Lubrication characteristics of a textured porous sliding bearing

Abstract

This study investigates the lubrication characteristics of a textured bearing sliding on a porous layer. To properly implement the porous layer effect in the bearing, the Brinkman-extended-Darcy model is employed, which results in a modified Reynolds equation. This equation is then evaluated to examine the effects of the porous layer’s material and geometric properties on the lubrication characteristics through the use of (semi-)analytic and numerical approaches. The cavitation effect is reflected in the numerical treatment of the model equation, and an adaptive finite difference scheme is employed to efficiently model the bearing system with finite dimples that may be far smaller than the overall bearing dimensions. Along with the variation trends of the bearing performance with respect to the material and geometric properties of the porous layer, the pattern effects of textured surfaces on the bearing performance indices (load capacity and equivalent friction coefficient) are also thoroughly investigated. Here, the pattern comprised a set of regularly spaced identical dimples either in semi-spherical or semi-ellipsoidal shape, and the effects of size and orientation of the dimples are comprehensively explored to separately identify their influence on the bearing performance indices.

Introduction

Most of the machinery used in modern industry strongly demands the use of environmentally conscious or energy-efficient components, including various moving small components, which are directly or indirectly in contact with each other. The resulting friction and wear at the contact–sliding interface may be the most important factor affecting the energy efficiency for these moving components. Surface texturing technology is one prominent method to reduce friction and wear and thus improves the energy efficiency, by changing lubrication characteristics at the interface between the components in relative movement. Maintaining appropriate lubrication on a sliding surface can be accomplished by using a porous material. In the oil-impregnated bearings widely used in many machines where longevity is sought, a thin porous layer is coated on the bearing surface to reserve excessive oil and enhance the bearing reliability.

Studies on hydrodynamic lubrication have a long history. Among these studies, the Reynolds equation, which was derived by introducing assumptions such as the incompressibility of the lubricant, thin-film geometry, and negligence of body force and fluid inertia along with the continuity conditions into the Navier–Stokes equation, has been one of the most widely used theories for studying lubricant behaviors of various applications. This Reynolds equation is also used to analyze the lubrication characteristics of textured surfaces.^1–5

In a porous region, however, the standard form of Reynolds equation cannot be directly employed, and it should be modified with the constitutive equation that governs the fluid behavior in such a porous region. The fluid flow behavior in porous media was theoretically explained first by Darcy. ⁶ Using the Darcy equation, Morgan and Cameron ⁷ theoretically studied the hydrodynamic lubrication problem in short porous journal bearings. Later, Cameron et al. ⁸ conducted an experiment to support the validity of the analytic solution obtained in their earlier study. ⁷ Capone ⁹ attempted to obtain the solution for the flow dynamic lubrication problem with a long porous bearing using the theoretical approach used by Morgan and Cameron. ⁷ Then, Rouleau ¹⁰ enabled the boundary conditions to be satisfied at the open ends by imposing a zero-pressure condition. Rhodes and Rouleau ¹¹ examined the performance of a narrow porous bearing with sealed ends using the Darcy equation. Murti^12–14 used the preceding results to analyze long porous bearings and then extended the problem to short, finite porous bearings. Tichy ¹⁵ proposed a lubrication model sliding on the thin film on a porous layer using the Darcy equation combined with the Reynolds equation.

The rheological effects of an interfacial plane between the porous medium and free flow region were first studied by Beavers and Joseph. ¹⁶ They performed an experiment to identify the boundary conditions at the interfacial plane. In their experiment, they observed slip flow in the boundary layer of the interfacial plane and thereby confirmed that the Darcy equation is not sufficient to reflect the velocity slip phenomenon on the lubrication problems coupled with porous media. Murti ¹⁷ derived a modified Reynolds equation using the criterion proposed by Beavers and Joseph ¹⁶ and introduced the velocity slip phenomenon into the analysis of narrow porous bearings. Chandra et al. ¹⁸ also considered the velocity slip phenomenon to theoretically identify the static and dynamic characteristics of a plane porous journal bearing. Also, Srinivasan ¹⁹ performed an analysis on axially undefined porous journal bearings using a similar theoretical basis to that of Beavers and Joseph. ¹⁶

