An Analytical Approach for Analysis of Slider Bearings with Non-Newtonian Lubricants

Abstract

In this study, a regular perturbation technique is utilized to derive the modified Reynolds equation which is applicable to power-law lubricant. The performance of slider bearings including pressure distributions, velocity distributions, film thickness, load capacity, flow rate, shear force, and friction coefficient is also derived analytically for various ξ, flow indices (n), and outlet film thicknesses (H₀). These analytical solutions are clear to find the effects of the operation parameters rather than numerical methods. It can be simply and fast used for engineers. Subsequently, these proposed analytical solutions are used to analyze the lubrication performance of slider bearing with the power-law fluids.

1. Introduction

In all geometric-shaped sliders, wedge sliders provide the simplest support for fluid lubrication, and thus they are usually used as components of thrust slider bearings. With the development of modern machines, polymer oil, especially engine oil for vehicles such as multigrade crank case oil, is usually added to the lubricating oil in order to effectively reduce the friction loss of machine parts. However, such lubricants containing polymer belong to the non-Newtonian fluid type. As a result, the non-Newtonian characteristics of lubricants have become important. In recent years, there have been a great number of models used to describe non-Newtonian fluids. Among them, the power-law model is the simplest and most widely used. Safar and Shawki [1] divided fluids into three different categories: (1) Newtonian fluid, n = 1, which is assumed as the type of ordinary pure fluid and gas; (2) dilatant fluids or shear-thickening fluids, n>1, which exhibit an increase in apparent viscosity with increasing shear rate; (3) pseudoplastic fluids, n<1, which are characterized by linearity at extremely low and extremely high shear rates.

With regard to investigating slider bearings, many scholars [2 –4] have derived closed-form solutions of Newtonian fluid for different slider bearings shapes. Hamrock [5] analyzed pressure distributions, velocity distributions, maximum pressure, film thickness, load capacity per unit width, flow rate, and friction coefficient of fixed-incline slider bearings with Newtonian fluid. As mentioned above, the closed-form solutions focus on simple geometry shape and Newtonian fluid. However, many experiments have shown that base oil blended with long-chain additives to a Newtonian fluid gives the most preferable lubricant results and can improve the load-carrying capacity and reduce the friction parameter [6, 7]. Therefore, the use of non-Newtonian fluids as lubricants has become a more important issue with the industrial development. The non-Newtonian fluids are difficult to get closed-form solution, so many researchers used numerical method to solve bearings performances including pressure distributions and film thickness distributions. But closed-form formulas can clearly find the effects of the operation parameters rather than numerical methods. It can be simply used for engineers.

For slider bearings with non-Newtonian fluids investigation, Dien and Elrod [8] employed the regular perturbation technique to expand the pressure and the velocity into series forms and then substituted them into the Navier-Stokes equation to derive a modified Reynolds equation for a power-law fluid model. Furthermore, the model was used to analyze the lubrication performance of journal bearings. Buckholz [9] used the power-law fluid model to analyze the load capacity and friction coefficient of slider bearings. Das [10] developed a set of algebraic equations in order to obtain the pressure gradient for any value of the power-law index. He used inclined and parabolic infinitely wide lubricated slider bearings to interpret this mathematical development and analyzed the variation of optimum load capacity, coefficient of friction, and so forth, with respect to simultaneous changes of the inlet-outlet film height ratio and of the power-law exponent of lubricants. Naduvinamani et al. [11] analyzed the lubrication performance of slider bearings with different shapes by using the coupling stress fluid model. They also discussed the influences of coupling stress and surface roughness parameters on load capacity and friction coefficient and temperature effect on slider bearings of different shapes. Lin [12] proposed an approximation closed-form solution for the MHD power-law film-shape slider bearings, in which special bearing characteristics of the inclined-plane shape and the parabolic-film profile can also be included.

As mentioned above, the lubrication performance analysis of power-law fluid on slider bearings is almost always obtained by numerical methods. The closed-form solutions of the hydrodynamic lubrication with power-law fluids were absent. Therefore, this paper proposes some analytical solutions for the power-law fluid model of slider bearings. Subsequently, we use these proposed solutions to analyze the lubrication performance of slider bearing with the power-law fluids.

