Effect of Surface Roughness and Couple Stresses on Squeeze Films between Curved Annular Plates

Abstract

The combined effects of couple stresses and surface roughness patterns on the squeeze film characteristics of curved annular plates are studied. The Stokes (1966) couple stress fluid model is included to account for the couple stresses arising due to the presence of microstructure additives in the lubricant. In the context of Christensen's (1969) stochastic theory for the lubrication of rough surfaces, two types of one-dimensional roughness patterns (circumferential and radial) are considered. The governing modified stochastic Reynolds type equations are derived for these roughness patterns. Expressions for the mean squeeze film characteristics are obtained. Numerical computations of the results show that the circumferential roughness pattern on the curved annular plate results in more pressure buildup whereas performance of the squeeze film suffers due to the radial roughness pattern for both concave and convex pads. Further the squeeze film time is longer (shorter) for the circumferential (radial) roughness patterns. Improved squeeze film characteristics are predicted for the couple stress lubricant.

1. Introduction

Squeeze film characteristics play an important role in many applications, namely, lubrication of machine elements and artificial joints. In view of their wide rang of applications, numerous theoretical and experimental studies have been conducted [1–7]. Later, Parkins and Woollam [8] conducted an experimental study of the behavior of an oscillating oil squeeze film. Most of the theoretical studies on squeeze film lubrication between plane parallel plates or between curved circular plates are based on the Newtonian constitutive approximation for the lubricants. However, this approximation is not a satisfactory engineering approach for most of the practical problems in lubrication. The modern lubricants exhibiting non-Newtonian behaviour are the fluids containing long chain polymer additives. The microcontinuum theory derived by Stokes [9] is the simplest generalization of the classical theory of fluids, which allows for polar effects such as the presence of antisymmetrical stresses, couple stress, and body couples. Many investigators have used couple stress fluid theory to analyze the performance of various bearing systems [10–14]. These studies have led to the predictions such as higher load carrying capacity, lower coefficient of friction, and delayed time of approach in comparison with the Newtonian case.

In most of the theoretical investigations of hydrodynamic lubrication, it has been assumed that the bearing surfaces are smooth. This is an unrealistic assumption for the bearings operating with small film thickness. In recent years, a considerable amount of tribological research has been devoted to the study of surface roughness on hydrodynamic lubrication. This is mainly because of the fact that all solid surfaces are rough to some extent and generally the height of roughness asperities is of the same order of magnitude as the mean separation between lubricated contacts. In the literature, several lubrication models accounting for surface roughness effects have been proposed in order to seek a more realistic representation of bearing surfaces. Burton [15] studied the effect of surface roughness on the load supporting characteristics of a lubricant film by postulating the sinusoidal variations in film thickness. Christensen [16] developed a stochastic model for the study of hydrodynamic lubrication of rough surfaces. Prakash and Christensen [17] used the stochastic theory to study the surface roughness effects on squeeze film lubrication between two rectangular plates. The hydrodynamic lubrication of rough journal bearings was studied by Christensen and Tonder [18]. Bujurke and Naduvinamani [19] have included the effect of surface roughness on the squeeze film lubrication between anisotropic porous rectangular plates and predicted the sensitivity of the maximum load to the roughness parameter. Many investigators used this theory to analyze the lubrication characteristics of various bearing systems such as slider bearings [20, 21], journal bearings, and squeeze film bearings [22, 23]. Since the effect of couple stresses is significant and the roughness cannot be avoided, it is worth to investigate the combined effects of these on the bearing performance.

The aim of this paper is to extend the earlier analysis of squeeze films between curved annular plates studied by Gupta and Vora [24] and Bujurke et al. [25] to include the effect of surface roughness and couple stresses fluid. The stochastic method for rough surface developed by Christensen is adopted. The generalized Reynolds type equation is derived, and later the Christensen stochastic roughness model is introduced to take into account the surface roughness effects. Expressions for mean squeeze film pressure, mean load carrying capacity, and squeeze film time are obtained.

