On lower-dimensional models in lubrication, Part B: Derivation of a Reynolds type of equation for incompressible piezo-viscous fluids

2020 ◽  
pp. 135065012097380
Author(s):  
Andreas Almqvist ◽  
Evgeniya Burtseva ◽  
Kumbakonam Rajagopal ◽  
Peter Wall
Keyword(s):  
Film Thickness ◽  
Reynolds Equation ◽  
Constitutive Relations ◽  
Viscous Fluids ◽  
Dimensional Model ◽  
Pressure Dependent Viscosity ◽  
Constant Viscosity ◽  
Dimensional Models ◽  
Dependent Viscosity ◽  
Lower Dimensional

The Reynolds equation is a lower-dimensional model for the pressure in a fluid confined between two adjacent surfaces that move relative to each other. It was originally derived under the assumption that the fluid is incompressible and has constant viscosity. In the existing literature, the lower-dimensional Reynolds equation is often employed as a model for the thin films, which lubricates interfaces in various machine components. For example, in the modelling of elastohydrodynamic lubrication (EHL) in gears and bearings, the pressure dependence of the viscosity is often considered by just replacing the constant viscosity in the Reynolds equation with a given viscosity-pressure relation. The arguments to justify this are heuristic, and in many cases, it is taken for granted that you can do so. This motivated us to make an attempt to formulate and present a rigorous derivation of a lower-dimensional model for the pressure when the fluid has pressure-dependent viscosity. The results of our study are presented in two parts. In Part A, we showed that for incompressible and piezo-viscous fluids it is not possible to obtain a lower-dimensional model for the pressure by just assuming that the film thickness is thin, as it is for incompressible fluids with constant viscosity. Here, in Part B, we present a method for deriving lower-dimensional models of thin-film flow, where the fluid has a pressure-dependent viscosity. The main idea is to rescale the generalised Navier-Stokes equation, which we obtained in Part A based on theory for implicit constitutive relations, so that we can pass to the limit as the film thickness goes to zero. If the scaling is correct, then the limit problem can be used as the dimensionally reduced model for the flow and it is possible to derive a type of Reynolds equation for the pressure.

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 On lower-dimensional models in lubrication, Part A: Common misinterpretations and incorrect usage of the Reynolds equation

2020 ◽  
pp. 135065012097379
Author(s):  
Andreas Almqvist ◽  
Evgeniya Burtseva ◽  
Kumbakonam Rajagopal ◽  
Peter Wall
Keyword(s):  
Stokes Equation ◽  
Reynolds Equation ◽  
Elastohydrodynamic Lubrication ◽  
Lubricant Film ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Constant Viscosity ◽  
Incompressible Navier Stokes Equation ◽  
Lower Dimensional

Most of the problems in lubrication are studied within the context of Reynolds’ equation, which can be derived by writing the incompressible Navier-Stokes equation in a dimensionless form and neglecting terms which are small under the assumption that the lubricant film is very thin. Unfortunately, the Reynolds equation is often used even though the basic assumptions under which it is derived are not satisfied. One example is in the mathematical modelling of elastohydrodynamic lubrication (EHL). In the EHL regime, the pressure is so high that the viscosity changes by several orders of magnitude. This is taken into account by just replacing the constant viscosity in either the incompressible Navier-Stokes equation or the Reynolds equation by a viscosity-pressure relation. However, there are no available rigorous arguments which justify such an assumption. The main purpose of this two-part work is to investigate if such arguments exist or not. In Part A, we formulate a generalised form of the Navier-Stokes equation for piezo-viscous incompressible fluids. By dimensional analysis of this equation we, thereafter, show that it is not possible to obtain the Reynolds equation, where the constant viscosity is replaced with a viscosity-pressure relation, by just neglecting terms which are small under the assumption that the lubricant film is very thin. The reason is that the lone assumption that the fluid film is very thin is not enough to neglect the terms, in the generalised Navier-Stokes equation, which are related to the body forces and the inertia. However, we analysed the coefficients in front of these (remaining) terms and provided arguments for when they may be neglected. In Part B, we present an alternative method to derive a lower-dimensional model, which is based on asymptotic analysis of the generalised Navier-Stokes equation as the film thickness goes to zero.