2. Theoretical Analysis

In all geometric-shaped sliders, wedge sliders provide the simplest support for fluid lubrication. As shown in Figure 1, a wedge slider is composed of a bottom plane surface that moves with u_b speed and a fixed-incline top surface. The region between them is full of lubricant. The oil film thickness is h at any point, the inlet clearance is h₀ + s_h, the outlet clearance is h0, and the film thickness h at the x coordinate within the lubrication region can be expressed as

h = h_{0} + s_{h} (1 - \frac{x}{l})

(1)

or in dimensionless form as

H = H_{0} + 1 - X .

(2)

Figure 1:

Geometry of a slider bearing.

Because the fluid film is thin, the variation of pressure across the film thickness is negligible. It is also assumed that the fluid inertia and the body force are neglected. In addition, the viscosity and density are assumed to be pressure independent, and the flow is assumed to be steady and laminar. The lubricant in the system is taken to be a non-Newtonian power-law lubricant.

Mathematically, the viscosity equation of power-law fluid [9] is expressed as

η = m {| \frac{\partial u}{\partial z} |}^{n - 1},

(3)

where η is the viscosity, ∂u/∂z is the shear rate, m is the consistency index, and n is the flow index.

On the hypothesis mentioned above, the Navier-Stokes equation can be simplified as

\frac{\partial p}{\partial x} = \frac{\partial}{\partial z} (η \frac{\partial u}{\partial z}) .

(4)

The boundary conditions are

z = 0, u = u_{b}

(5a)

z = h, u = 0 .

(5b)

To derive a modified Reynolds equation which is suitable for power-law fluid, the regular perturbation technique is used to expand all dependent variables into series type with a small amount ε (dimensionless amplitude), and then substitute it into (4) and integrate the equation with the boundary conditions (5a) and (5b); the velocity is obtained as

u = u_{b} (1 - \frac{z}{h}) - \frac{1}{2 n m} {(\frac{h}{u_{b}})}^{n - 1} \frac{\partial p}{\partial x} z (h - z) .

(6)

The equation of continuity is

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial z} = 0 .

(7)

Substituting the value of u from (6) into (7) and integrating across the film, the steady modified Reynolds equation for power-law fluid can be derived as

\frac{d}{d x} (\frac{h^{n + 2}}{n m} \frac{\partial p}{\partial x}) = 6 {u_{b}}^{n} \frac{d h}{d x} .

(8)

This equation is only suitable for power-law fluid with higher Couette-dominate and power-law fluid with any Couette-Poiseuille component. Note that, as n = 1.0 and m = η, (8) can be simplified into the classical Reynolds equation.

Equation (8) can be integrated as

\frac{d p}{d x} = 6 n m \frac{{u_{b}}^{n} (h - h_{m})}{h^{n + 2}},

(9)

where h_m is the film thickness at maximum pressure; that is, dp/dx = 0. The dimensionless form of (9) is

\frac{d P}{d X} = 6 n ξ^{n - 1} \frac{H - H_{m}}{H^{n + 2}},

(10)

where

ξ = {(\frac{m}{η})}^{1 / (n - 1)} \frac{u_{b}}{s_{h}} .

(11)

The boundary conditions for (10) are

P = 0, H = H_{0} + 1, at X = X_{in} (= 0),

(12a)

P = 0, H = H_{0}, at X = X_{end} (= 1) .

(12b)

Integrating (10) gives

P = 6 n ξ^{n - 1} (\frac{H^{- n}}{n} - \frac{H^{- n - 1} H_{m}}{n + 1}) + A .

(13)

Making use of boundary conditions (12a) and (12b) gives

H_{m} = \frac{n + 1}{n} {\frac{H_{0} (H_{0} + 1) [{H_{0}}^{n} - {(H_{0} + 1)}^{n}]}{{H_{0}}^{n + 1} - {(H_{0} + 1)}^{n + 1}}},

(14)

A = \frac{- 6 ξ^{n - 1}}{{(H_{0} + 1)}^{n + 1} - {H_{0}}^{n + 1}} .

(15)

Substituting (14) and (15) into (13) gives

P = \frac{- 6 ξ^{n - 1}}{H^{n}} {\frac{H_{0} (H_{0} + 1) [{H_{0}}^{n} - {(H_{0} + 1)}^{n}]}{H [{H_{0}}^{n + 1} - {(H_{0} + 1)}^{n + 1}]} - 1} - \frac{6 ξ^{n - 1}}{{(H_{0} + 1)}^{n + 1} - {H_{0}}^{n + 1}} .