2. Mathematical Formulation and Solution of the Problem

Figure 1 shows the configuration of the squeeze film geometry under consideration. The upper rough curved annular plate having circular pocket approaches the lower annular flat plate with velocity 𝑑ℎ₀/𝑑𝑡. The lubricant in the film region is assumed to be Stokes [9] couple stress fluid. It is also assumed that body forces and body couples are absent. Under the usual assumptions of hydrodynamic lubrication applicable to thin films, the equations of motion for couple stress fluid and continuity equation in the film region are given by 𝜕𝑝𝜕𝜕𝑟=𝜇²𝑢𝜕𝑧²𝜕−𝜂⁴𝑢𝜕𝑧⁴,𝜕𝑝1𝜕𝑧=0,𝑟𝜕(𝑟𝑢)+𝜕𝑟𝜕𝑤𝜕𝑧=0, (1)(2)(3) where 𝑢 and 𝑤 are the velocity components in 𝑟 and 𝑧 directions, respectively, 𝑝 is the fluid film, 𝜇 is the Newtonian viscosity, and 𝜂 is the material constant characterizing the couple stress and is of dimension of momentum. The ratio (𝜂/𝜇) has the dimension of length square and hence characterizes the material length of the fluid.

Figure 1 

Squeeze film configuration of curved circular plate with a concentric circular pocket.

To represent the surface roughness, the expression for the film thickness is considered to be of two parts: 𝐻=ℎ(𝑟,𝑡)+ℎ_𝑠(𝑟,𝜃,𝜉), (4) where ℎ(𝑟,𝑡)=ℎ₀(𝑡)𝑒^−𝛽𝑟2,   𝑎≤𝑟≤𝑏 denote the nominal smooth part of the film geometry while ℎ_𝑠 is the part due to the surface roughness measured from the nominal level and is regarded as a randomly varying quantity of zero mean, ℎ₀(𝑡) is the thickness at the center of the film, 𝑎 and 𝑏 are, respectively, the inner and outer radii, 𝑟 is the radial coordinate, and 𝛽 is the curvature parameter; convex film can be generated for 𝛽<0 and concave for 𝛽>0.

The relevant boundary conditions for the velocity components are 𝜕𝑢=0,𝑤=0,²𝑢𝜕𝑧²=0,at𝑧=0,𝑢=0,𝑤=𝑑ℎ₀,𝜕𝑑𝑡²𝑢𝜕𝑧²=0,at𝑧=𝐻. (5) Since 𝑝 is independent of 𝑧, the solution of (1) subject to boundary conditions (5) is 1𝑢=2𝜇𝑑𝑝𝑧𝑑𝑟²2−𝑧𝐻+𝑙²))1−cosh((2𝑧−𝐻𝑙/2(),cosh𝑙𝐻/2 (6) where 𝑙=(𝜇/𝜂)^1/2 is the couple stress parameter.

Integration of (3) across the fluid film and the use of boundary conditions (5) and an expression (6) for 𝑢 give the modified Reynolds equation in the following form: 1𝑟𝑑()𝑑𝑟𝑟𝑓𝐻,𝑙𝑑𝑝𝑑𝑟=12𝜇𝑑ℎ₀,𝑑𝑡 (7) where 𝑓(𝐻,𝑙)=𝐻³−12𝑙²2𝐻−𝑙tanh𝑙𝐻2. (8) Taking the stochastic average of (7), we get 1𝑟𝑑𝑓()𝑑𝑟𝑟𝐸𝐻,𝑙𝑑𝑝𝑑𝑟=12𝜇𝑑ℎ₀,𝑑𝑡 (9) where 𝐸(⋅)=^∞_−∞(⋅)𝑓ℎ_𝑠𝑑ℎ_𝑠. (10) We assume that [16] 𝑓ℎ_𝑠=3532𝑐⁷𝑐²−ℎ²_𝑠³−𝑐<ℎ_𝑠<𝑐,0elsewhere, (11) where 𝑐=3𝜎 and 𝜎 is the standard deviation.

