On The Role of a Non-Newtonian Fluid in Short Bearing Theory

The importance of rheological properties of lubricants has arisen from the realization that non-Newtonian fluid effects are manifested over a broad range of lubrication applications. In this paper a theoretical investigation of short journal bearings performance characteristics for non-Newtonian power-law lubricants is given. A modified form of the Reynolds’ equation for hydrodynamic lubrication is studied in the asymptotic limit of small slenderness ratio (i.e., bearing length to diameter, L/D = λ→0). Fluid film pressure distributions in short bearings of arbitrary azimuthal length are studied using matched asymptotic expansions in the slenderness ratio. The merit of the short bearing approach used in solving a modified Reynolds’ equation by the method of matched asymptotic expansions is emphasized. Fluid film pressure distributions are determined without recourse to numerical solutions to a modified Reynolds’ equation. Power-law rheological exponents less than and equal to one are considered; power-law fluids exhibit reduced load capacities relative to the Newtonian fluid. The cavitation boundary shape is determined from Reynolds’ free surface condition; and the boundary shape is shown to be independent of the bearing eccentricity ratio.

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Two Dimensional Modified Reynolds Equation Including Pressure Dependent Viscosity Effect

ASME/STLE 2012 International Joint Tribology Conference ◽

10.1115/ijtc2012-61174 ◽

2012 ◽

Author(s):

Jung Gu Lee ◽

Alan Palazzolo

Keyword(s):

Reynolds Equation ◽

Elastohydrodynamic Lubrication ◽

Fluid Film ◽

Maximum Pressure ◽

Elastic Solids ◽

Pressure Distributions ◽

Pressure Dependent Viscosity ◽

Modified Reynolds Equation ◽

Dependent Viscosity ◽

Modified Equation

The Reynolds equation plays an important role for predicting pressure distributions for fluid film bearing analysis, One of the assumptions on the Reynolds equation is that the viscosity is independent of pressure. This assumption is still valid for most fluid film bearing applications, in which the maximum pressure is less than 1 GPa. However, in elastohydrodynamic lubrication (EHL) where the lubricant is subjected to extremely high pressure, this assumption should be reconsidered. The 2D modified Reynolds equation is derived in this study including pressure-dependent viscosity, The solutions of 2D modified Reynolds equation is compared with that of the classical Reynolds equation for the ball bearing case (elastic solids). The pressure distribution obtained from modified equation is slightly higher pressures than the classical Reynolds equations.

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A Calculation Method to Investigate the Effects of Geometric Parameters and Operational Conditions on the Static Characteristics of Aerostatic Spherical Bearings

Journal of Tribology ◽

10.1115/1.4041365 ◽

2018 ◽

Vol 141 (2) ◽

Cited By ~ 2

Author(s):

Hailong Cui ◽

Huan Xia ◽

Dajiang Lei ◽

Xinjiang Zhang ◽

Zhengyi Jiang

Keyword(s):

Experimental Data ◽

Calculation Method ◽

Reynolds Equation ◽

Geometric Parameters ◽

Static Characteristics ◽

Optimal Parameters ◽

Film Pressure ◽

Operational Conditions ◽

Pressure Distributions ◽

Calculation Results

In this paper, a calculation method based on matlab partial differential equations (PDE) tool is proposed to investigate the static characteristics of aerostatic spherical bearings. The Reynolds equation of aerostatic spherical bearings is transformed into a standard elliptic equation. The effects of geometric parameters and operational conditions on the film pressure, bearing film force, and stiffness are studied. The axial and radial eccentricities result in different film pressure distributions; the bearing film force and stiffness are significantly influenced by geometric parameters and operational conditions. The relative optimal parameters are confirmed based on the calculation results. A comparison between the numerical and experimental results is also presented. The highest relative error between the numerical results and the experimental data is 11.3%; the calculation results show good agreements with the experimental data, thus verifying the accuracy of the calculation method used in this paper.

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Nonlinear Effects in a Plain Journal Bearing: Part 1—Analytical Study

Journal of Tribology ◽

10.1115/1.2920659 ◽

1991 ◽

Vol 113 (3) ◽

pp. 555-561 ◽

Cited By ~ 30

Author(s):

F. K. Choy ◽

M. J. Braun ◽

Y. Hu

Keyword(s):

