9.1 Introduction to Fluid-Film Lubrication

Hydrodynamic lubrication is the engineering discipline of separating two solid surfaces in relative motion using a continuous fluid film generated by the motion itself. Unlike boundary lubrication (where surface chemistry dominates) or solid lubrication (Chapter 8), hydrodynamic lubrication relies on viscous flow to create pressures that support the applied load without direct asperity contact.

The fundamental requirement for hydrodynamic action is a converging wedge — a geometry where the gap between the surfaces decreases in the direction of motion. As the moving surface drags viscous fluid into this converging region, the fluid cannot escape fast enough, pressure builds, and the surfaces are forced apart. This self-generating pressure is the essence of every journal bearing, thrust bearing, and (in modified form) elastohydrodynamic gear contact.

Without hydrodynamic lubrication, modern machinery — turbines, compressors, electric motors, automotive engines — could not operate at the speeds and loads we demand. Bearing designers, lubrication engineers, and failure analysts must all master the concepts in this chapter.

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9.2 The Stribeck Curve and Lubrication Regimes

The Stribeck curve is the central organizing diagram of tribology. It plots the coefficient of friction $\mu$ against a dimensionless speed parameter, revealing three distinct lubrication regimes.

9.2.1 The Hersey Number

The original Stribeck parameter is the Hersey number:

$$\mathrm{Hersey} = \frac{\eta N}{P}$$

where:

• $\eta$ = dynamic viscosity (Pa·s or reyn)
• $N$ = rotational speed (rev/s)
• $P$ = average bearing pressure (Pa)

Increasing the Hersey number (higher speed, higher viscosity, lower load) moves the operating point rightward on the Stribeck curve toward full-film lubrication.

9.2.2 Three Regimes

1. Boundary lubrication (left side of curve): Metal-to-metal asperity contact. Friction is governed by surface chemistry, adsorbed films, and additives (ZDDP, MoDTC). $\mu$ is high and relatively constant (0.08–0.15 for steel).

1. Mixed lubrication (center, minimum friction): Partial asperity contact with partial film support. Friction reaches a minimum as adhesive contact diminishes faster than viscous drag increases. This is the most complex regime to analyze.

1. Hydrodynamic (full-film) lubrication (right side): Complete fluid separation. Friction increases with speed (viscous shear in the film). Wear is negligible. $\mu$ is low but rising with speed.

9.2.3 The Lambda Ratio

A more physically meaningful parameter is the specific film thickness (lambda ratio) from Chapter 8:

$$\lambda = \frac{h_{\min}}{\sqrt{R_{q1}^2 + R_{q2}^2}}$$

Design target: $\lambda > 3$ for reliable hydrodynamic operation with minimal wear.

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9.3 Journal Bearing Geometry and Operation

The journal bearing (sleeve bearing, plain bearing) is the archetypal hydrodynamic bearing. A cylindrical shaft (journal) rotates inside a slightly larger cylindrical bore (sleeve) with radial clearance $c = R_{bore} - R_{journal}$.

9.3.1 Eccentricity and Film Profile

At rest, the journal rests on the bore (eccentricity $e = c$). As speed increases, hydrodynamic pressure lifts the journal toward the center. The eccentricity ratio is:

$$\varepsilon = \frac{e}{c}$$

where $e$ is the distance between journal and bore centers, and $c$ is the radial clearance. At rest: $\varepsilon = 1$. At full speed with light load: $\varepsilon \to 0$.

The minimum film thickness occurs at the point of closest approach:

$$h_{\min} = c(1 - \varepsilon)$$

The attitude angle $\phi$ is the angle between the load line and the minimum-film-thickness line. It increases with speed and decreases with load.

9.3.2 Pressure Distribution

The pressure profile around the bearing is computed from the Reynolds equation (Section 9.5). Key features:

• Positive pressure in the converging wedge (supporting the load).
• Zero pressure in the diverging region (cavitation boundary).
• Peak pressure occurs upstream of the minimum film thickness point.

