Pressure Viscosity Relationship Research Articles

High-pressure rheological properties of polyalphaolefin and ester oil blends and their impact on lubrication

Different base oils are often blended to optimize lubrication effectiveness. For blends of polyalphaolefins (PAOs) and ester oils, while their boundary lubrication performance is well-studied, studies on their rheological properties and elastohydrodynamic lubrication (EHL) are limited. This study investigates the viscosity and EHL properties of PAO 8 and pentaerythritol tetra(2-ethylhexanoate) (PEB 8) blends at various blending ratios, pressures, and temperatures. Results showed that under certain conditions, viscosities of the blends are lower than either base oil. This phenomenon was explained from a molecular perspective. The pressure–viscosity relationship was fitted using the Doolittle-Tait free-volume model. Additionally, studies of EHL performance and simulations of different oils correlate the pressure–viscosity to EHL characteristics. These findings provide insights for future lubricant formulation.

Flow Governed by Generalised Brinkman’s Equation through an Inclined Porous Channel

The unidirectional flow of a fluid with pressure-dependent viscosity through a porous structure is considered when the viscosity–pressure relationship is an exponential function of a pressure power function in order to investigate effects of the viscosity–pressure relation on the flow characteristics. The flow is governed by the generalized Brinkman’s equation with constant permeability, and a model flow domain of flow down an inclined porous channel is chosen for the sake of studying flow behaviour. Although the current work considers flow in a constant permeability porous structure, it does represent the first step in studying the more general flow through a variable permeability porous channel. The arising governing equations are solved numerically using MATLAB (version R2022a) and the flow is simulated to illustrate the effects of fluid properties, as well as flow and medium parameters, on the velocity profiles and shear stress. The results obtained should represent a baseline and a benchmark with which future experimental and theoretical work can be compared.

Newton-GMRES-Method for the Scritinization of electric double layer and surface roughness on EHL line contact problem

The electric double layer phenomenon exists on the solid interface under the water-liquid condition. The water molecules are ionized and adhered in the interface forming the sturn layer is a diffused layer in which molecules can move with the movement of bulk of molecules. Because of these two characteristics, the boundary layer of water molecules is called the electric double layer. The aim of present study is to explore the impact of two fraction surfaces of electric double layer (EDL) on a thin water lubricating film on an elastohydrodynamic lubrication (EHL) line contact problem with a sinusoidal surface roughness. The governing modified Reynolds and film thickness equations are based on mathematical model of electro-viscosity of asymmetrical electrical double layer is analyzed numerically. The viscosity-pressure relation of water and theoretical evaluation pertaining to the effect of electric double layer on film-thickness and pressure distribution of EHL with water film of line contact problem is discussed in detail. The effect of zeta potential on film thickness and pressure is determined using Newton’s-GMRES method with Daubechies D6 wavelet as a pre-conditioner. The results predict that, EDL has less impact on pressure distribution and significant impact on film thickness. The obtained results are compared with results of Dowson and Higginson which are comparable.

Evaluation of Methods for Viscosity Simulations of Lubricants at Different Temperatures and Pressures: A Case Study on PAO-2

The behavior of lubricants at operational conditions, such as at high pressures, is a topic of great industrial interest. In particular, viscosity and the viscosity-pressure relation are especially important for applications and their determination by computational simulations is very desirable. In this study, we evaluate methods to compute these quantities based on fully atomistic molecular dynamics simulations, which are computationally demanding but also have the potential to be most accurate. We used the 9,10-dimethyloctadecane molecule, main component of PAO-2 base oil as the lubricant for our tests. The methods used for the viscosity simulations are the Green-Kubo equilibrium molecular dynamics (EMD-GK) and nonequilibrium molecular dynamics (NEMD), at pressures of up to 1.0 GPa and various temperatures (40-150 degrees Celsius). We present the theory behind these methods and investigate how the simulation parameters affect the results obtained, to ensure viscosity convergence with respect to the simulation intervals and all other parameters. We show that, by using each method in its regime of applicability, we can achieve good agreement between simulated and measured values. NEMD simulations at high pressures captured zero shear viscosity successfully; while at 40 degrees Celsius, EMD-GK is only applicable to pressures up to 0.3 GPa, where the viscosity is lower. In NEMD, longer and multiply repeated simulations improve the confidence interval of viscosity, which is essential at lower pressures. Another aspect of these methods is the choice of the utilized force field for the atomic interactions. This was investigated by selecting two different commonly used force fields.