Brinkman ²⁰ pointed out that the Darcy equation ignores viscosity shearing stress on the fluid volume element. He argued that the Darcy equation cannot be effectively applied to a highly permeable porous medium because the error due to viscosity shearing stress rapidly increases with increasing permeability. To make up for this gap, he proposed the Brinkman-extended-Darcy equation that takes the viscous force on the fluid volume element into account. Later, Hamdan and Barron ²¹ compared the Darcy model and the Brinkman-extended-Darcy model in their study. Neale and Nader ²² found that the Brinkman-extended-Darcy equation can cause a remarkable change in the lubrication characteristics due to the inclusion of a viscous shear term that can influence the free flow region, even if the porous layer is thin. Later, Vafai and Kim ²³ obtained an exact solution for the interfacial fluid flow problem between the porous medium and the free flow region. Using this solution, Nield ²⁴ verified that shear stress continuity condition is not necessary at the interface. Lin and Hwang ²⁵ employed the Brinkman-extended-Darcy model in their short porous journal bearing analysis in the hydrodynamic lubrication regime.

On the other hand, Ochoa-Tapia and Whitaker^26,27 performed both theoretical and empirical studies on the stress jump condition and effective viscosity in the boundary layer between the porous medium and the free flow region by introducing a volume averaging technique. Li ²⁸ proposed a fluid flow model to solve the thin-film lubrication problem on porous media by extensively using the Brinkman-extended-Darcy equation incorporated with the stress jump condition and the viscosity ratio.

In this study, we performed an analysis on the planar bearing with a textured surface sliding over a porous layer using the modified Reynolds equation incorporated with the Brinkman-extended-Darcy model. ²⁰ The material properties including permeability, stress jump factor, and viscosity ratio are varied, dependently or independently, to examine their effects on the bearing characteristics including the load capacity (LC) and the equivalent friction coefficient (EFC). To find valid physical parametric ranges for the modified Reynolds equation, a parametric compatibility condition is also derived here (Appendix 1). The modified Reynolds equation along with this parametric compatibility condition is derived in a (semi-)analytic manner and solved/evaluated through the use of several numerical methods.

The patterns textured on a surface in this study are modeled as regularly spaced semi-ellipsoidal or semi-spherical dimples, and the size and orientation of the dimple are changed to examine their effects on the aforementioned bearing characteristics.

In a flow analysis of a patterned thin channel that includes diverging portions, treatment of cavitation may be an important ingredient for physically plausible results. In this study, the cavitation phenomenon is reflected through a numerical relaxation method, ²⁹ where the Gauss–Seidel iteration method is extensively used.

Governing equations

The planar bearing system under consideration is composed of a textured surface abutting a flat surface coated with a thin porous layer. The schematics of the present bearing system are shown in Figure 1(a) and (b).

Figure 1.

Schematic of the textured/porous planar bearing model: (a) three-dimensional shape and (b) cross-sectional view in two dimensions.

As shown in Figure 1(a), in the present planar bearing model, a thin porous layer, with thickness δ, is covered over the upper surface of the bottom plate moving with velocity components of $u_1$ and $v_1$ in the x and y directions, respectively. A finite number of dimples are textured over the lower surface of the mating top plate moving with velocity components of $u_2$ and $v_2$ in the x and y directions, respectively.