(16)

From (2), the location where the pressure gradient is equal to zero gives

X_{m} = H_{0} + 1 - H_{m} .

(17)

Then, substituting (14) into (17) gives

X_{m} = \frac{1}{n} {\frac{- n {(H_{0} + 1)}^{n + 1} - H_{0} (H_{0} + 1) [{H_{0}}^{n} - {(H_{0} + 1)}^{n}]}{{H_{0}}^{n + 1} - {(H_{0} + 1)}^{n + 1}}} .

(18)

From (16) and (17), the maximum pressure can be derived as

P_{m} = \frac{6 ξ^{n - 1} n^{n} {[{H_{0}}^{n + 1} - {(H_{0} + 1)}^{n + 1}]}^{n}}{{H_{0}}^{n} {(H_{0} + 1)}^{n} {[{H_{0}}^{n} - {(H_{0} + 1)}^{n}]}^{n} {(1 + n)}^{n + 1}} - \frac{6 ξ^{n - 1}}{{(H_{0} + 1)}^{n + 1} - {H_{0}}^{n + 1}} .

(19)

The dimensionless load-carrying capacity of the oil film per unit length is

W = \int_{0}^{1} P d X .

(20)

Substituting (16) and (2) into (20) gives

W = 6 ξ^{n - 1} {\frac{[{H_{0}}^{1 - n} - {(H_{0} + 1)}^{1 - n}]}{n - 1} - H_{0} (H_{0} + 1) [2 - \frac{{H_{0}}^{2 n} + {(H_{0} + 1)}^{2 n}}{{H_{0}}^{n} {(H_{0} + 1)}^{n}}] {\times (n [{H_{0}}^{n + 1} - {(H_{0} + 1)}^{n + 1}])}^{- 1} + \frac{1}{{H_{0}}^{n + 1} - {(H_{0} + 1)}^{n + 1}}} .

(21)

The dimensionless volume flow rate per unit width is

Q = \int_{0}^{1} U d Z,

(22)

where U is the dimensionless of velocity:

U = 1 - \frac{Z}{H} - \frac{1}{2 n} ξ^{1 - n} \frac{d P}{d X} H^{n - 1} Z (H - Z) .

(23)

Substituting (23) into (22) gives

\begin{array}{l} Q = \frac{H}{2} - \frac{H^{n + 2}}{12 n} ξ^{1 - n} \frac{d P}{d X} \\ = \frac{n + 1}{2 n} {\frac{H_{0} (H_{0} + 1) [{H_{0}}^{n} - {(H_{0} + 1)}^{n}]}{{H_{0}}^{n + 1} - {(H_{0} + 1)}^{n + 1}}} . \end{array}

(24)

The shear force components per unit width acting on the Z = 0 are

F_{b} = \frac{f_{b}}{η_{0} u_{b}} \frac{s_{h}}{l} = - \int_{0}^{1} (\frac{H}{2} \frac{d P}{d X} + \frac{1}{H}) d X,

(25)

F_{b} = 3 n ξ^{n - 1} [\frac{{H_{0}}^{- n + 1} - {(H_{0} + 1)}^{- n + 1}}{- n + 1}] + 3 n ξ^{n - 1} (\frac{n + 1}{n^{2}}) [2 - \frac{{H_{0}}^{2 n} + {(H_{0} + 1)}^{2 n}}{{H_{0}}^{n} {(H_{0} + 1)}^{n}}] \times [\frac{H_{0} (H_{0} + 1)}{{H_{0}}^{n + 1} - {(H_{0} + 1)}^{n + 1}}] + \ln \frac{H_{0}}{H_{0} + 1} .

(26)

The shear force components per unit width acting on the Z = 1 are

F_{a} = \frac{f_{a}}{η_{0} u_{b}} \frac{s_{h}}{l} = - \int_{0}^{1} (\frac{H}{2} \frac{d P}{d X} - \frac{1}{H}) d X,

(27)

F_{a} = 3 n ξ^{n - 1} [\frac{{H_{0}}^{- n + 1} - {(H_{0} + 1)}^{- n + 1}}{- n + 1}] + 3 n ξ^{n - 1} (\frac{n + 1}{n^{2}}) [2 - \frac{{H_{0}}^{2 n} + {(H_{0} + 1)}^{2 n}}{{H_{0}}^{n} {(H_{0} + 1)}^{n}}] \times [\frac{H_{0} (H_{0} + 1)}{{H_{0}}^{n + 1} - {(H_{0} + 1)}^{n + 1}}] - \ln \frac{H_{0}}{H_{0} + 1} .

