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# Petroff's Law - shear stress in the lubricant

## Description

In the design of fluid bearings, the Sommerfeld number (S), or bearing characteristic number, is a dimensionless quantity used extensively in hydrodynamic lubrication analysis. The Sommerfeld number is very important in lubrication analysis because it contains all the variables normally specified by the designer.

The Sommerfeld number is named after Arnold Sommerfeld (1868–1951).

Derivation
Petroff’s Law

Petroff’s method of lubrication analysis, which assumes a concentric shaft and bearing, was the first to explain the phenomenon of bearing friction. This method, which ultimately produces the equation known as Petroff’s Law, is useful because it defines groups of relevant dimensionless parameters, and predicts a fairly accurate coefficient of friction, even when the shaft is not concentric.

Considering a vertical shaft rotating inside a bearing, it can be assumed that the bearing is subjected to a negligible load, the radial clearance space is completely filled with lubricant, and that leakage is negligible. The surface velocity of the shaft is: U = 2*π*r*N where N is the rotational speed of the shaft in rev/s.

The shear stress in the lubricant can be represented as shown here, assuming a constant rate of shear.

Related formulas

## Variables

 τ shear stress in the lubricant assuming a constant rate of shear. (pascal) π pi r shaft radius (m) μ absolute viscosity of the lubricant (pascal*s) N rotational speed of the shaft (revs/s) (1/s) c radial clearance (m)