In order to derive a proper lubrication equation for the model explained above, the following Brinkman-extended-Darcy constitutive model ²⁰ is introduced first

$$\begin{array}{cc}\multicolumn{1}{c}{0\le z<\delta :}& \frac{\partial p}{\partial x}=\tilde{\mu }\frac{{\partial }^2}{\partial x^2}\tilde{u}-\frac{\mu }{k}(\tilde{u}-u_1),\hfill \\ \multicolumn{1}{c}{}& \frac{\partial p}{\partial y}=\tilde{\mu }\frac{{\partial }^2}{\partial y^2}\tilde{v}-\frac{\mu }{k}(\tilde{v}-v_1)\hfill \end{array}$$

(1)

$$\delta \le z<h:\frac{\partial p}{\partial x}=\mu \frac{{\partial }^2}{\partial x^2}u,\frac{\partial p}{\partial y}=\mu \frac{{\partial }^2}{\partial y^2}v$$

(2)

where $\tilde{\mu }$ and k are the effective viscosity and the permeability of the porous layer, respectively; $\mu$ is the viscosity of the lubricant; $\tilde{u}$ and $\tilde{v}$ are the velocity components of the lubricant inside the porous layer in the x and y directions, respectively; whereas u and v are those in the free flow region. The boundary and interfacial continuity conditions in the thickness direction of the lubricant film can be fully described by

$$\tilde{u}(x,y,0)=u_1,\tilde{v}(x,y,0)=v_1$$

(3)

$$\tilde{u}(x,y,\delta )=u(x,y,\delta ),\tilde{v}(x,y,\delta )=v(x,y,\delta )$$

(4a)

$$\tilde{\mu }\frac{\partial \tilde{u}}{\partial z}-\mu \frac{\partial u}{\partial z}=\beta \frac{\mu }{\sqrt{k}}(\tilde{u}-u_1),\tilde{\mu }\frac{\partial \tilde{v}}{\partial z}-\mu \frac{\partial v}{\partial z}=\beta \frac{\mu }{\sqrt{k}}(\tilde{v}-v_1)$$

(4b)

$$u(x,y,h)=u_2,v(x,y,h)=v_2$$

(5)

where $\beta$ is the stress jump factor at the interface between the porous layer and the free flow region.^26,27 Equations (3) and (5) are the boundary conditions for fluid velocity at z = 0 and h, respectively; equation (4a) is the continuity condition for fluid velocity at the interface between the porous layer and the free flow region; and equation (4b) is the shear stress jump condition at the interface proposed by Ochoa-Tapia and Whitaker.^26,27

To make the present analysis general and convenient, the following dimensionless parameters are introduced

$$\begin{array}{l}H=\frac{h}{h^{*}},K=\frac{k}{l^{*2}},P=\raisebox{1ex}{$p{h}^{*}{}^2$}\!\left/ \!\raisebox{-1ex}{$\raisebox{1ex}{$\mu $}\!\left/ \!\raisebox{-1ex}{$\raisebox{1ex}{$u^{*}$}\!\left/ \!\raisebox{-1ex}{$l^{*}$}\right.$}\right.$}\right.,U=\frac{u}{u^{*}},\tilde{U}=\frac{\tilde{u}}{u^{*}},V=\frac{v}{u^{*}},\tilde{V}=\frac{\tilde{v}}{v^{*}},\\ X=\frac{x}{l^{*}},Y=\frac{y}{l^{*}},Z=\frac{z}{h^{*}},\Delta =\frac{\delta }{h^{*}},A={X|}_{Z=H}\times {Y|}_{Z=H}\end{array}$$

(6)

Using equation (6) and the mass conservation law to an infinitesimal fluid element in the film, the following modified Reynolds equation can be obtained

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{\partial }{\partial x}\left(G_3\frac{\partial P}{\partial x}\right)}& +\frac{\partial }{\partial y}\left(G_3\frac{\partial P}{\partial y}\right)=\frac{\partial }{\partial x}{G}_1{U}_1+\frac{\partial }{\partial y}{G}_1{V}_1\hfill \\ \multicolumn{1}{c}{}& +\frac{\partial }{\partial x}{G}_2{U}_2+\frac{\partial }{\partial y}{G}_2{V}_2\hfill \end{array}$$

(7)

where

$$\begin{array}{cc}\multicolumn{1}{c}{G_1=}& F_2\sqrt{K}\alpha \{\cosh \left(\frac{\Delta }{\sqrt{K}\alpha }\right)-1\}+\frac{F_8}{2}(H^2-{\Delta }^2)\hfill \\ \multicolumn{1}{c}{}& +F_5(H-\Delta )+\Delta \hfill & \hfill \end{array}$$

