1: Introduction 2: Bearing Classification and Selection 3: Surface Topography 4: Lubricant Properties 5: Bearing Materials 6: Viscous Flow 7: Reynolds Equation 8: Hydrodynamic Thrust Bearings - Analytical Solutions 9: Hydrodynamic Thrust Bearings - Numerical Solutions 10: Hydrodynamic Journal Bearings - Analytical Solutions 11: Dynamically Loaded Journal Bearings 12: Hydrodynamic Journal Bearings - Numerical Solutions 13: Hydrodynamic Squeeze Film Bearings 14: Hydrostatic Lubrication 15: Hydrodynamic Bearings - Considering Fluid Inertia 16: Gas-Lubricated Thrust Bearings 17: Gas-Lubricated Journal Bearings 18: Hydrodynamic Lubrication of Nonconformal Surfaces 19: Simplified Solutions for Stresses and Deformations 20: General Solution for Stresses and Deformations in Dry Contacts 21: Elastohydrodynamic Lubrication of Rectangular Conjunctions 22: Elastohydrodynamic Lubrication of Ellipitcal Conjunctions 23: Film Thicknesses for Different Regimes of Fluid Film Lubrication 24: Rolling-Element Bearings 25: Additional Elastohydrodynamic Lubrication Applications 26: Non-Newtonian Fluid Effects in Elastohydrodynamic Lubrication 27: Thermo Elastohydrodynamic Lubrication.

Fundamentals of fluid film lubrication

The flow of a single-phase fluid through a rough-walled rock fracture is discussed within the context of fluid mechanics. The derivation of the ‘cubic law’ is given as the solution to the Navier-Stokes equations for flow between smooth, parallel plates - the only fracture geometry that is amenable to exact treatment. The various geometric and kinematic conditions that are necessary in order for the Navier-Stokes equations to be replaced by the more tractable lubrication or Hele-Shaw equations are studied and quantified. In general, this requires a sufficiently low flow rate, and some restrictions on the spatial rate of change of the aperture profile. Various analytical and numerical results are reviewed pertaining to the problem of relating the effective hydraulic aperture to the statistics of the aperture distribution. These studies all lead to the conclusion that the effective hydraulic aperture is less than the mean aperture, by a factor that depends on the ratio of the mean value of the aperture to its standard deviation. The tortuosity effect caused by regions where the rock walls are in contact with each other is studied using the Hele-Shaw equations, leading to a simple correction factor that depends on the area fraction occupied by the contact regions. Finally, the predicted hydraulic apertures are compared to measured values for eight data sets from the literature for which aperture and conductivity data were available on the same fracture. It is found that reasonably accurate predictions of hydraulic conductivity can be made based solely on the first two moments of the aperture distribution function, and the proportion of contact area.

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Hydraulic conductivity of rock fractures

Fluid flow through rock joints is commonly described by the parallel plate model where the volume flow rate varies as the cube of the joint aperture. However, deviations from this model are expected because real joint surfaces are rough and contact each other at discrete points. To examine this problem further, a computer simulation of flow between rough surfaces was done. Realistic rough surfaces were generated numerically using a fractal model of surface topography. Pairs of these surfaces were placed together to form a “joint” with a random aperture distribution. Reynolds equation, which describes laminar flow between slightly nonplanar and nonparallel surfaces, was solved on the two-dimensional aperture mesh by the finite-difference method. The solution is the local volume flow rate through the joint. This solution was used directly in the cubic law to get the so-called “hydraulic aperture.” For various surface roughnesses (fractal dimensions) the hydraulic aperture was compared to the mean separation of the surfaces. At large separations the surface topography has little effect. At small separations the flow is tortuous, tending to be channeled through high-aperture regions. The parameter most affecting fluid flow through rough joints is the ratio of the mean separation between the surfaces to the root-mean-square surface height. This parameter describes the distance the surface asperities protrude into the fluid and accounts for most of the disagreement with the parallel plate model. Variations in the fractal dimension produce only a second-order effect on the fluid flow. For the range of joint closures expected during elastic deformation these results show that the actual flow rate between rough surfaces is about 70–90% of that predicted by the parallel plate model.

Fluid flow through rock joints: The effect of surface roughness

https://cyberleninka.org/article/n/544115.pdf

Hydrodynamic lubrication of textured surfaces: A review of modeling techniques and key findings

A model is presented to study the potential use of micro-surface structure in the form of micro pores to improve tribological properties of reciprocating automotive components. The Reynolds equation and the equation of motion are solved simultaneously for a simplified “piston/cylinder” system with surface texturing. The solution provides the time behavior of both the clearance and the friction force between the “piston ring” and “cylinder liner” surfaces. It is shown that surface texturing can efficiently be used to maintain hydrodynamic effects even with nominally parallel surfaces. It is also shown that optimum surface texturing may substantially reduce the friction losses in reciprocating automotive components. Presented at the 56th Annual Meeting Orlando, Florida May 20–24, 2001

Friction-Reducing Surface-Texturing in Reciprocating Automotive Components

An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication

Application of Average Flow Model to Lubrication Between Rough Sliding Surfaces

A Numerical Solution of the Elastohydrodynamic Film Thickness in an Elliptical Contact

The pressure and deformation profiles between two colliding lubricated cylinders are obtained by solving the coupled, time-dependent elastohydrodynamic equations with an iterative procedure. The analysis includes several effects which were not considered in a previous solution by Christensen [4], namely, the effect of surface velocities due to local deformation, the effect of the lubricant compressibility, and the effect of a lubricant with composite pressure-viscosity coefficients. It is found that the local approach velocity plays an important role during final stages of normal approach. It causes the lubricant to be entrapped within the contact region, and both the pressure and deformation profiles appear to converge to the Hertzian profile. The use of a smaller pressure-viscosity coefficient at high pressures reduces the sharp pressure gradient at the center of the contact, and produces a much milder variation of load with respect to the film thickness. The effect of compressibility of the lubricant is found to be relatively small.

The Pressure and Deformation Profiles Between Two Normally Approaching Lubricated Cylinders

The Christensen theory of stochastic models [7] for hydrodynamic lubrication of rough surfaces is extended to elastohydrodynamic lubrication between two rollers. The Grubin-type equation including asperity effects in the inlet region is derived. Solutions for the reduced pressure at the entrance as a function of the ratio of the average nominal film thickness to the r.m.s. surface roughness (in terms of standard deviation σ), have been obtained numerically. Results were obtained for purely transverse as well as purely longitudinal surface roughness for cases with or without slip. The reduced pressure is shown to decrease slightly by considering longitudinal surface roughness. The transverse surface roughness, on the other hand, has a slight beneficial effect on the average film thickess at the inlet. The same approach was used to study the effect of surface roughness on lubrication between rigid rollers and lubrication of an infinitely-wide slider bearing. Results of these two cases show that the effects of surface roughness are similar to those found in elastohydrodynamic contacts.

H. S. Cheng

Papers

An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication

Application of Average Flow Model to Lubrication Between Rough Sliding Surfaces

A Numerical Solution of the Elastohydrodynamic Film Thickness in an Elliptical Contact

The Pressure and Deformation Profiles Between Two Normally Approaching Lubricated Cylinders

The effect of surface roughness on the average film thickness between lubricated rollers