The relevant boundary conditions for the pressure are 𝐸𝑝𝐸𝑝=0at𝑟=𝑎,=0at𝑟=𝑏. (12) In the context of stochastic theory, the following two types of roughness structures are of special interest.

2.1. Circumferential Roughness

In this model, the roughness is assumed to have the form of long narrow ridges and valleys running in 𝜃-direction. The film thickness can be described by a function of the following form: 𝐻=ℎ(𝑟,𝑡)+ℎ_𝑠(𝑟,𝜉). (13) Then the modified Reynolds equation (9) takes the following form: 1𝑟𝑑𝑟𝑑𝑟𝐸()𝑝1/𝑓𝐻,𝑙𝑑𝐸𝑑𝑟=12𝜇𝑑ℎ₀𝑑𝑡. (14)

2.2. Radial Roughness

In this model, the roughness is assumed to have the form of long narrow ridges and valleys running in 𝑟-direction. The film thickness can be described by a function of the following form: 𝐻=ℎ(𝑟,𝑡)+ℎ_𝑠(𝜃,𝜉), (15) and in this case the modified Reynolds equation (9) is 1𝑟𝑑𝑓()𝑝𝑑𝑟𝑟𝐸𝐻,𝑙𝑑𝐸𝑑𝑟=12𝜇𝑑ℎ₀𝑑𝑡. (16)

3. Solution of the Problem

Equations (14) and (16) together can be written in the following form: 1𝑟𝑑()𝑝𝑑𝑟𝑟𝐺𝐻,𝑙𝑑𝐸𝑑𝑟=12𝜇𝑑ℎ₀𝑑𝑡, (17) where 𝐺()=𝐸1𝐻,𝑙𝑓(𝐻,𝑙)⁻¹𝐸𝑓(forcircumferentialroughness,𝐻,𝑙)forradialroughness. (18) Integrating (17) twice with respect to 𝑟 and making use of the boundary conditions (12), the mean squeeze film pressure in the film region is obtained as 𝐸𝑝=𝜇ℎ̇₀^𝑏_𝑟6𝑟𝐺()∫𝐻,𝑙𝑑𝑟−^𝑏𝑟(1/𝑟𝐺(𝐻,𝑙))𝑑𝑟∫^𝑏_𝑎(1/𝑟𝐺(𝐻,𝑙))𝑑𝑟^𝑏𝑎6𝑟𝐺().𝐻,𝑙𝑑𝑟 (19) Introducing the following nondimensional parameters and variables 𝐻=ℎ+ℎ_𝑠,ℎℎ=ℎ₀=𝑒^−𝛽𝑅2,ℎ_𝑠=ℎ_𝑠ℎ₀,𝑟𝑅=𝑏,𝐸𝑝ℎ𝑝=³⁰𝑏²𝜇ℎ̇₀,𝑐𝐶=ℎ₀𝑙,𝜏=ℎ₀,𝛽=𝛽𝑏² (20) we get the nondimensional mean film pressure in the following form: 𝑝=¹_𝑅6𝑅𝐺−∫𝐻,𝜏𝑑𝑅_1𝑅1/𝑅𝐺𝐻,𝜏𝑑𝑅∫¹_𝑎/𝑏1/𝑅𝐺𝐻,𝜏𝑑𝑅^1𝑎/𝑏6𝑅𝐺𝐻,𝜏𝑑𝑅, (21) and the load carrying capacity of squeeze film is found by integrating pressure over the plate: 𝐸(𝑊)=2𝜋^𝑏_𝑎𝑟𝐸𝑝𝑑𝑟, (22) which in nondimensional form is given by 𝐸(𝑊)ℎ𝑊=³⁰2𝜋𝜇ℎ̇₀𝑏⁴=¹_𝑎/𝑏3𝑅³𝐺∫𝐻,𝜏𝑑𝑅−3^1𝑎/𝑏𝑅/𝐺𝐻,𝜏𝑑𝑅²∫¹_𝑎/𝑏1/𝑅𝐺.𝐻,𝜏𝑑𝑅 (23) The time taken to attain the film thickness ℎ₀₂ from an initial film thickness ℎ₀₁ under a constant mean load 𝑊 is obtained from (23) as 1Δ𝑡=ℎ²₀₂−ℎ²₀₁𝑊𝐸(𝑊)𝜋𝜇𝑏⁴, (24) which in nondimensional form is (𝑊)ℎΔ𝑇=Δ𝑡𝐸²₀𝜋𝜇𝑏⁴=1ℎ²₀₂−1ℎ²₀₁𝑊, (25) where ℎ₀₂=ℎ₀₂/ℎ₀,  ℎ₀₁=ℎ₀₁/ℎ₀, and 𝐺=𝐸1𝐻,𝜏𝐹𝐻,𝜏⁻¹𝐸𝐹forcircumferentialroughness,𝐻,𝜏forradialroughness. (26)