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

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.

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

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.

Determination of Viscosity Versus Pressure by Means of a Clearance Seal

This paper describes the construction and testing of a simple, experimental tool setup that enables determination of the pressure–viscosity relationship for high viscosity oils. Comparing the determined pressure–viscosity relationship with a reference rheometer measuring the viscosity at ambient pressure yields reasonable agreement. The computed viscosity at elevated pressures was well represented by the Chu and Cameron model.

Effect of surface roughness and deformation on Rayleigh step bearing under thin film lubrication

PurposeThe aim of this paper is to study the effect of deterministic roughness and small elastic deformation of surface on flow rates, load capacity and coefficient of friction in Rayleigh step bearing under thin film lubrication.Design/methodology/approachReynolds equation, pressure-density relationship, pressure-viscosity relationship and film thickness equation are discretized using finite difference method. Progressive mesh densification (PMD) method is applied to solve the related equations iteratively.FindingsThe nature and shape of roughness play a significant role in pressure generation. It has been observed that square roughness dominates the pressure generation for all values of minimum film thickness. Deformation more than 100 nm in bounding surfaces influences the film formation and pressure distribution greatly. Divergent shapes of film thickness in step zone causes a delay of pressure growth and reduces the load capacity with decreasing film thickness. The optimum value of film thickness ratio and step ratios have been found out for the maximum load capacity and minimum coefficient of friction, which are notably influenced by elastic deformation of the surface.Practical implicationsIt is expected that these findings will help in analysing the performance parameters of a Rayleigh step bearing under thin film lubrication more accurately. It will also help the designers, researchers and manufacturers of bearings.Originality/valueMost of the previous studies have been limited to sinusoidal roughness and thick film lubrication in Rayleigh step bearing. Effect of small surface deformation due to generated pressure in thin film lubrication is significant, as it influences the performance parameters of the bearing. Different wave forms such as triangular, sawtooth, sinusoidal and square formed during finishing operations behaves differently in pressure generation. The analysis of combined effect of roughness and small surface deformation has been performed under thin film lubrication for Rayleigh step bearing using PMD as improved methods for direct iterative approach.

Asymptotic analysis of the fluid flow with a pressure-dependent viscosity in a system of thin pipes

We consider the incompressible fluid with a pressure-dependent viscosity flowing through a multiple pipe system. The viscosity–pressure relation is given by the Barus law commonly used in the engineering applications. Assuming that the ratio between pipes thickness and its length is small, we propose a rigorous asymptotic approach based on the concept of the transformed pressure. As a result, we obtain new macroscopic model describing the effective behavior of the fluid in the system. In particular, the generalized version of the Kirchhoff’s law is derived giving the explicit formula for the junction pressure. The error estimate for the asymptotic approximation is also provided. Mathematical analysis presented here can be applied to a general viscosity–pressure relation satisfied by other empiric laws.

Stratification of lubricating material in radial bearings

The technique of calculating the radial sliding bearings of infinite and finite length running on a double-layer stratified lubricant is presented. The technique is based on the use of the self-simulated variable that allows obtaining the exact self-similar problem solution both in polar and in cylindrical coordinates, as well as the parameter value characterizing the interface of the stratified layers. These tasks are complicated by the simultaneous consideration of the lubricant viscosity-pressure relation, the presence of the friction-adapted supporting profile of the bearing bush, and the effect of the axial lubricant feed in the finite-length bearing. As a result, computational models are obtained for two-layer stratified lubricants the numerical analysis of which allows establishing the impact of variables on the basic bearings performance - components of the bearing capacity vector, friction forces, as well as optimal values of the reference profile parameter, the lubricant supply, and its viscous ratio in the stratified layers.