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9.4 The Sommerfeld Number

The Sommerfeld number is the primary dimensionless parameter for journal bearing design:

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

where:

• $r$ = journal radius (m)
• $c$ = radial clearance (m)
• $\eta$ = dynamic viscosity (Pa·s)
• $N$ = rotational speed (rev/s)
• $P$ = bearing pressure = $W/(2rL)$ (Pa), with $W$ = radial load and $L$ = bearing length

9.4.1 Physical Meaning

• $(r/c)^2$: Geometric factor — tight clearance (small $c$) increases $S$.
• $\eta N / P$: The Hersey number — higher viscosity, higher speed, or lower load increases $S$.

Higher $S$ means better lubrication: lower eccentricity, thicker minimum film, lower friction.

9.4.2 Raimondi–Boyd Charts

For $L/D = 1$ and full Sommerfeld (360°) bearings, Raimondi–Boyd charts provide:

• Eccentricity ratio $\varepsilon$ vs. $S$
• Minimum film thickness ratio $h_{\min}/c$ vs. $S$
• Attitude angle $\phi$ vs. $S$
• Friction coefficient $f$ vs. $S$

These charts are the standard design tool for preliminary journal bearing analysis.

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9.5 The Reynolds Equation

The Reynolds equation is the fundamental PDE governing pressure generation in thin fluid films. For steady, incompressible, isoviscous flow in a thin film:

$$\frac{\partial}{\partial x}\left(h^3 \frac{\partial p}{\partial x}\right) + \frac{\partial}{\partial z}\left(h^3 \frac{\partial p}{\partial z}\right) = 6\eta U \frac{\partial h}{\partial x}$$

where $h(x,z)$ is the film thickness, $p$ is pressure, $\eta$ is viscosity, and $U$ is the surface velocity.

9.5.1 Physical Interpretation

The right-hand side is non-zero only where $\partial h/\partial x \neq 0$ — i.e., in the converging wedge. This is the mathematical statement that pressure is generated only where the gap narrows in the direction of motion.

9.5.2 One-Dimensional Form

For an infinitely long bearing ($\partial p/\partial z = 0$):

$$\frac{d}{dx}\left(h^3 \frac{dp}{dx}\right) = 6\eta U \frac{dh}{dx}$$

This ODE, with appropriate boundary conditions ($p = 0$ at inlet and outlet; cavitation at $dp/dx = 0$), yields the classical half-Sommerfeld pressure distribution.

9.5.3 Assumptions and Limitations

The Reynolds equation assumes:

• Laminar flow (Reynolds number based on film thickness $< 2000$).
• Negligible inertia (low modified Reynolds number).
• Constant viscosity (isothermal).
• No slip at walls.

Violations occur at very high speeds (turbulence), very high pressures (EHL, variable viscosity), or with textured/slippery surfaces.

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9.6 Elastohydrodynamic Lubrication (EHL)

In concentrated contacts (gears, rolling bearings, cam followers), the contact pressures are so high (0.5–3 GPa) that:

1. Elastic deformation of the surfaces significantly modifies the geometry.
2. Pressure-viscosity effect dramatically increases the oil viscosity.

This combined phenomenon is elastohydrodynamic lubrication (EHL).

9.6.1 Pressure-Viscosity Relationship

The Barus equation models viscosity increase with pressure:

$$\eta(p) = \eta_0 e^{\alpha_{pv} p}$$

where $\eta_0$ is the ambient viscosity and $\alpha_{pv}$ is the pressure-viscosity coefficient (typically 10–25 GPa⁻¹ for mineral oils).

At $p = 1$ GPa with $\alpha_{pv} = 20$ GPa⁻¹: $\eta(p) = \eta_0 e^{20} \approx 5 \times 10^8 \eta_0$. The oil behaves nearly as a solid in the contact inlet.