Equilibrium Position ◽

Reynolds Equation ◽

Journal Bearing ◽

Fluid Film ◽

Liquid Oxygen ◽

Solution Technique ◽

Space Applications ◽

Linear Solution ◽

Film Pressure ◽

Linear And Nonlinear

Hydrodynamic/hydrostatic journal bearings have been widely used in various types of high speed rotating machinery. For space applications, the issue of using cryogenic fluids as working lubricants has steadily gained in significance. The primary goal of this paper is to model the nonlinearities that occur in a hydrodynamic journal bearing with both cryogenic and oil lubricants. Results will be examined through bearing fluid film pressure distribution and bearing linear and nonlinear stiffness characteristics. The numerical model that couples a variable property Reynolds equation with the dynamics of the rotor is solved by means of a finite difference solution technique. The procedure for the fluid film pressure solution involves an iterative scheme that solves the Reynolds equation coupled with the equations of state for liquid oxygen (LO2). The pressure curve is then integrated to calculate bearing supporting forces. A two-dimensional Newton-Raphson iteration method is used to locate the journal equilibrium position from which both linear and nonlinear bearing stiffness are evaluated by means of the small perturbation technique. The effects of load on the linear/nonlinear plain journal bearing characteristics are analyzed and presented in a parametric form. The relationship between the accuracy of the linear solution and the various orders (3rd, 5th, and 7th power for ΔX) of the nonlinear approximation are also discussed. The validity of both linear and nonlinear solutions at various distances from the journal equilibrium position is also examined. A complete parametric study on the effects of load, temperature, operating speed, and shaft misalignment will be given in Part 2 of this paper.

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Analysis of power law and log law velocity profiles in the overlap region of a turbulent wall jet

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2004.1400 ◽

2005 ◽

Vol 461 (2058) ◽

pp. 1889-1910 ◽

Cited By ~ 16

Author(s):

Noor Afzal

Keyword(s):

Power Law ◽

Asymptotic Expansions ◽

Friction Velocity ◽

Matched Asymptotic Expansions ◽

Large Reynolds Number ◽

Wall Jet ◽

Law Of The Wall ◽

Log Law ◽

Turbulent Wall ◽

Overlap Region

The two-dimensional turbulent wall jet on a flat surface without free stream is analysed at a large Reynolds number, using the method of matched asymptotic expansions. The open mean equations of the turbulent boundary layer are analysed in the wall and wake layers by the method of matched asymptotic expansions and the results are matched by the Izakson–Millikan–Kolmogorov hypothesis. In the overlap region, the outer wake layer is governed by the velocity defect law (based on U m , the maximum velocity) and the inner layer by the law of the wall. It is shown that the overlap region possesses a non-unique solution, where the power law region simultaneously exists along with the log law region. Analysis of the power law and log law solutions in the overlap region leads to self-consistent relations, where the power law index, α , is of the order of the non-dimensional friction velocity and the power law multiplication constant, C , is of the order of the inverse of the non-dimensional friction velocity. The lowest order wake layer equation has been closed with generalized gradient transport model and a closed form solution is obtained. A comparison of the theory with experimental data is presented.

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Non-Newtonian Lubrication Theory for Rough Surfaces: Application to Rigid and Elastic Rollers

Journal of Mechanical Engineering Science ◽

10.1243/jmes_jour_1982_024_028_02 ◽

1982 ◽

Vol 24 (3) ◽

pp. 147-154 ◽

Cited By ~ 3

Author(s):

P. Sinha ◽

J. B. Shukla ◽

C. Singh ◽

K. R. Prasad

Keyword(s):

Surface Roughness ◽

Elastic Deformation ◽

Newtonian Fluid ◽

Power Law ◽

Reynolds Equation ◽

Stochastic Theory ◽

Power Law Model ◽

Pure Rolling ◽

Lubrication Characteristics ◽

Rigid Surfaces

To predict the consequences of the interaction of the lubricant rheology with surface roughness, the stochastic theory of lubrication for Newtonian fluid is modified to take into account the non-Newtonian behaviour of the lubricant, by considering the power law model. Generalized forms of the Reynolds equation for two types of roughness arrangements, viz., longitudinal and transverse, are derived. These equations are subsequently used to study the lubrication characteristics of infinitely long rough roller bearings, and two particular cases, namely pure rolling (rigid surfaces) and rolling with elastic deformation, are discussed.

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Theoretical Pressure Distribution in Journal Bearings

Journal of Applied Mechanics ◽

10.1115/1.3641774 ◽

1961 ◽

Vol 28 (4) ◽

pp. 497-506 ◽

Cited By ~ 1

Author(s):

Kichiye Habata

Keyword(s):

Reynolds Equation ◽

Variable Viscosity ◽

Slenderness Ratio ◽

Approximate Solutions ◽

Journal Bearings ◽

Oil Viscosity ◽

Pressure Distributions ◽

Hill’S Method ◽

Ritz’S Method ◽

Theoretical Pressure

By assuming oil viscosity constant, Reynolds’ equation for journal bearings has been solved in a manner similar to Hill’s method. Two approximate solutions using E. O. Waters’ method and Ritz’s method have been added. Numerical computations have been carried out for a centrally supported 120-deg bearing with a unity slenderness ratio. Isobarriers have been determined from the pressure distributions. In order to show a justification for assuming the viscosity constant, the Reynolds equation was solved for the infinitely long bearing with variable viscosity, and the solution compared with that of Sommerfeld.

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