Rupture Point Movement in Journal Bearings

Bara, Richard J.

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Rupture Point Movement in Journal Bearings

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Two most important events in the history of lubrication theory are attributed to Reynolds and Sommerfeld. Reynolds derived the governing equations for lubricating films in simplifying the Navier-Stokes equations considering thin-film effects. Sommerfeld obtained a closed form analytical solution to the Reynolds equation for the long bearing (one-dimensional case) with fixed constant eccentricity which results in a point symmetric pressure profile compared to an arbitrary (ambient) level. In attempting to reconcile with experimental evidence, Gumbel advanced the argument that sub-ambient pressure in a fluid film is not possible. On the basis that the fluid film would rupture, he put forth that the sub-ambient portion of the Sommerfeld solution should be discarded, a proposition that is commonly recognized as the half-Sommerfeld solution (of Gumbel). Ever since Gumbel suggested this improvement, much interest remains regarding the physical process of rupture in bearing lubricating films. In lubrication literature, cavitation is used interchangeably with rupture to indicate a condition in which an abundance of a gas phase, essentially ambient air, is present in a portion of the bearing clearance. A cogent two-phase morphology for addressing cavitation in long bearings is postulated in order to predict time-dependent fluid behavior from an initial state that is a generalization of Gumbelâ€™s half-Sommerfeld solution. The ultimate steady-state is presumed to satisfy the hypothesis of Swift and Stieber that an ambient condition is reached by the rupture point at an unspecified location simultaneously with a vanishing pressure gradient. A trans-rupture continuity equation, as proposed by Olsson, determines a formula for the speed of a moving rupture point requiring a specific model of the two-phase flow in the rupture region. Employing an adhered film model, sequential application of Olssonâ€™s equation to the rupture points of the intermediate states between the half-Sommerfeld and Swift-Stieber states renders an interpretation of a time-dependent progression towards a steady-state solution. Closed form analytical formulas, which readily combine to provide an exact solution to the Reynolds equation are derived with the start (formation point) of the full-film other than the customary bearing maximum gap and with the rupture point at any assigned intermediate location. Each valid solution for an intermediate state yields an invariant flux that must satisfy a window of constraints to exclude the possibility of sub-ambient pressures. A complete set of such valid solutions exists for each fixed eccentricity and can be depicted as a contour plot of the invariant flux with formation and rupture points as coordinates. The method can readily be extended to two-dimensions, offering a promising alternative to the Elrod cavitation algorithm, which is commonly used in more comprehensive bearing analyses.

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