The oil film in a closing gap

1962 ◽  
Vol 266 (1326) ◽  
pp. 312-328 ◽  
Keyword(s):  
High Pressure ◽  
Film Thickness ◽  
Pressure Coefficient ◽  
Lubricant Film ◽  
Polished Surface ◽  
Maximum Pressure ◽  
Pressure Dependent Viscosity ◽  
Special Cases ◽  
Dry Contact ◽  
Dependent Viscosity

This paper presents a solution to the elasto-hydrodynamic problem of normal approach of two cylindrical bodies separated by a lubricating film. Analytic solutions are found for the special cases of constant viscosity and rigid material and also for pressure-dependent viscosity. The more general case accounting for elastic deformation of the bodies with constant or pressure dependent viscosity was solved by using an iterative numerical process with the help of an electronic computer. It is found that a very high pressure may be developed in the lubricant film at a finite separation of the cylinders. As the film thickness is further reduced, the value of the maximum pressure decreases and as the film thickness approaches zero, the pressure distribution converges to the Hertzian dry contact form. For a given load applied to the cylinders, the value of the maximum pressure reached depends to a large extent upon the value of the parameter oc E , i.e. the product of the pressure coefficient of viscosity and the equivalent Young’s modulus of the elastic system. Also, once the pressure has reached a sufficiently high value it becomes extremely sensitive to an increase in load; a small increase in load will produce a large increase in maximum pressure. A number of experiments were performed in order to check some of the theoretical predictions made. In these experiments a loaded steel ball was allowed to approach the polished surface of various materials whose surfaces were covered by a lubricant film, and the plastic deformations produced in the surface were then measured. These tests showed clearly the influence of the lubricant in that in every case the depth of the impressions with lubricant was significantly larger than the corresponding ones produced under Hertzian, dry contact impacts. The experimental results indicate a correlation between maximum pressure and the value of ol E and its sensitivity to increase in load at high pressure as predicted by the theory.

A Bounded Variable Approach to the Optimum Slider Bearing

1968 ◽  
Vol 90 (1) ◽  
pp. 240-242 ◽  
Author(s):  
C. J. Maday
Keyword(s):  
Film Thickness ◽  
Load Capacity ◽  
Inequality Constraints ◽  
Maximum Load ◽  
Slider Bearing ◽  
Lagrange Equations ◽  
Variable Approach ◽  
Pressure Dependent Viscosity ◽  
One Step ◽  
Dependent Viscosity

Contemporary methods for treating inequality constraints in the calculus of variations are employed to determine the maximum load-capacity one-dimensional slider bearing using a lubricant with pressure-dependent viscosity. A lower bound on the minimum film thickness is put into equational form to facilitate the use of the Euler-Lagrange equations, the corner conditions, and the Weierstrass E-function. It is found that, for typical lubricants, the slider bearing contains only one step separting two values of the film thickness. It is shown also that there exist cases for which a solution cannot be obtained to describe a real situation.

Elastohydrodynamic Theory of Spherical Bodies in Normal Approach

1970 ◽  
Vol 92 (1) ◽  
pp. 145-153 ◽  
Author(s):  
H. Christensen
Keyword(s):  
Film Thickness ◽  
Elastic Deformation ◽  
Numerical Solutions ◽  
Maximum Pressure ◽  
Mineral Oils ◽  
Elasticity Equations ◽  
Metallic Contacts ◽  
Pressure Dependent Viscosity ◽  
Approach Velocity ◽  
Dependent Viscosity

The elastohydrodynamic problem of normal approach of two spherical bodies is studied and the lubrication and elasticity equations governing this type of motion are established. Numerical solutions to the general case accounting for elastic deformation of the bodies and pressure dependent viscosity are presented. It is found that for values of central film thickness that are not too small, the load and relative approach velocity is much more influenced by the increase of viscosity with pressure than by the effects of elastic distortion. Once the separation of the two surfaces becomes small enough, however, the effects of elastic deformation will profoundly influence all aspects of the motion. The transition film thickness HT at which this change takes place is sharply defined and for metallic contacts lubricated with mineral oils quite small, even compared to the surface roughness. Very high pressure—considerably in excess of the Hertzian maximum pressure corresponding to the load—can be generated by the normal approach motion. The maximum value of pressure is generated when film thickness reaches its transition value HT for the load in question. For loads sufficiently large to generate a high enough pressure in the oil film a small increase in load will cause a large increase in maximum pressure. Once the pressure has reached a high enough value it becomes extremely sensitive to a further increase in load.

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 Asymptotic Modeling of the Thin Film Flow with a Pressure-Dependent Viscosity

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Eduard Marušić-Paloka ◽  
Igor Pažanin
Keyword(s):  
Thin Film ◽  
Film Thickness ◽  
Asymptotic Analysis ◽  
Film Flow ◽  
Thin Film Flow ◽  
Asymptotic Modeling ◽  
Governing System ◽  
Expansion Technique ◽  
Pressure Dependent Viscosity ◽  
Dependent Viscosity

We study the lubrication process with incompressible fluid taking into account the dependence of the viscosity on the pressure. Assuming that the viscosity-pressure relation is given by the well-known Barus law, we derive an effective model using asymptotic analysis with respect to the film thickness. The key idea is to conveniently transform the governing system and then apply two-scale expansion technique.

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