4. Results and Discussion

Using Christensen’s [16] stochastic theory for rough surface and Stokes [9] couple stress fluid model for lubricant, we analyze the combined effects of surface roughness and couple stress on the characteristics of squeeze film lubrication between curved circular plate and a flat plate. The couple stress parameter 𝑙 (=(𝜇/𝜂)^1/2) is the dimension of length-squared and may be regarded as the chain length of polar additives in the lubricant. Hence, it provides the microstructure “size” effect on the mechanism of interaction of the lubricant with the bearing geometry.

The squeeze film characteristics are the functions of the various nondimensional quantities such as roughness parameter 𝐶 (=𝑐/ℎ₀), couple stress parameter 𝜏 (=𝑙/ℎ₀), and curvature parameter 𝛽 (=𝛽𝑏²). The negative values of 𝛽 (𝛽<0) produce convex pad geometry, and positive values of 𝛽 (𝛽>0) generate that of concave pad.

Figures 2 and 3 show the variation of nondimensional pressure 𝑝 as a function of the nondimensional radial coordinate 𝑅 for different values of roughness parameter 𝐶 for both concave and convex pads. The dotted curves in the graphs correspond to the smooth case (𝐶=0). It is observed that the point of maximum pressure is asymmetrically located and is shifted toward the outer edge for concave pad (𝛽>0) and this shift is toward the inner edge for convex pad (𝛽<0). This shift of the pressure peak is due to the wedge effect and is toward the minimum film thickness. The important observation is that the effect of roughness is to increase the pressure for circumferential roughness pattern and to decrease it for radial roughness pattern as compared to the corresponding smooth cases. The variation of nondimensional mean squeeze film pressure 𝑝 as function of 𝑅 for different values of 𝜏 for both concave (𝛽=0.25) and convex (𝛽=−0.25) pad geometries is shown in Figures 4 and 5 for radial and circumferential roughness patterns, respectively. The dotted curves in the graphs correspond to Newtonian case (𝜏→∞). It is found that the effect of couple stress is to increase the squeeze film pressure as compared to the corresponding Newtonian case. Further, the increase in 𝑝 is more pronounced in the case of concave pad geometries as compared to the convex pad geometries for both circumferential and radial roughness patterns.

Figure 2 

Variation of nondimensional mean squeeze film pressure 𝑝 as a function of 𝑅 for different values of roughness parameter 𝐶 with 𝜏=5 for radial roughness pattern.

Figure 3 

Variation of nondimensional mean squeeze film pressure 𝑝 as a function of 𝑅 for different values of roughness parameter 𝐶 with 𝜏=5 for circumferential roughness pattern.