(8a)

$$\begin{array}{cc}\multicolumn{1}{c}{G_2=}& F_3\sqrt{K}\alpha \{\cosh \left(\frac{\Delta }{\sqrt{K}\alpha }\right)-1\}\hfill \\ \multicolumn{1}{c}{}& +\frac{F_9}{2}(H^2-{\Delta }^2)+F_6(H-\Delta )\hfill \end{array}$$

(8b)

$$\begin{array}{cc}\multicolumn{1}{c}{}& G_3=\Delta K-\frac{1}{6}(H^3-{\Delta }^3)-K^{\frac{3}{2}}\alpha \sinh \left(\frac{\Delta }{\sqrt{K}\alpha }\right)\hfill \\ \multicolumn{1}{c}{}& -F_1\sqrt{K}\alpha \{\cosh \left(\frac{\Delta }{\sqrt{K}\alpha }\right)-1\}\hfill \\ \multicolumn{1}{c}{}& -F_4(H-\Delta )-\frac{F_7}{2}(H^2-{\Delta }^2)\hfill \end{array}$$

(8c)

In equation (8), $\alpha$ is the viscosity ratio, defined as $\alpha =\sqrt{\tilde{\mu }/\mu }$ and expressions of $F_1-F_9$ are described in Appendix 2. In many studies,^20,22 the viscosities of the fluid inside the porous layer and in the free flow region are assumed to be identical, that is, $\tilde{\mu }=\mu$ (or $\alpha =1$ ). However, Ochoa-Tapia and Whitaker^26,27 proposed that the square of the viscosity ratio is inversely proportional to the porosity of the porous layer ( $\epsilon$ ), that is, ${\alpha }^2\cong 1/\epsilon$ , based on the observation that the superficial and intrinsic velocities and pressures are related by the volume fraction of voids (or porosity). ²⁶ This viscosity ratio/porosity relationship is employed in this study.

Note that these equations are not the same as those in Li ²⁸ due to the differences in the bearing configuration and the manner of derivation; however, they are consistent with each other.

As can be clearly seen in Equation (8), $G_1$ , $G_2$ , and $G_3$ are the parameters to be determined with full knowledge of their arguments K, $\beta$ , $\alpha$ , $\Delta$ , and H.

The two most important figures of merit for the performance measure of the plain sliding bearing (bearing performance indices) may be the LC and the EFC. Here, the LC (or ${\overline{F}}_N$ ) is defined as a dimensionless quantity representing the dimensionless normal force component exerting the load on the bearing, such that

$$\mathrm{LC}={\overline{F}}_N=\underset{A}{\int }\mathrm{PdA}$$

(9)

where the integral domain A means the area of the textured bearing surface.

The EFC (or ${\mu }_e$ ) is defined, following Amonton’s law, ³⁰ as the ratio of the friction (or shear) force component to the normal force component. The shear force component can be obtained by integrating the shear stress over the whole bearing surface. The shear stresses inside the porous layer and in the free flow region can be defined, respectively, as

$$\begin{array}{cc}\multicolumn{1}{c}{}& 0\le Z<\Delta :{\overline{\tau }}_{\mathrm{xz}}={\alpha }^2\frac{\partial }{\partial Z}\tilde{U},{\overline{\tau }}_{\mathrm{yz}}={\alpha }^2\frac{\partial }{\partial Z}\tilde{V}\hfill \\ \multicolumn{1}{c}{}& \Delta \le Z<H:{\overline{\tau }}_{\mathrm{xz}}=\frac{\partial U}{\partial Z},{\overline{\tau }}_{\mathrm{yz}}=\frac{\partial V}{\partial Z}\hfill \end{array}$$

(10)

The force components in two tangential directions at the upper channel surface (i.e. two friction force components) can be obtained, namely

$${\overline{F}}_{\mu x}=\int {{\tilde{\tau }}_{\mathrm{xz}}|}_{Z=H}\mathrm{dA}$$

