acts
uniformly over the outer surface of the slipper and on the exposed part of
the lubricant film, so the lubricant exerts no resultant force normal to the
boundaries. We are therefore led to conclude that the bearing surfaces are
incapable of supporting any load – except when they actually touch each
other, but such contact of course defeats the purpose of lubrication.
This conclusion, however, depends on the assumptions of constant density
and viscosity. If a film of lubricant could in fact be maintained between the
slipper and the plate, the energy needed to overcome the viscous resistance
would be dissipated as heat, and the temperature of the lubricant would
increase in the direction of flow. Consequently there would be not only a
variation of viscosity but also a decrease of density in the flow direction
and thus an increase of q. (We may disregard any variation of temperature,
and therefore viscosity, across the very small clearance c, and, for a liquid
lubricant, the effect of pressure on density may also be neglected.) In such
circumstances a parallel slider bearing could in fact support a load. This
would require the pressure in the lubricant to increase from the ambient
value p0 at one end of the slipper, pass through a maximum, and return to
p0 at the other end of the slipper, as shown in Fig. 6.18. At the peak value
dp/dx = 0 and there, by eqn 6.34, q = Vc/2; negative; but downstream of
this position q would be greater, thereby making dp/dx negative, whereas
upstream dp/dx would be positive. (Although the derivation of eqn 6.34
involves the assumption of constant density, a small, non-zero, value of
δp/δx has a negligible effect on that equation, and does not undermine the
present argument.) In addition, the small clearance c might not remain uniform
even for surfaces initially parallel. Significant variations of c may be
caused by quite tiny distortions of the surfaces, particularly as a result of
temperature changes.