Effect of Load Variation on Elastohydrodynamic Lubrication Film Collapse

Elastohydrodynamic line contact simulations have been carried out in the present study. A practical situation of transient EHL film collapse has been analyzed. The aim is to observe the effect of variation of maximum Hertzian pressure (PH) on transient behavior of EHL film thickness (H).The analysis is based upon classical Reynolds equation considering time variation. The simulation results pertaining to EHL film thickness calculated using linear pressure-viscosity relationship have been compared for different values of load. It has been observed that film thickness reduces with increase in load. Similar results are obtained using exponential pressure-viscosity relationship and compared with those for linear pressure-viscosity. The EHL equations are solved by discretizing Reynolds equation and load equilibrium equation along with other equations using Newton-Raphson technique with the help of a computer code.

Elastohydrodynamic Lubrication Model With Limiting Shear Stress Behavior Considering Realistic Shear-Thinning and Piezo-Viscous Response

This paper addresses a largely ignored aspect pertaining to the elastohydrodynamic lubrication (EHL) traction behavior of fragile lubricants which undergo transition to glassy state at typical EHL contact zone pressures. For such lubricants, a conventional EHL model predicts extremely high and unrealistic values of traction coefficient, especially under near pure rolling conditions where thermal effect is negligible. Therefore, an EHL model incorporating the effect of limiting shear stress and the associated wall slip phenomenon is presented herein. Unlike the other such investigations involving limiting shear stress behavior, the present model employs Carreau-type power-law based models to describe the rheology of lubricants below the limiting shear stress along with realistic pressure-viscosity relationships (WLF and Doolittle-Tait). The use of Carreau-type shear-thinning model in this analysis allows the simultaneous prediction of minimum film thickness and traction coefficient for lubricants which shear-thin in the inlet zone and exhibit limiting shear stress behavior in the contact zone, a feature absent in the existing EHL models utilizing ideal visco-plastic or some other unrealistic rheological model. Using published experimental data pertaining to the shear-thinning and pressure-viscosity response of two fragile lubricants (L100 and LVI260), it has been demonstrated that the present model can explain the appearance of plateau in the experimental traction curve. Also, the influence of shear-thinning parameters and the pressure-viscosity coefficient on the predicted limiting shear stress zone has been studied.

On the helical pipe flow with a pressure-dependent viscosity

We address the flow of incompressible fluid with a pressure-dependent viscosity through a pipe with helical shape. The viscosity-pressure relation is defined by the Barus law. The thickness of the pipe and the helix step are assumed to be of the same order and considered as the small parameter. After transforming the starting problem, we compute the asymptotic solution using curvilinear coordinates and standard perturbation technique. The solution is provided in the explicit form clearly showing the influence of viscosity-pressure dependence and pipe's geometry on the effective flow.

Asymptotic Modeling of the Thin Film Flow with a Pressure-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.

Effect of an Improved Yasutomi Pressure-Viscosity Relationship on the Elastohydrodynamic Line Contact Problem

This paper presents the application of an improved Yasutomi correlation for lubricant viscosity at high pressure in a Newtonian elastohydrodynamic line contact simulation. According to recent experimental studies using high pressure viscometers, the Yasutomi pressure-viscosity relationship derived from the free-volume model closely represents the real lubricant piezoviscous behavior for the high pressure typically encountered in elastohydrodynamic applications. However, the original Yasutomi correlation suffers from the appearance of a zero in the function describing the pressure dependence of the relative free volume thermal expansivity. In order to overcome this drawback, a new formulation of the Yasutomi relation was recently developed by Bair et al. This new function removes these concerns and provides improved precision without the need for an equation of state. Numerical simulations have been performed using the improved Yasutomi model to predict the lubricant pressure-viscosity, the pressure distribution, and the film thickness behavior in a Newtonian EHL simulation of a squalane-lubricated line contact. This work also shows that this model yields a higher viscosity at the low-pressure area, which results in a larger central film thickness compared with the previous piezoviscous relations.