Figure 4 

Variation of nondimensional mean squeeze film pressure 𝑝 as a function of 𝑅 for different values of couple stress parameter 𝜏 with 𝐶=0.2 for radial roughness pattern.

Figure 5 

Variation of nondimensional mean squeeze film pressure 𝑝 as a function of 𝑅 for different values of couple stress parameter 𝜏 with 𝐶=0.2 for circumferential roughness pattern.

The variation of nondimensional mean load carrying capacity 𝑊 as a function of the normalized curvature parameter 𝛽 for different values of roughness parameter 𝐶 for two values of 𝑎/𝑏 for both concave and convex pads is depicted in Figures 6 and 7 for radial and circumferential roughness patterns, respectively. The dotted curves in the graphs correspond to the smooth case. It is observed that 𝑊 increases (decreases) for concave (convex) pad geometries as 𝛽 increases. Further it is observed that 𝑊 decreases for increasing values of 𝑎/𝑏 in both cases. The effect of roughness is to increase the load carrying capacity in the case of circumferential roughness pattern and to decrease it in the case of radial roughness pattern as compared to smooth case.

Figure 6 

Nondimensional mean load carrying capacity 𝑊 as a function of normalized curvature parameter 𝛽 for various values of roughness parameter 𝐶 with different values of 𝑎/𝑏 and 𝜏=5 for radial roughness pattern.

Figure 7 

Nondimensional mean load carrying capacity 𝑊 as a function of normalized curvature parameter 𝛽 for various values of roughness parameter 𝐶 with different values of 𝑎/𝑏 and 𝜏=5 for circumferential roughness pattern.

Figures 8 and 9 show the variation of nondimensional mean load carrying capacity 𝑊 as a function of normalized curvature parameter 𝛽 (for radial and circumferential roughness patterns, resp.) for various values of couple stress parameter 𝜏 with different values of 𝑎/𝑏 for both concave and convex pads. The effect of pocket size on load carrying capacity is characterized by the parameter 𝑎; the load carrying capacity of the bearing decreases with the increase of pocket radius 𝑎. The mean load carrying capacity increases (decreases) for concave (convex) pads with increasing values of the curvature parameter, 𝛽. It is also observed that the effect of couple stresses is to increase the 𝑊 for both types of roughness patterns as compared to the corresponding Newtonian case.

Figure 8 

Nondimensional mean load carrying capacity 𝑊 as a function of normalized curvature parameter 𝛽 for various values of couple stress parameter 𝜏 with different values of 𝑎/𝑏 and 𝐶=0.2 for radial roughness pattern.

Figure 9 

Nondimensional mean load carrying capacity 𝑊 as a function of normalized curvature parameter 𝛽 for various values of couple stress parameter 𝜏 with different values of 𝑎/𝑏 and 𝐶=0.2 for circumferential roughness pattern.

The variation of nondimensional squeeze film time Δ𝑇 with the nondimensional final central film thickness ℎ₀₂ for different values of 𝜏 is depicted in Figures 10 and 11 for radial and circumferential roughness patterns, respectively. It is observed that Δ𝑇 is large for couple stress lubricant as compared to the corresponding Newtonian case for both concave and convex pads geometries. Figure 12 shows the variation of Δ𝑇 with ℎ₀₂ for various values of 𝐶 for both types of roughness patterns with 𝛽=0.2 (concave pad geometry). It is observed that squeeze film time is lengthened for the circumferential roughness pattern as compared to the smooth case (𝐶=0), whereas the effect of radial roughness pattern is to reduce Δ𝑇 as compared to the corresponding smooth case. However, the numerical computations showed that the effect of 𝐶 on the variations of Δ𝑇 With ℎ₀₂ is marginal for convex pad geometries.

Figure 10 

Variation of nondimensional squeeze time Δ𝑇 with ℎ02 for different values of couple stress parameter 𝜏 for fixed values of 𝑎/𝑏=0.4 and 𝐶=0.2 for radial roughness pattern.

