Theoretical Foundations

The foundation of a fluid film-bearing analysis emanates from the boundary layer theory of fluid mechanics. The governing differential equation was first formulated by Osborne Reynolds in 1886 and is known as Reynolds' equation in his honor. It has been only in the last 30 years or so that general solutions have been obtained, and this has been primarily due to the use of numerical methods applied to the digital computer. References 2 and 3 go into the details of contemporary numerical solutions and are recommended for those interested in the analytical aspects of lubrication.

Principal Assumptions Reynolds' equation can be derived from the Navier-Stokes equation of fluid mechanics, and a number of textbooks are available that comprehensively describe the derivation.4 The primary assumptions are as follows:

• Laminar flow conditions prevail, and the fluids obey a Newtonian shear stress distribution where the shear stress is proportional to the velocity gradient.

• Inertial forces, resulting from acceleration of the liquid, are small relative to the viscous shear forces and may be neglected.

• The pressure across the film is constant since the fluid films are so thin.

• The height of the fluid film is small relative to other geometric dimensions, and so the curvature of the fluid film can be ignored.

• The viscosity of the liquid remains constant. In most cases, this is a reasonable assumption since it has been repeatedly demonstrated that, if the average viscosity is used, little error is introduced and the complexity of the analysis is considerably reduced.

Derivaton of Reynold's Equation of Lubrication Assume that the rotating journal has a peripheral velocity U. Consider an elemental volume in the clearance space of the bearing and establish equilibrium (see Figure 3). Note that since inertial forces are neglected, the volume is in equilibrium by the pressure and shear forces acting upon it, so there is no acceleration. As shown in Figure 4,p is the pressure and t is the shear stress acting upon the volume. Summing forces in the x direction

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