Effects of the Lubricant Piezo-Viscous Properties on EHL Line and Point Contact Problems

This paper presents the application of the free-volume viscosity model in a Newtonian elastohydrodynamic line and point contact simulation using a more effective multigrid approach. According to recent experimental studies using high pressure viscometers, the free volume-based pressure–viscosity relationship closely represents the realistic piezo-viscous behavior for the high pressure typically encountered in elastohydrodynamic applications [1]. The effects of different pressure–viscosity relationships, including the exponential model, the Roelands model, and the free-volume model are investigated through an example with poly-alpha-olefin lubricant. It is found that the real pressure–viscosity behavior predicted by the free-volume model yields a higher viscosity at the low-pressure area, which results in a larger central film thickness. The fact that film thickness is formed mainly by the entraining action at the inlet area significantly weighs the importance of viscosity variation from different models in this area. The inlet area is a low-pressure area, and accordingly, the real viscosity of the lubricant predicted by Doolittle model undergoes a rapid increase in a convex function, being apparently larger than the Roelands one. Furthermore, the Doolittle model leads to higher pressure spike amplitude than that observed using the Roelands model. To solve the problem, a full multigrid approach has been used upon the assumptions of isothermal condition. Multigrid is more effective because it uses coarser grid levels to remove errors of different frequencies, which could be more quickly smoothed away than those on simply the fine grid alone. The developed coarse grid correction cycle proves to be an efficient tool to solve the EHL problem for a wide range of load conditions.

Influence of the rheological behaviour of the lubricant on the appearance of pitting in elastohydrodynamic regime

ABSTRACTThis paper analyses the influence of lubricant behaviour on the appearance of pitting. It attempts to study the effect of viscosity–pressure relation, compressibility, film thickness–roughness relation and friction coefficient on pitting failure of the contacting elements. To explain these effects, we first deal with the influence of the oil on the lubrication of the contact using elastohydrodynamic theory and secondly two multiaxial fatigue criteria are used, Crossland criterion and Dang Van criterion, to evaluate the influence of the rheology on the appearance of pitting. Finally, different applications are presented together with a discussion on the results obtained.

Effect of lubricant selection on EHL performance of involute spur gears

As the gear oils usually undergo shear-thinning even in the inlet zone, an accurate EHL analysis requires realistic rheological models. This is necessary for gear oil selection so as to prevent scuffing failure. This paper demonstrates the effect of rheology on the EHL characteristics of spur gears using full transient thermal EHL simulations with Carreau shear-thinning model and Doolittle's free volume based pressure-viscosity relationship. The PDMS oil considered here is found to exhibit severe film thinning with 74% thinner EHL film as compared to a moderately shear-thinning PAO oil which, on the other hand, undergoes a larger thermal reduction.

Rheological behavior’s effect on the work performance of oil film

A 3D model of hydrostatic turntable’s oil chamber is established to investigate the lubricants performance with different rheological properties by using FLUENT software and the finite volume method. Newtonian oil and non-Newtonian oil’s performance under varied speeds are compared on the large size hydrostatic turntable system considering the temperatureviscosity relationship and pressure-viscosity relationship. The results show that the property of non-Newtonian fluid viscosity influenced by shear rate largely affects the lubricants performance for most oil added polymer additives. Lubricants cannot simply be regarded as Newtonian fluid. The shear thickening non-Newtonian fluid has a better work property. The results are important to design a large size and high-speed hydrostatic support system, choose lubricant oils, and investigate oil film’s work properties.

New film thickness formula for shear thinning fluids in thin film elastohydrodynamic lubrication line contacts

A new central film thickness formula pertaining to thin film elastohydrodynamic lubrication (EHL) line contacts has been developed for Carreau-type shear-thinning lubricants using an extensive set of full EHL simulations. The shear-thinning correction factors available in the literature are based on EHL simulations carried out for a particular value of load and piezo-viscous coefficient using the simplest exponential pressure–viscosity relationship. In the present work, therefore, the load and piezo-viscous coefficient are varied over a wide range so as to arrive at a more generic and accurate film thickness equation. Also, the pressure–viscosity relation employed herein is the Doolittle's free volume-based viscosity model, which is capable of replicating the experimental pressure–viscosity data with a high degree of accuracy. It has been demonstrated that the present film thickness equation accurately captures the recently discovered scale and load sensitivity effects.