Remarks on the Reynolds problem of elastohydrodynamic lubrication

1993 ◽  
Vol 4 (1) ◽  
pp. 83-96 ◽  
Author(s):  
José-Francisco Rodrigues
Keyword(s):  
Free Boundary ◽  
A Priori ◽  
Elastohydrodynamic Lubrication ◽  
Small Data ◽  
Existence Of A Solution ◽  
Pressure Dependent Viscosity ◽  
Uniqueness Of The Solution ◽  
Non Local ◽  
Dependent Viscosity ◽  
The Mathematical Model

The mathematical model of the flow of a viscous lubricant between elastic bearings leads to the study of a highly non-linear and non-local elliptic variational inequality. We discuss the existence of a solution by using an a prioriL∞-estimate. This method allows us to solve a large class of problems, including those arising from the linear Hertzian theory, and yields new existence results for the cases of a pressure-dependent viscosity or the inclusion of a load constraint. For small data the uniqueness of the solution holds, and we show that in the cylindrical journal bearing problem with small eccentricity ratio, the free boundary is given by two disjoint differentiable arcs close to the free boundary of the first-order approximate solution.

Two Dimensional Modified Reynolds Equation Including Pressure Dependent Viscosity Effect

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.

scholarly journals Nonlocal Problem for a Third-Order Equation with Multiple Characteristics with General Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 110
Author(s):  
Abdukomil Risbekovich Khashimov ◽  
Dana Smetanová
Keyword(s):  
Quadratic Forms ◽  
Nonlocal Problem ◽  
General Boundary ◽  
Fredholm Integral Equations ◽  
Third Order ◽  
Existence Of A Solution ◽  
Energy Integrals ◽  
Multiple Characteristics ◽  
Uniqueness Of The Solution ◽  
Non Local

The article considers third-order equations with multiple characteristics with general boundary value conditions and non-local initial data. A regular solution to the problem with known methods is constructed here. The uniqueness of the solution to the problem is proved by the method of energy integrals. This uses the theory of non-negative quadratic forms. The existence of a solution to the problem is proved by reducing the problem to Fredholm integral equations of the second kind. In this case, the method of Green’s function and potential is used.

A Simplified Model for the Elastohydrodynamic Traction Between Rollers

1976 ◽  
Vol 98 (3) ◽  
pp. 357-361 ◽  
Author(s):  
Charles W. Allen
Keyword(s):  
Preliminary Analysis ◽  
High Pressures ◽  
Elastohydrodynamic Lubrication ◽  
Rolling Contact ◽  
Experimental Investigations ◽  
Pressure Dependent Viscosity ◽  
Linear Shear ◽  
Dependent Viscosity ◽  
Low Pressures ◽  
Good Agreement

The elastohydrodynamic lubrication of two rollers in combined rolling and sliding is considered. A simple rheological model to predict the traction is presented. The model is based upon an exponential pressure dependent viscosity at low pressures and a linear shear stress/pressure relationship at high pressures. The slope of the latter is assumed to be a function of the rolling and sliding velocities. The model is used to analyze the traction data of two recently published experimental investigations of other authors and good agreement is achieved in most cases. The model should be of considerable use to designers in the preliminary analysis of rolling contact systems in combined rolling and sliding.

scholarly journals On the one-dimensional nonlinear elastohydrodynamic lubrication

1994 ◽  
Vol 50 (3) ◽  
pp. 353-372 ◽  
Author(s):  
Daniel Goeleven ◽  
Van Hien Nguyen
Keyword(s):  
Mathematical Model ◽  
Variational Inequality ◽  
Numerical Analysis ◽  
Elastohydrodynamic Lubrication ◽  
Existence Of A Solution ◽  
One Dimensional ◽  
Abstract Theorem ◽  
The One ◽  
The Mathematical Model ◽  
Lubricant Viscosity

In this paper the authors prove an abstract theorem for solutions of a variational inequality on a cone and use it to study the free boundary problem of elastohydrodynamic lubrication from mechanical engineering. The mathematical model is set in a one-dimensional geometry. The existence of a solution for every non-negative lubricant viscosity is proved, and some properties useful for the numerical analysis are obtained.

Oil Film Thickness and Shape for a Ball Sliding in a Grooved Raceway

1972 ◽  
Vol 94 (3) ◽  
pp. 199-208 ◽  
Author(s):  
N. Thorp ◽  
R. Gohar
Keyword(s):  
General Theory ◽  
Film Thickness ◽  
Contact Area ◽  
Elastohydrodynamic Lubrication ◽  
Surface Velocity ◽  
Oil Film ◽  
Pressure Dependent Viscosity ◽  
Ball Surface ◽  
Lubricated Contact ◽  
Dependent Viscosity

The behavior in the lubricated contact area of a driven ball sliding in a conforming glass groove, is studied. Interferometry is used to measure the oil film. Coupled ball surface velocity components are provided by angling the drive, while loads and speeds are varied in order to cover a range of conditions from undistorted surfaces to elastohydrodynamic lubrication. A general theory for lubrication with, no distortion and pressure-dependent viscosity, is developed and compared with experiment. Ball spin is found to have only a small effect on the oil film thickness.

