Petroff Friction⚠ unverified

Understanding the Parameters

• Bearing number $\mu N/P$ — the same Hersey number as in the Stribeck curve. Higher viscosity or speed (or lower load) thickens the film and raises viscous friction — the counter-intuitive hallmark of full-film lubrication.

• Radius-to-clearance ratio $r/c$ — typically 500–1000 (clearance is ~0.001 of the radius). Because the oil is sheared across the tiny clearance $c$, a smaller clearance means a steeper velocity gradient and more shear friction — hence $f \propto r/c$. This large ratio is why the small bearing number still yields a real friction coefficient.

• Concentric assumption — Petroff's law assumes the journal runs nearly centred in the bearing (light load, thick uniform film), so the oil film is an even annulus sheared like a simple Couette flow. Real loaded bearings run eccentric (see Minimum Film Thickness), so Petroff over/under-estimates and the full Sommerfeld Number analysis is used — but Petroff captures the physics and magnitude.

Derivation (Approaching a Proof)

Model the lightly-loaded journal as a shaft of radius $r$ rotating concentrically inside a bearing, separated by a uniform oil film of thickness (clearance) $c$. The oil is sheared in simple Couette flow: the surface speed of the journal is $U = 2\pi r N$, sheared across the gap $c$, so the shear stress is

$$\tau = \mu\,\frac{U}{c} = \mu\,\frac{2\pi r N}{c}.$$

This shear acts over the journal surface area $A = 2\pi r L$ ($L$ = bearing length), giving a friction force and hence a friction torque about the axis:

$$T_f = \tau\,A\,r = \mu\,\frac{2\pi r N}{c}\,(2\pi r L)\,r = \frac{4\pi^2 \mu N r^3 L}{c}.$$

The friction coefficient is the friction force divided by the load, $f = F_f/W$, where the load is $W = P\,(2 r L)$ (unit load × projected area) and the friction force is $F_f = T_f/r$:

$$f = \frac{T_f/r}{W} = \frac{4\pi^2 \mu N r^2 L/c}{P\,(2 r L)} = 2\pi^2\,\frac{\mu N}{P}\,\frac{r}{c}.$$

The whole result flows from Newtonian (Couette) shear of the film — no empirical constant. It is exact for the concentric, light-load case and defines the rising, linear-in-$\mu N/P$ hydrodynamic branch that the Stribeck curve joins onto.

Dimensional check. $\dfrac{\mu N}{P}\dfrac{r}{c} = \dfrac{(\text{Pa}\cdot\text{s})(1/\text{s})}{\text{Pa}}\cdot\dfrac{\text{m}}{\text{m}} = 1$ (dimensionless). ✓

History and Development

Nikolai Petrov (Petroff) published this analysis in 1883 — the first theoretical treatment of hydrodynamic bearing friction, predating and complementing Reynolds' 1886 lubrication theory and Tower's experiments. It remains the textbook starting point (Shigley) for journal-bearing friction and the theoretical basis of the Stribeck hydrodynamic branch; the Sommerfeld Number generalises it to loaded, eccentric bearings.

Related Concepts: Sommerfeld Number, Stribeck Curve, Journal Bearing Load Capacity, Bearing Power Loss, Minimum Film Thickness, Hydrodynamic Film Pressure

Notes: Concentric (light-load) journal; loaded bearings run eccentric — use Sommerfeld Number/ Raimondi–Boyd charts. Friction is viscous (rises with $\mu N/P$). Unit load $P = W/(2rL)$; feeds directly into Bearing Power Loss.