(28)

\bar{μ} = μ \frac{l}{s_{h}} = - \frac{f_{b}}{w} \frac{l}{s_{h}} = - \frac{F_{b}}{W} .

(29)

The friction coefficient can be expressed as

3. Results and Discussion

To further understand the effects of non-Newtonian fluids on the lubrication performance of slider bearings, this paper discusses the steady state problem of linear contact of hydrodynamic lubrications. Figure 2 shows the influences of different flow indices on pressure and oil film thickness distributions of slider bearings. When the H0 is given, the oil film thickness distribution can be calculated by using (2). It can also be shown from (16) that the pressure is the function of ξ, n, H₀, and X. Thus, if ξ and H0 remain fixed, the pressure distribution will vary with the n. The greater the n is, the greater the pressure distribution is. These phenomena can be explained by (3). The equivalent viscosity is $m {| \partial u / \partial z |}^{n - 1}$ . The greater the n is, the greater the equivalent viscosity is, which also enlarges the normal load under the same slider bearing shape.

Figure 2:

Pressure distributions and film shapes versus n.

Figure 3 shows the influences of different flow indices on velocity distributions along the film thickness at X = 0.4 and X = 0.9, respectively, under ξ = 10.0 and H₀ = 1.0 conditions. It can be seen from (10) and (14) that the pressure gradient is the function of ξ, n, H₀, and X. Therefore, the greater the n is, the greater the pressure gradient is. But, it can also be seen from (23) that the dimensionless velocity distribution along the film thickness direction is the function of ξ, n, H₀, pressure gradient, and X. If ξ, H₀, and X remain fixed, then the smaller the n is, the greater the velocity distribution is.

Figure 3:

Velocity distributions versus n.

Figure 4 shows the influences of different flow indices on maximum pressure (p_m) and the film thickness at maximum pressure (h_m) at ξ = 10.0 and H₀ = 1.0. It can be shown from (14) that the h_m is the function of n and H0. Thus, if H0 remains fixed, the h_m will vary with the n. The greater the n is, the smaller the h_m is, and the greater the x_m is. It can also be shown from (19) that the p_m is the function of ξ, n, and H0. Thus, if ξ and H0 remain fixed, the p_m will vary with the n. The greater the n is, the greater the p_m is.

Figure 4:

P_m and H_m versus n.

Note that the dimensionless ξ is a function of u_b and s_h. Therefore, if the shoulder height remains fixed, the ξ increases with the increasing speed. In addition, the shoulder height parameter implied a dimensionless parameter. Figure 5 shows the influences of different ξ values on p_m of slider bearings for three different flow indices (n = 0.5, 1.0, 1.5) and H₀ = 1.0. Under the condition of H = H₀ + 1-X, Hamrock [5] solved the integrated form of the Reynolds equation for fixed-incline slider bearings with Newtonian fluid to obtain a close form solution of maximum pressure which is only the function of H0. If H0 is fixed, then the p_m will be fixed. When the lubricant is a non-Newtonian fluid, it can also be shown from (19) that the p_m is the function of ξ, n, and H0. Thus, if n and H0 remain fixed, the p_m will vary with ξ. When n>1.0, the greater the ξ is, the greater the p_m is. However, when n<1.0, the greater ξ is, the smaller p_m is.

Figure 5:

P_m versus ξ with three different lubricants.

Table 1 shows the influences of different ξ values on the W of slider bearings for two different flow indices (n = 0.5, 1.5) and H₀ = 1.0. Under the condition of H = H₀ + 1-X, Hamrock [5] solved the integrated form of the Reynolds equation for fixed-incline slider bearings with Newtonian fluid to obtain a close form solution of W which is only the function of H0. If H0 is fixed, then W will be fixed. When the lubricant is a non-Newtonian fluid, it can also be shown from (21) that the W is the function of ξ, n, and H0. Thus, if n and H0 remain fixed, W will vary with the ξ. When n>1.0, the greater ξ is, the greater W is. However, when n<1.0, the greater the ξ is, the smaller the W is.

Table 1:

Load-carrying capacity versus ξ with two different lubricants (H₀ = 1.0).

