Abstract: This study presents an analysis of journal bearing lubricated with couple stress fluids considering the effects of a layer adhered to bearing surface. The modified classical Reynold’s equation is derived considering the effects of surface layer and couple stress fluids. In the present study, the effects of couple stresses on the steady state journal bearing performance characteristics are analyzed based on Stokes micro-continuum theory. The Reynold’s boundary conditions are used in the analysis. Results of non-dimensional load capacity and coefficient of friction are presented.
INTRODUCTION
Structure and properties of fluid film are important aspects in the analysis of journal bearing lubrication. Szeri (2010) advocated the modification of structure of fluid film to a composite film that combines both high and low viscosity fluid. Composite film bearings provide significant savings in power loss due to viscous friction in hydrodynamic bearings. Tichy (1995) developed a rheological model for thin film lubrication in which surface layers appears to have higher viscosity compared to conventional viscosity.
The increasingly severe technological requirements led to considerable thermal and mechanical deformations and consequently compromise minimum film thickness in bearings (Mansouri et al., 2007). The couple stresses fluid takes into account the properties of lubricants with additives. Couple stress fluid model based on micro-continuum theory is the simplest generalization of classical theory of fluids. The experimental study of Oliver (1988) predicted load enhancement and friction reduction effects in a short journal bearing due to dissolved polymer in lubricant. Bujurke and Naduvinamani (1991) presented characteristics of narrow porous journal bearing lubricated with couple stress fluids. Couple stress fluid theory predicts improvement in load capacity and reduction in coefficient of friction in journal bearings compared to those predicted using Newtonian fluids. Lin (1997) investigated the effects of couple stresses on lubrication of finite journal bearing. Mokhiamer et al. (1999) presented a study of couple stress fluid taking into account the elasticity of the liner in journal bearing. Li and Chu (2004) and Elsharkawy (2005) derived both porous media and couple stress model to study the effects of lubricant additives on the bearing performance characteristics. Siddiqui et al. (2006) presented an analytical study of an infinite slider bearing lubricated with Powell-Eyring fluid.
The purpose of this study is to investigate the non-dimensional load capacity and coefficient of friction in long journal bearing using both surface layer and couple stress fluid effects. A modified Reynold’s equation is derived considering Newtonian surface layer and Stokes’ couple stress fluid model. The non-dimensional pressure and shear stress expressions are derived. Reynold’s boundary conditions are used to solve the pressure distribution. The influence of surface layer and couple stress effects on the bearing characteristics are presented.
ANALYSIS
In this theoretical study, the influence of Newtonian surface layer and Stokes’ couple stress fluid film on the journal bearing load capacity and coefficient of friction are investigated. The schematic of journal bearing with surface layer and couple stress fluid film is shown in Fig. 1.
Using the assumptions of thin film lubrication theory, the momentum equations for Newtonian surface layer (0≤y≤δ) and the equations of motion for Stokes’ couple stress fluid film (δ≤y≤h) are:
| (1) |
| (2) |
The boundary conditions are that velocities and viscous shear stresses are continuous at the interface, the couple stresses vanish at the interface, and no-slip at the bearing surfaces:
| (3) |
| (4) |
| Fig. 1: | Geometry of journal bearing with surface layer and couple stress fluid film |
| (5) |
Integrating the Eq. 1 and 2, using the boundary conditions in Eq. 3 and 5, the non-dimensional velocity distribution in the surface layer (0≤Y≤Δ) and couple stress fluid film (Δ≤Y≤H) is expressed as:
| (6) |
| (7) |
Where:
| (8) |
| (9) |
| (10) |
| (11) |
| (12) |
The equation of continuity across the film is:
| (13) |
Simplifying the equation of continuity across the film, yields:
| (14) |
Where:
| (15) |
| (16) |
For Δ = 0, G1 and G2 in Eq. 15 and 16 reduce to:
| (17) |
For Δ = 0 and λ = 0, G1 and G2 in Eq. 17 reduce to:
| (18) |
The Reynold’s boundary conditions are:
| (19) |
Integrating the Eq. 14 and substituting the first boundary condition given in Eq. 19, yields the non-dimensional pressure profile as:
| (20) |
Substitution of the Reynold’s boundary conditions for non-dimensional pressure at film rupture in Eq. 20 and simplifying results in Q as:
| (21) |
Substituting the pressure gradient boundary condition given in Eq. 19 in the expression for non-dimensional pressure gradient in Eq. 14, results in:
| (22) |
The Newton-Raphson iterative procedure is used to solve simultaneously both θr and Q using Eq. 21 and 22.
The radial and tangential non-dimensional load capacity obtained by integration of non-dimensional pressure along and perpendicular to line of centers are expressed as:
| (23) |
The non-dimensional load capacity is expressed as:
| (24) |
The non-dimensional shear stress in the journal bearing at Y = H is obtained as:
| (25) |
The non-dimensional friction force on the journal surface is obtained by integrating the shear stress along the journal surface as:
| (26) |
The non-dimensional friction coefficient is calculated as:
RESULTS AND DISCUSSION
A journal bearing considering the effects of surface layer and couple stress fluids is considered in the analysis. The parameters used in the analysis of journal bearing are: eccentricity ratio (ε) = 0.5; couple stress parameter (λ) = 0.1, 0.2, 0.3, 0.4; dynamic viscosity ratio of surface layer to couple stress fluid film (α) = 10, 100 and non-dimensional surface layer thickness (Δ) = 0.025, 0.05, 0.075, 0.1, 0.2, 0.3, 0.4. The influence of couple stress parameter and high viscosity surface layer on the load capacity enhancement and coefficient of friction reduction for journal bearing are analyzed.