(11a)

$${\overline{F}}_{\mu y}=\int {{\tilde{\tau }}_{\mathrm{yz}}|}_{Z=H}\mathrm{dA}$$

(11b)

which result in the EFC as

$${\mu }_e=\frac{\sqrt{{\overline{F}}_{\mu x}^2+{\overline{F}}_{\mu y}^2}}{{\overline{F}}_N}$$

(12)

Discussion of results

Effects of viscosity ratio, stress jump factor, and permeability on the lubrication characteristics of the planar journal bearing with a porous layer

In order to trace the effects of the viscosity ratio ( $\alpha$ ), stress jump factor ( $\beta$ ), and permeability (K) on the lubrication characteristics (LC and EFC) of the sliding bearing with a porous layer separately, we attempt to conduct a series of case studies as designed below:

• Case I: ( $\alpha$ fixed) the effect of $\beta$ for a few representative values of K.

• Case II: ( $\beta$ fixed) the effect of K for a few representative values of $\alpha$ .

The detailed geometric and operational conditions used in this study are listed in Table 1.

Table 1.

The geometric and operational conditions for case studies (dimensionless).

Radius of semi-spherical dimple ( $R_d$ )	0.2
Velocity of the lower surface in x direction ( $U_1$ )	1
Velocity of the upper surface in x direction ( $U_2$ )	0
Velocity of the lower surface in y direction ( $V_1$ )	0
Velocity of the upper surface in y direction ( $V_2$ )	0
Thickness of porous layer ( $\Delta$ )	0.1
Length of channel	10

In this sub-section, a total of 25 semi-spherical dimples with dimensionless radius of 0.2 are assumed to be patterned on the top plane of the bearing. Under the same conditions, the LC and EFC values for nonporous bearings ( $\Delta =0$ ) are computed first as references, which are 0.875 and 113.882, respectively.

For the discretization of the analysis domain into meshes, we employed an adaptive mesh refinement scheme. This scheme can provide a way for adapting the numerical precision/resolution based on the requirements for the underlying problem in specific analysis domains that need precision while leaving the other domains at lower levels of precision or resolution. In this study, we divided the analysis domain (bearing surface) into three subdivision surface groups: (1) surfaces of dimples that require the finest meshes, (2) surfaces far away from the dimples that allow the coarsest meshes, and (3) surfaces in the near neighborhood of the dimples that require middle-class meshes. Due to this meshing scheme, the mesh number in each sub-surface tends to vary depending on the configuration of the pattern. For all case studies performed in this article, the total mesh number was kept large (not smaller than 16,000) enough to ensure the convergence and precision of the present analysis.

1. Case I

Figure 2(a) and (b) shows the velocity and shear stress variations as a function of fluid film thickness for several stress jump parameter values ( $\beta$ ) with $\alpha =1.8$ , K = 4.96 × 10⁻⁴, dP/dX = −10, and H = 1. Here, the dimensionless thickness of the porous layer is set to be 0.1.

As expected, Figure 2(a) clearly shows that the maximum velocity in the free flow region increases with $\beta$ . In addition, the velocity profiles in the porous layer (0 < Z < 0.1) vary differently from those of the Darcy flow, where the velocity is assumed to be constant. Furthermore, the configuration of velocity profiles in the porous layer is substantially different from that of the velocity profile in the associated region for the nonporous bearing in that the former is parabolic with respect to the horizontal axis (U), whereas the latter is parabolic with respect to the vertical axis (Z). Apparently, this difference is attributed to the stress jump phenomenon at the interface between the porous layer and free flow region (Z = 0.1). Figure 2(b) shows that shear stress significantly varies in the porous layer compared with stress variation in the free flow region. Furthermore, the stress jump phenomena are clearly exhibited at the interface. As suggested by its name, stress jump increases with $\beta$ . However, in the free flow region, not only the profiles but also the values of shear stress appear to be virtually independent of $\beta$ .

