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See this article to know more about the Reynolds equation history and available solutions. This equation was first obtained by Osborne Reynolds and that’s why this equation holds his name. As it will be shown later, this fact allows to eliminate several terms in the Navier-Stokes equations and significantly simplify the problem in hand. The major simplification comes from the geometrical consideration, namely, from the fact that the characteristic length in the direction across the film (see the figure above) is usually much smaller compared to the dimensions along the film (e.g., radii of curvature of the contacting bodies or the linear size).
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Reynolds equation derivation starts with the simplification of the above Navier-Stokes equations. Dimensions in a lubricated contact, see Szeri. Derivation of Reynolds Equation Figure 1.
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