Hydrodynamic lubrication theory and film formation

Hydrodynamic theory (Osborne Reynolds, 1886, after Tower's experiments) explains how a pressure is generated in a converging oil film, so that a shaft can float on oil.

Film formation in a journal bearing

   at rest        starting        running
   _____           _____          _____
  (  O  )         ( O   )        (  O  )
 shaft on        climbs up      floats on a
 bottom          the bearing    wedge of oil
                 wall           (offset by e)

1. At rest, the journal rests on the bottom of the bearing, with oil squeezed out.
2. When the shaft starts to rotate it climbs the bearing wall in boundary lubrication.
3. As speed increases, the rotating journal drags oil by viscosity into the narrowing space (converging wedge). The oil is pushed along the wedge and pressure builds up.
4. At a certain speed the pressure supports the load; the journal lifts and takes an eccentric position, with its centre displaced by $e$ from the bearing centre and slightly to one side of the load line.

Assumptions of Reynolds

• The lubricant is Newtonian and incompressible, with constant viscosity.
• Flow is laminar; inertia and body forces are neglected.
• Pressure does not change across the film thickness.
• Film thickness is very small compared with the radius, so curvature is neglected.
• Surfaces are rigid and smooth, and there is no slip at the boundaries.
• (For the simplest case) no side leakage, i.e. infinitely long bearing.

Terms

• Radial clearance $c = R - r$ (bearing radius minus journal radius).
• Eccentricity ratio $\varepsilon = e/c$.
• Minimum film thickness $h_0 = c - e = c(1 - \varepsilon)$.
• Sommerfeld number (bearing characteristic number)

$$S = \left(\frac{r}{c}\right)^2 \frac{\mu N}{p}$$

where $\mu$ is viscosity (Pa.s), $N$ speed (rev/s) and $p = W/(ld)$ the projected-area pressure. $S$ is dimensionless. For a given $l/d$, the design charts (Raimondi and Boyd) give $\varepsilon$ (or $h_0/c$), friction variable, flow and temperature rise as functions of $S$. A high $S$ (viscous oil, high speed, light load) means a thick film and small eccentricity; a low $S$ means large eccentricity and a thin film.