Sommerfeld Number⚠ unverified

Understanding the Parameters

• Bearing number $\mu N/P$ — viscosity × speed ÷ load, the same Hersey number driving the Stribeck Curve and Petroff Friction. High $\mu N/P$ = thick film, low friction risk of instability; low = thin film, toward mixed/boundary and wear.

• Clearance ratio $(r/c)^2$ — note the square, distinguishing $S$ from Petroff's $r/c$. The clearance ratio $r/c \approx 500$–$1000$, so $(r/c)^2$ is a huge multiplier; small changes in clearance strongly shift $S$. Clearance is a critical, tightly-toleranced design variable.

• What $S$ predicts — a large $S$ (light load, high speed, viscous oil) → thick film, journal nearly centred (low eccentricity), safe but higher friction. A small $S$ (heavy load, low speed) → thin film, high eccentricity, approaching metal contact. The Raimondi–Boyd charts plot minimum-film-thickness ratio, eccentricity, friction variable $f\,r/c$, and flow against $S$.

Derivation (Approaching a Proof)

The Sommerfeld number emerges from non-dimensionalising the Reynolds equation — the governing PDE of hydrodynamic lubrication, which balances the pressure-driven and shear-driven (wedge) flows in the thin film:

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

Scaling the film thickness by the clearance ($h \sim c$), the surface speed by $U \sim 2\pi r N$, the pressure by the unit load $P$, and lengths by $r$, the equation becomes dimensionless with all the physical parameters collapsing into a single group:

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

This is dynamic similarity: any two geometrically similar bearings with the same $S$ have the same dimensionless pressure field, hence the same eccentricity, film ratio, and friction variable. It relates to Petroff's friction ($f\,r/c = 2\pi^2 S$ in the concentric limit) and lets a single set of charts cover all bearings. $S$ is inversely related to how heavily loaded the bearing is: heavy load (large $P$) → small $S$ → thin film.

Dimensional check. $\dfrac{\mu N}{P}\left(\dfrac{r}{c}\right)^2 = \dfrac{(\text{Pa}\cdot\text{s})(1/\text{s})}{\text{Pa}}\cdot(\text{m/m})^2 = 1$. ✓

History and Development

Arnold Sommerfeld derived the number (and an analytical journal-bearing solution) in 1904, building on Reynolds' 1886 lubrication theory. Raimondi and Boyd (1958) published the definitive dimensionless design charts keyed to $S$, still the standard for journal-bearing design in Shigley and ISO 7902. The Sommerfeld number is the organising parameter of the entire field.

Related Concepts: Petroff Friction, Journal Bearing Load Capacity, Minimum Film Thickness, Stribeck Curve, Hydrodynamic Film Pressure, Reynolds Number

Notes: Note the square on $r/c$ (vs $r/c$ in Petroff). Unit load $P = W/(2rL)$. Use with Raimondi–Boyd charts for film thickness, eccentricity, friction, and flow. Small $S$ = heavily loaded / thin film.