Effect of Surface Roughness on Elastohydrodynamic Line Contact

1982 ◽  
Vol 104 (3) ◽  
pp. 401-407 ◽  
Author(s):  
B. C. Majumdar ◽  
B. J. Hamrock
Keyword(s):  
Surface Roughness ◽  
Film Thickness ◽  
Strain Analysis ◽  
Elastohydrodynamic Lubrication ◽  
Asperity Contact ◽  
Pressure Dependent Viscosity ◽  
Surface Irregularities ◽  
Dependent Viscosity ◽  
Contact Pressures ◽  
Effect Of Surface

A numerical solution of an elastohydrodynamic lubrication (EHL) contact between two long, rough surface cylinders is obtained. A theoretical solution of pressure distribution, elastohydrodynamic load, and film thickness for given speeds and for lubricants with pressure-dependent viscosity, material properties of cylinders, and surface roughness parameters is made by simultaneous solution of an elasticity equation and the Reynolds equation for two partially lubricated rough surfaces. The pressure due to asperity contact is calculated by assuming a Gaussian distribution of surface irregularities. The elastic deformation is found from hydrodynamic and contact pressures by using plane strain analysis. The effect of surface roughness on EHL loads, speeds, and central film thicknesses is studied. The results indicate that for a constant central film thickness (1) increasing the surface roughness decreases the EHL load and (2) there is little variation in minimum film thickness as the surface roughness is increased.

Nondimensional Presentation of Frictional Tractions in Elastohydrodynamic Lubrication—Part II: Starved Conditions

1975 ◽  
Vol 97 (3) ◽  
pp. 412-421 ◽  
Author(s):  
J. F. Archard ◽  
K. P. Baglin
Keyword(s):  
Elastic Modulus ◽  
Film Thickness ◽  
Sliding Friction ◽  
Elastohydrodynamic Lubrication ◽  
Rolling Friction ◽  
Analytic Treatment ◽  
Low Elastic Modulus ◽  
Pressure Dependent Viscosity ◽  
Dependent Viscosity ◽  
Negligible Influence

Part I of this paper presented a broad semi-analytic treatment of frictional tractions in nondimensional terms; this was confined to the fully flooded situation and the present paper extends the analysis to include starved conditions. As in Part I three major conditions are considered in detail: classical (isoviscous, undeformed) low elastic modulus (isoviscous, heavily deformed) and high elastic modulus (pressure dependent viscosity, heavily deformed). The influence of starvation is presented as a series of correction curves for the rolling and sliding friction derived for fully flooded conditions. Starvation influences friction both through the extent to which the gap between the surfaces is filled by lubricant and through its influence upon the film thickness. Both factors affect rolling friction which is therefore markedly reduced by starvation so mild that there is negligible influence upon the film thickness. In contrast, sliding friction (arising either in the main pressure zone or the cavitated region) is most strongly influenced by the film thickness and is therefore markedly affected only by relatively severe starvation.

scholarly journals Analysis of the Reynolds Equation for Lubrication in Case of Pressure-Dependent Viscosity

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Eduard Marušić-Paloka ◽  
Sanja Marušić
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Reynolds Equation ◽  
Homogenization Method ◽  
Nonlocal Effects ◽  
Pressure Dependent Viscosity ◽  
Uniqueness Of The Solution ◽  
Dependent Viscosity

We study the Reynolds equation, describing the ow of a lubricant, in case of pressure-dependent viscosity. First we prove the existence and uniqueness of the solution. Then, we study the asymptotic behavior of the solution in case of periodic roughness via homogenization method. Some interesting nonlocal effects appear due to the nonlinearity.

scholarly journals A non-local free boundary problem arising in a theory of financial bubbles

2014 ◽  
Vol 372 (2028) ◽  
pp. 20130404 ◽  
Author(s):  
Henri Berestycki ◽  
Regis Monneau ◽  
José A. Scheinkman
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Existence And Uniqueness ◽  
Obstacle Problem ◽  
Speculative Bubbles ◽  
Financial Bubbles ◽  
Uniqueness Of The Solution ◽  
Non Local ◽  
The Cost

We consider an evolution non-local free boundary problem that arises in the modelling of speculative bubbles. The solution of the model is the speculative component in the price of an asset. In the framework of viscosity solutions, we show the existence and uniqueness of the solution. We also show that the solution is convex in space, and establish several monotonicity properties of the solution and of the free boundary with respect to parameters of the problem. To study the free boundary, we use, in particular, the fact that the odd part of the solution solves a more standard obstacle problem. We show that the free boundary is and describe the asymptotics of the free boundary as c , the cost of transacting the asset, goes to zero.

Nondimensional Presentation of Frictional Tractions in Elastohydrodynamic Lubrication—Part I: Fully Flooded Conditions

1975 ◽  
Vol 97 (3) ◽  
pp. 398-410 ◽  
Author(s):  
J. F. Archard ◽  
K. P. Baglin
Keyword(s):  
Elastic Modulus ◽  
Elastohydrodynamic Lubrication ◽  
Analytic Solutions ◽  
Glassy Polymers ◽  
Exponential Relationship ◽  
Low Elastic Modulus ◽  
Pressure Dependent Viscosity ◽  
Wide Range ◽  
Computer Based ◽  
Dependent Viscosity

Using several sources, analytic and semi-analytic solutions for frictional tractions of a lubricated line contact are presented in the appropriate non-dimensional form which is similar to that previously used by Moes for film thickness. A Newtonian lubricant with an exponential relationship between viscosity and pressure is assumed and, at this stage, the treatment is confined to fully flooded conditions. The components of frictional tractions arising from rolling (Poisseiulle) and sliding (Couette) flows are distinguished and sliding tractions in the outlet cavitated region are separated from those in the main pressure zone. Three main regimes of lubrication are studied: classical (isoviscous, undeformed), low elastic modulus (isoviscous, heavily deformed) and high elastic modulus (pressure dependent viscosity, heavily deformed). The results presented here provide a broad background of approximate results, covering a very wide range of conditions against which the results of more precise computer-based analyses can be judged. Thus the treatment reveals the existence of a range of conditions (typical of the lubrication of glassy polymers by hydrocarbon lubricants) which has been little studied and is, as yet, imperfectly understood.

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