Approximate analytical solution to Reynolds equation for finite length journal bearings

Abstract

The understanding of the behavior of hydrodynamic bearings requires the analysis of the fluid film between two solid surfaces in relative motion. The differential equation that governs the movement of this fluid, called the Reynolds equation, arises from the integration over the film thickness of the continuity equation, previously combined with the Navier–Stokes equation. An order of magnitude analysis, which is based on the relative value of the dimensions of the bearing, produces two dimensionless numbers that govern the behavior of the system: the square of the aspect ratio, length over diameter ( L/ D) 2, and the eccentricity ratio ( η). An analytical solution of the Reynolds equation can only be obtained for particular situations as, for example, the isothermal flow of Newtonian fluids and values of L/ D→0 or L/ D→∞. For other conditions, the equation must be solved numerically. The present work proposes an analytical approximate solution of the Reynolds equation for isothermal finite length journal bearings by means of the regular perturbation method. ( L/ D) 2 is used as the perturbation parameter. The novelty of the method lays in the treatment of the Ocvirk number as an expansible parameter. The zero-order solution of the Reynolds equation (obtained for L/ D→0), which matches the Ocvirk solution, may be used to describe the behavior of finite length journal bearings, up to L/ D∼1/8–1/4, and relatively small eccentricities. The first-order solution obtained with the proposed method gives an analytical tool that extends the description of pressure and shear-stress fields up to L/ D∼1/2 and η∼1/2 (or combinations of larger eccentricities with smaller aspect ratios, or vice versa). Moreover, the friction force and load-carrying capacity are accurately described by the proposed method up to L/ D∼1 and η very near to 1.