Page 192 - Tribology in Machine Design

Sliding-elemen t be a rings 17 7

U 2(dh/dx) normal to the top area, that may be taken as dx dz because of its
very small inclination in bearings. In Fig. 5.3, case (b), flow at velocity U 2 is
shown through the projected area (dh/dx)dxdz, which is shaded. Either
analysis gives the same product of velocity and area. Hence the total flows
qi inwards through the lower boundary of the geometric space and q 2
outwards through the upper boundary area, are respectively

Continuity with an incompressible fluid requires that the total inward flow
across the boundaries equals the total outward flow, or

For the case of a compressible fluid (gas bearings), mass flows instead of
volume flows wound be equated. A relationship between density and
pressure must be introduced. With substitution from eqns (5.3) and (5.4)
into eqn (5.5), selective differentiation, and elimination of the product dx dz,
the result is

With rearrangement

The last two terms are nearly always zero since there is rarely a change in
the surface velocities U and W, which represents the stretch-film case. The
stretch-film case can occur when there is a lubricating film separating a wire
from the die through which it is being drawn. Reduction in the diameter of
the wire gives an increase in its surface velocity during its passage through
the die.
This basic equation of hydrodynamic lubrication was developed for a less
general case in 1886 by Osborne Reynolds. As usual, the eqn (5.7) and its
reduced forms in any coordinate system shall be referred to as the Reynolds