ξ	W
ξ	n = 0.5	n = 1.5

2.0	0.06831	0.27687
4.0	0.04830	0.39155
6.0	0.03944	0.47954
8.0	0.03415	0.55373
10.0	0.03055	0.61909
12.0	0.02789	0.67818
14.0	0.02582	0.73252
16.0	0.02415	0.78309
18.0	0.02277	0.83060
20.0	0.02160	0.87553

Figure 6 shows the influences of different H0 values on the p_m and the h_m for three different flow indices (n = 0.5, 1.0, 1.5) and ξ = 10.0. It can be shown from (14) that the h_m is the function of n and H0. Thus, if the n remains fixed, the h_m will vary with H0. Therefore, the greater the H0 is, the greater the h_m is. In addition, the greater the n is, the smaller the h_m is. It can also be shown from (19) that the p_m is the function of ξ, n, and H0. Thus, if ξ and the n remain fixed, the p_m will vary with the H0. The greater the H0 is, the smaller the wedge effect is, and the smaller p_m is.

Figure 6:

P_m and H_m versus H0 with three different lubricants.

Table 2 shows the influences of different H0 values on W and Q for three different flow indices (n = 0.5, 1.0, 1.5) and ξ = 10.0. The W is a function of ξ, n, and H0, as shown in (21). Thus, if ξ and n remain fixed, W will vary with H0. Therefore, the greater the H0 is, the smaller the wedge effect is and the smaller the W is. The Q is a function of n and H0 as shown in (24). Thus, if the flow indices remain fixed, the Q will vary with H0. The greater the H0 is, the greater the Q is. In addition, the greater the n is, the smaller the Q is. Moreover, it can be shown from (21) that the W is the function of ξ, n, and H0. Thus, if ξ and H0 remain fixed, the W will vary with n. The greater the n is, the greater the W is. It can be also shown from (24) that the Q is the function of n and H0. Thus, if H0 remains fixed, the Q will vary with the n. The greater the n is, the smaller the Q is.

Table 2:

Load-carrying capacity and volume flow rate versus H0 with three different lubricants (ξ = 10.0).

H0	n = 0.5		n = 1.0		n = 1.5
H0	W	Q	W	Q	W	Q

1.0	0.030547	0.6796	0.158880	0.66667	0.619090	0.65439
2.0	0.008178	1.2081	0.032791	1.2000	0.098553	1.1920
3.0	0.003488	1.7202	0.011807	1.7143	0.029962	1.7085
4.0	0.001853	2.2268	0.005528	2.2222	0.012367	2.2177
5.0	0.001119	2.7310	0.003020	2.7273	0.006111	2.7235
6.0	0.000736	3.2340	0.001827	3.2308	0.003400	3.2276
7.0	0.000514	3.7361	0.001188	3.7333	0.002059	3.7306
8.0	0.000376	4.2377	0.000816	4.2353	0.001328	4.2329
9.0	0.000285	4.7390	0.000584	4.7368	0.000899	4.7347
10.0	0.000222	5.2401	0.000433	5.2381	0.000633	5.2361

Figure 7 shows the influences of different H₀ values on the shear force and the friction coefficient for three different flow indices (n = 0.5, 1.0, 1.5) and ξ = 10.0. It can be shown from (26) and (28) that shear force is the function of, n, and H₀. Thus, if the n and ξ remain fixed, shear force will vary with H₀. The greater the H₀ is, the greater the shear force on bottom surface is. It can be shown from (21), (26), and (29) that friction coefficient is the function of ξ, n, and H₀. Thus, if n and ξ remain fixed, friction coefficient will vary with H₀. Therefore, the greater the H₀ is, the greater the friction coefficient is. The greater the n is, the smaller the friction coefficient is.

Figure 7:

Shear force and friction coefficient versus H₀.

4. Conclusions

In this paper, a regular perturbation technique is utilized to derive the modified Reynolds equation which is applicable to power-law lubricants. The performance of slider bearings is also derived analytically for various ξ, n, X, and H0. The pressure and velocity distribution are the function of ξ, n, H₀, and X. The load-carrying capacity and shear force are a function of ξ, n, and H0. The volume flow rate and h_m are a function of n and H0. These analytical solutions can clearly find the effects of the operation parameters rather than numerical methods. The researcher can use these equations to analyze the performance of slider bearings with power-law lubricants.

Footnotes

Nomenclature

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to express their appreciation to the National Science Council (NSC 101-2221-E-214-013) in Taiwan for financial support.

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