Figure 2a and b show the non-dimensional load capacity (W) with variation in non-dimensional surface layer thickness (Δ = 0.025-0.1 and Δ = 0.1-0.4) for various values of (1) couple stress parameter (λ) and (2) dynamic viscosity ratio of surface layer to couple stress fluid film (α).
The non-dimensional load capacity (W) increases (1) with increase in couple stress parameter (λ) and (2) increase in non-dimensional surface layer thickness (Δ) in the range 0.025-0.1. The non-dimensional load capacity (W) for α = 100 increases significantly with increase in non-dimensional surface layer thickness (Δ = 0.1-0.4). The non-dimensional load capacity (W) increases with increase in non-dimensional surface layer thickness (Δ = 0.1-0.4) for α = 10 and λ = 0.1-0.2. The non-dimensional load capacity (W) increases with increase in non-dimensional surface layer thickness (Δ = 0.1-0.3) for α = 10 and λ = 0.3-0.4. It is observed that non-dimensional load capacity (W) increases with increase in couple stress parameter (λ) especially for higher values of dynamic viscosity ratio of surface layer to couple stress fluid film (α = 100).
Figure 3a and b show the coefficient of friction (Cf) with variation in non-dimensional surface layer thickness (Δ = 0.025-0.1 and Δ = 0.1-0.4). The coefficient of friction (Cf) decreases (i) with increase in couple stress parameter (λ) and (ii) increase in non-dimensional surface layer thickness (Δ) in the range 0.025-0.1.
| Fig. 2(a-b): | Non-dimensional load capacity (ε = 0.5) |
| Fig. 3(a-b): | Coefficient of friction (ε = 0.5) |
For a given value of non-dimensional surface layer thickness (Δ), coefficient of friction decreases with increase in (1) couple stress parameter (λ) and (2) dynamic viscosity ratio of surface layer to couple stress fluid film (α). Also for a given value of couple stress parameter (λ), the reduction in coefficient of friction with non-dimensional surface layer thickness (Δ = 0.3-0.4) is higher for α = 100 compared to α =10.
CONCLUSION
The present study evaluates on improvement in load capacity and reduction in friction coefficient for a journal bearing considering couple stress parameter and high viscosity surface layer. Modified Reynold’s equation is derived for long journal bearing taking into consideration of surface layer and couple stress effects. Both surface layer and couple stress fluid film improve the characteristics of journal bearing. The conclusions based on the analysis presented in this study are:
| • | The non-dimensional load capacity (W) increases significantly with increase in non-dimensional surface layer thickness (Δ) for higher values of ratio of dynamic viscosity of surface layer and couple stress fluid film (α). The non-dimensional load capacity (W) also increases significantly with increase in couple stress parameter (λ) for higher values of dynamic viscosity ratio of surface layer to couple stress fluid film (α) and non-dimensional surface layer thickness (Δ) |
| • | The coefficient of friction (Cf) decreases with increase in non-dimensional surface layer thickness (Δ) and couple stress parameter (λ), for higher values of dynamic viscosity ratio of surface layer to couple stress fluid film (α) |
High viscosity bearing surface layer and couple stress effects increase the load carrying capacity and reduce the coefficient of friction in a journal bearing.
ACKNOWLEDGMENT
This research study is funded by Exploratory Research Grant Scheme of Ministry of Higher Education (ERGS-MOHE) Malaysia under grant ERGS/1/2011/TK/UTP/02/40. The authors greatly appreciate the support provided by Universiti Teknologi PETRONAS for this research.
NOMENCLATURE
| C | = | Radial clearance (m) |
| f | = | Friction force, N; F = fC/μuRL |
| h, H | = | Film thickness (m), H = h/C |
| L | = | Length of the journal bearing (m) |
| p | = | Pressure distribution, (N m-2), P = pC2/μujR |
| q | = | Volume flow rate per unit length along film thickness, (m2 sec-1), Q = q/ujC |
| R | = | Journal radius (m) |
| u | = | Velocity component along θ direction (m sec-1), U = u/uj |
| ui, i = 1, 2 | = | Velocity component along θ direction in thin fluid film layer and fluid film layer with couple stress fluid respectively (m sec-1) |
| u12 | = | Velocity component along θ direction at the interface of thin fluid film layer and fluid film layer with couple stress fluid respectively (m sec-1) |
| uj | = | Journal velocity along θ direction (m sec-1) |
| w | = | Static load, N; W = wC2/μujR2L |
| Wε, Wφ | = | Non-dimensional radial and tangential static load for journal bearing |
| x | = | Coordinate along circumferential (x) direction, m; θ = x/R |
| y | = | Coordinate along radial (y) direction (m), Y = y/c |
| α | = | Dynamic viscosity ratio of surface layer to couple stress fluid film; α = μ1/μ2 |
| δ | = | Thickness of surface layer (m), Δ = δ/C |
| ε | = | Journal bearing eccentricity ratio |
| μi, i = 1, 2 | = | Dynamic viscosity of surface layer and fluid film layer with couple stress fluid respectively (Ns m2) |
| η | = | Material constant for couple stress (kg sec-1) |
| λ | = | Couple stress parameter; |
| θ | = | Angular coordinate measured from the position of maximum film thickness in journal bearing |
| θr | = | Angular extent of film rupture for journal bearing |
| τ | = | Shear stress component (N m-2), Π = τC/μuj |
| ω | = | Angular velocity of journal bearing (rad sec-1) |
| r | = | Extent of outlet film in journal bearing measured |
| ε | = | Along the radial direction |
| φ | = | Along the tangential direction |