Fluid Mechanics

The equations employed to describe the flow of lubricants in bearings result from simplifications of the governing equations of fluid mechanics. It is appropriate, therefore, to devote a chapter to summarizing pertinent results from fluid mechanics. This discussion will not be limited only to concepts necessary to understand the classical theory of lubrication. A more than elementary discussion of fluid behavior is called for here, as various nonlinear effects will be studied in later chapters.

Our discussion begins with the mathematical description of motion, followed by the definition of stress. Cauchy's equations of motion will be obtained by substituting the rate of change of linear momentum of a fluid particle and the forces acting on it, into Newton's second law of motion. This will yield three equations, one in each of the three coordinate directions. But these three equations will contain twelve unknowns: three velocity components, (u, v, w) and nine stress components (Txx, Txy,…, Tzz). To render the problem well posed, i.e., to have the number of equations agree with the number of unknowns so that solutions might be obtained, we will need to find additional equations. A fourth equation is easy to come by, by way of the principle of conservation of mass. The situation further improves on recognizing that only six of the nine stress components are independent, due to symmetry of the stress matrix. However, on specifying incompressibility of the fluid (incompressible lubricants are the only type treated in this chapter) a tenth unknown, the fluid pressure, makes its debut.