Mechanics of Fluids

Fundamentals of the theory of hydrodynamic lubrication 225
overcome the viscous resistance is dissipated as heat, and the consequent
variations in the viscosity and density of the lubricant cause the centre of
pressure to move forward. A centrally pivoted slipper is of course suitable
for movement in either direction.
We have seen that the heel of the slipper should not touch the bearing
plate because lubrication would then break down entirely. In practice there
is a lower limit, greater than zero, to the clearance at the heel. This limit is
largely governed by the inevitable lack of perfect smoothness of the surfaces
and the probable size of solid particles (e.g. grit) in the lubricant. If these
particles are unable to escape under the heel of the bearing, they collect at
this point and score the surfaces.
The tangential force (i.e. the resistance to the relative movement between
the slipper and the bearing plate) may also be calculated. The viscous shearing
stress at any point in the film of lubricant is given by τ = µ(∂u/∂y). The
velocity u is given by eqn 6.18
�
δp∗ �
y2 = µu + Ay + B
δx 2
It is again permissible to drop the asterisk from the symbol p ∗ since we consider
a horizontal slipper bearing, and again the inclination of the slipper is
assumed so small that any component of velocity in the vertical direction is
negligible. Since the rate of change of pressure with x is not constant, the
differential dp/dx must be used instead of the ratio δp/δx. The integration
constants A and B are determined by the conditions at the boundaries. Referring
to Fig. 6.19, the velocity of the lubricant must equal the velocity of the
bearing plate when y = 0. Substituting u = V and y = 0 gives B =−µV.
If the clearance is h at a particular value of x, the other boundary condition
is that u = 0 when y = h (since the slipper is considered stationary).
∴ dp h
dx
2
dp
= 0 + Ah − µV whence A =
2 dx
Thus the expression for velocity at any point becomes
h
2
+ µ
h V
dp y
dx
2 dp h µ
= µu + y + Vy − µV
2 dx 2 h
and differentiation with respect to y then gives an expression for the viscous
stress in the x direction at any value of y
µ ∂u
∂y =−dp
� �
h
− y −
dx 2 µV
(6.41)
h
The stress at the bearing plate is given by the value of this expression when
y = 0, that is, −(dp/dx)(h/2)−µV/h. (The sign convention for stress shows
that the stress on the bearing plate is given by µ(∂u/∂y) since the plate faces
the direction of increase of y.) Thus the total tangential force on the bearing
plate divided by the width of the plate is
� l�
�
dp h µV
− + dx
dx 2 h
0