9.6.2 EHL Film Thickness

The central film thickness in a line contact (Hamrock–Dowson):

$$h_c = 2.65 R^{0.54} (\eta_0 U)^{0.7} E^{*-0.03} W^{-0.13}$$

where $R$ is the equivalent radius, $U$ is the rolling velocity, $E^*$ is the reduced modulus, and $W$ is the load per unit width. Film thickness is weakly dependent on load ($W^{-0.13}$) but strongly dependent on speed and viscosity.

9.6.3 EHL Pressure Profile

The EHL pressure distribution features:

• A sharp pressure spike at the outlet (the pressure spike or flash temperature zone).
• A nearly flat region in the central contact (the Hertzian plateau modified by elasticity).
• A steep inlet pressure gradient where viscosity traps the fluid.

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9.7 Oil Degradation and Lubricant Life

Lubricating oils do not last forever. Degradation mechanisms include:

9.7.1 Thermal Oxidation

At temperatures above 80–100°C (lower for extended-drain oils), base oil molecules react with dissolved oxygen, forming acids, sludge, and varnish. Oxidation stability is improved by:

• Antioxidant additives (hindered phenols, amines).
• Synthetic base stocks (PAO, ester) with inherently better oxidation resistance.
• Cooler operating temperatures.

9.7.2 Shear Degradation

High shear rates in EHL contacts (10⁶–10⁷ s⁻¹) can break polymer viscosity index improvers (VII), permanently reducing viscosity. This is critical for multi-grade oils (5W-30, 10W-40) in high-performance engines and gearboxes.

9.7.3 Contamination

Water, fuel dilution, soot, and wear particles accelerate degradation and cause:

• Emulsification (water) → reduced film strength.
• Viscosity thinning (fuel) → boundary contact risk.
• Abrasive wear (particles) → accelerated component wear.

9.7.4 Monitoring

Oil condition monitoring uses:

• Viscosity change (±10% from new oil is a warning).
• Acid number (TAN) increase (oxidation products).
• Particle counting (ISO 4406 cleanliness codes).
• Ferrography (wear particle morphology).

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9.8 Bearing Design Procedure

A systematic journal bearing design follows these steps:

1. Define loads and speeds: Radial load $W$, speed $N$, temperature range.
2. Select oil grade: Based on viscosity at operating temperature (ISO VG 32, 46, 68, etc.).
3. Choose $L/D$ ratio: Typically 0.5–1.5. Short bearings ($L/D < 0.5$) have edge leakage; long bearings ($L/D > 2$) are inefficient.
4. Set clearance ratio: $c/r = 0.001$–$0.002$ (tight for precision, loose for dirty environments).
5. Compute Sommerfeld number $S$ at operating conditions.
6. Read Raimondi–Boyd charts for $\varepsilon$, $h_{\min}$, $f$.
7. Check $\lambda$ ratio — must exceed 3 for full-film operation.
8. Check temperature rise — friction power $P_f = f W r \omega$ must be dissipated without exceeding oil temperature limits.
9. Verify static load capacity at start-up (boundary conditions).

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9.9 Thrust Bearings and Tilting-Pad Designs

While journal bearings support radial loads, thrust bearings support axial loads using the same hydrodynamic wedge principle applied to flat or tapered surfaces.

Tilting-pad journal bearings use multiple pads that pivot to align with the oil wedge, providing:

• Excellent stability (no half-speed whirl).
• Adjustable stiffness and damping.
• Used in turbines, large motors, and high-speed machinery.

Squeeze-film dampers add a thin oil film between the bearing housing and the support structure, providing viscous damping that suppresses rotor vibration at critical speeds (Chapter 10).

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9.10 Lubrication Failure Modes

Design imperative: Always verify $\lambda > 3$ at the most severe operating condition (cold start, maximum load, minimum speed). The Stribeck curve tells you where you are; the Reynolds equation and Sommerfeld number tell you how to get there.