---
title: "Hydrodynamic lubrication theory and film formation"
subject: "Machine Design I"
chapter: 7
key: "semester_6/machine_design_i:hydrodynamic-lubrication-theory"
source: https://www.bctengineer.com/q/semester_6/machine_design_i/hydrodynamic-lubrication-theory
site: BCT Engineering
---

# Hydrodynamic lubrication theory and film formation

Machine Design I · Chapter 7: Lubrication and journal bearings

- Asked in: year not given, 6 marks
- Video: [L 13 Hydrodynamic Lubrication  | Tribology  and Terotechnology  | Mechanical](https://www.youtube.com/watch?v=odUVIux5w7U) — MECHANICAL & AUTOMOBILE ENGINEERING_LJIET
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/machine_design_i/hydrodynamic-lubrication-theory
- Whole chapter: https://www.bctengineer.com/md/semester_6/machine_design_i/chapter-7

## Question

Explain the hydrodynamic theory of lubrication and how an oil film forms in a journal bearing. State the assumptions made by Reynolds, define eccentricity ratio, minimum film thickness and Sommerfeld number, and give the relation between them.

## Short answer

Reynolds' theory assumes Newtonian incompressible lubricant, laminar flow, negligible inertia, constant viscosity, no side leakage. Film forms when rotating journal drags oil into converging wedge, generating pressure that lifts shaft. Eccentricity ratio ε = e/c; minimum film thickness h0 = c(1-ε); Sommerfeld number S = (r/c)² μN/p. Charts relate S to ε.

## Answer

**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

```text
   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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How does the oil film form once a journal starts rotating?
   A. Oil is forced in by a pump until the shaft floats
   B. The shaft stays at the bottom and oil is squeezed out
   C. The journal drags oil by viscosity into the converging wedge, building pressure that lifts it
   D. The oil expands due to heating and lifts the shaft
   Answer: C. Viscous drag carries oil into the narrowing space, pressure builds, and at sufficient speed it supports the load. (Key point: Film formation)

2. When running with a full hydrodynamic film, where is the journal centre?
   A. Exactly at the bearing centre
   B. Displaced by e from the bearing centre, slightly to one side of the load line
   C. On the load line directly below the bearing centre
   D. At the top of the bearing
   Answer: B. The journal takes an eccentric position offset by e and slightly to the side of the load line to form the wedge. (Key point: Journal position)

3. Which is one of Reynolds' assumptions?
   A. The flow is turbulent and compressible
   B. Inertia and body forces are important
   C. Pressure varies strongly across the film thickness
   D. The lubricant is Newtonian and incompressible with constant viscosity and laminar flow
   Answer: D. Reynolds assumed Newtonian, incompressible, constant viscosity, laminar flow, negligible inertia, and pressure constant across the film. (Key point: Reynolds assumptions)

4. Which statement about the Sommerfeld number S = (r/c)^2 mu N / p is correct?
   A. A high S means large eccentricity and a thin film
   B. S has units of Pa.s
   C. A high S (viscous oil, high speed, light load) means a thick film and small eccentricity
   D. S depends only on the clearance
   Answer: C. S is dimensionless; high S corresponds to a thick film and small eccentricity, low S to large eccentricity and a thin film. (Key point: Sommerfeld number)

## Related in Machine Design I

- [Journal bearing design with film thickness check](https://www.bctengineer.com/question.md?key=semester_6%2Fmachine_design_i%3Ajournal-bearing-minimum-film-thickness) — Machine Design I, chapter 7

## Related in other subjects

- [Hydrodynamic lubrication and Petroff's equation](https://www.bctengineer.com/question.md?key=semester_6%2Finternal_combustion_engines%3Ahydrodynamic-lubrication-petroff-numerical) — Internal Combustion Engines, chapter 6
- [Plate sliding on inclined oil film](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aincline-plate-oil-viscosity) — Fluid Mechanics, chapter 1
- [Nusselt theory of film condensation](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Anusselt-film-condensation-theory) — Heat Transfer, chapter 5