Figure 11 

Variation of nondimensional squeeze time Δ𝑇 with ℎ02 for different values of couple stress parameter 𝜏 for fixed values of 𝑎/𝑏=0.4 and 𝐶=0.2 for circumferential roughness pattern.

Figure 12 

Variation of nondimensional squeeze time Δ𝑇 with ℎ02 for different roughness parameters 𝐶 with 𝑎/𝑏=0.4 and 𝜏=5.

5. Conclusions

On the basis of the Stokes microcontinuum theory for couple stress fluids and the Christensen stochastic model for the rough surfaces, the combined effects of surface roughness and couple stresses on the squeeze film characteristics of curved annular plates are presented. The generalized stochastic non-Newtonian Reynolds-type equation has been derived. It is found that the effect of couple stresses is to increase the squeeze film pressure, the mean load carrying capacity, and squeeze film time as compared to the corresponding Newtonian case. These results are more pronounced for concave pads compared with convex pads. For the radial roughness pattern the influence of roughness is marginal whereas it is more pronounced for the circumferential roughness pattern. In the limiting case 𝜏→∞ and 𝐶=0 the analysis reduces to the analysis of classical squeeze film lubrication of curved annular plates studied by Gupta and Vora [24]. Hence the squeeze film bearings with curved annular plates having surface roughness and couple stress fluid as lubricant sustain larger load for a longer time as compared to the corresponding Newtonian case by which it improves the bearing performance.

Nomenclature

𝑎:	Inner radius of the plate
𝑏:	Outer radius of the plate
𝑐:	Maximum asperity deviation from the nominal film height
𝐶:	Nondimensional roughness parameter (=𝑐/ℎ0)
𝐸:	Expectancy operator
ℎ̇0:	(=𝑑ℎ0/𝑑𝑡)
ℎ:	Nominal film height (=ℎ0𝑒−𝛽𝑟2)
ℎ𝑠:	Deviation of film height from nominal level
ℎ:	Nondimensional film height (=ℎ/ℎ0)
ℎ01:	Nondimensional initial central film thickness (=ℎ01/ℎ0)
ℎ02:	Nondimensional final central film thickness (=ℎ02/ℎ0)
𝐻:	Film thickness (=ℎ+ℎ𝑠)
𝐻:	Nondimensional film thickness (=ℎ+ℎ𝑠)
𝑙:	Couple stress parameter (=(𝜇/𝜂)1/2)
𝑝:	Pressure in the film region
𝑝:	Nondimensional pressure in the fluid region (=𝐸(𝑝)ℎ30/𝑏2𝜇ℎ̇0)
𝑟:	Radial coordinate
𝑅:	Nondimensional radial coordinate (=𝑟/𝑏)
𝑢&𝑤:	Velocity components in 𝑟- and 𝑧-directions, respectively
Δ𝑡:	Time taken for a reduction in central film thickness from ℎ01 to ℎ02
Δ𝑇:	Nondimensional squeeze film time (=Δ𝑡𝐸(𝑤)ℎ20/2𝜋𝜇𝑏4)
𝐸(𝑊):	Mean load carrying capacity
𝑊:	Nondimensional mean load carrying capacity (=𝐸(𝑤)ℎ30/2𝜋𝜇|ℎ̇0|𝑏4)
𝛽:	Curvature parameter
𝛽:	Nondimensional curvature parameter (=𝛽𝑏2)
𝜇:	Viscosity of the lubricant
𝜂:	Material constant
𝜏:	Nondimensional couple stress parameter (=𝑙/ℎ0)
𝜉:	Random variable.

Acknowledgments

The author thanks the anonymous referees for their educating comments on the earlier version of the paper. The author wishes to thank the Principal of SDM College of Engineering and Technology, Dharwad, for his encouragement.