Figure 3(a)–(d) shows the variations of LC, with several K values and $\alpha$ equal to 1.2, 1.5, 1.8, and 2, respectively, with respect to $\beta$ for Case I. As shown consistently in all of these figures, the LC decreases as $\beta$ increases. Note that the LC values for the porous case are generally smaller than those for the nonporous case except for the cases with small $\beta$ and large K. Furthermore, this decreasing trend appears more notably as the permeability (K) decreases. As $\beta$ approaches the lowest value, the LC value tends to approach that of the nonporous case.

Figure 4(a)–(d) shows the EFC variations with respect to $\beta$ for Case I. As observed in Figure 2(b), the effect of stress jump at the interface between the porous layer and free flow region is not reflected on the bearing surface (Z = 1). This means that the effect of pressure on the EFC is greater than that of the friction force (or equivalently, the shear stress at the bearing surface), thereby the EFC value appears to be almost inversely proportional to the LC value. Accordingly, unlike the trend of LC variations, the EFC value increases with $\beta$ and the EFC variation range increases as K decreases. Such tendencies can be consistently observed throughout Figure 4(a)–(d), depicted with four different $\alpha$ values.

1. Case II.

Figure 5(a)–(c) shows the LC variations with respect to K for fixed $\beta$ with (a) 0.93, (b) 0.95, and (c) 0.98. The common feature that can be observed in these figures is that the LC values of porous bearings are smaller than those that are nonporous. Such a tendency can also be observed in Case I. Furthermore, as K increases, the LC value tends to decrease in its lower range (before K ^*) but increases after K ^*. The maximum value of LC increases with $\alpha$ and finally approaches that of the nonporous bearing.

Figure 2.

(a) Velocity profiles and (b) shear stress variations for various stress jump parameter values (with dP/dX = −10, H = 1, $\alpha =1.8$ , and K = 4.96 × 10−⁴).

Figure 3.

Load capacity variations for Case I with (a) $\alpha =1.2$ , (b) $\alpha =1.5$ , (c) $\alpha =1.8$ , and (d) $\alpha =2$ .

Figure 4.

Equivalent friction coefficient variations for Case I with (a) $\alpha =1.2$ , (b) $\alpha =1.5$ , (c) $\alpha =1.8$ , and (d) $\alpha =2$ .

Figure 5.

Variations in load capacity for Case II with (a) $\beta =0.93$ , (b) $\beta =0.95$ , and (c) $\beta =0.98$ .

Figure 6(a)–(c) shows the EFC variations with respect to K for Case II with the same set of $\beta$ values as in Figure 5. Similar to Case I, the EFC values are shown to be inversely proportional to the associated LC values due to the higher effect of LC compared with the friction force. The peak value of the EFC appears to be the largest at the lowest LC value with $\beta =0.98$ . In contrast to the trend in LC variations, the EFC value increases with K before $K^{*}$ and decreases with K after $K^{*}$ . The EFC value finally approaches that of the nonporous bearing.

Figure 6.

Variation in equivalent friction coefficient for Case II with (a) $\beta =0.93$ , (b) $\beta =0.95$ , and (c) $\beta =0.98$ .

Case studies of dimple size on the lubrication characteristics of a porous bearing textured with either semi-ellipsoidal or semi-spherical dimples

In this sub-section, the effects of dimple size on the lubrication characteristics of a porous bearing are investigated, where the bearing is textured to contain either semi-ellipsoidal or semi-spherical dimples. The geometric and operating conditions used in this section are listed in Table 2, and the parameter values for the porous layer are shown in Table 3. For the cases with semi-ellipsoidal dimples, only the length of the major axis (the axis aligned in the x direction) is changed, whereas the lengths of the other two minor axes (the other two axes aligned in y and z directions, respectively) are kept constant for computational convenience. Therefore, the carved semi-ellipsoids are in the shape of prolate ellipsoids of revolution (prolate spheroids) in this study. For notational convenience and unification, the length of the major axis is to be called, hereafter, the “radius” for a semi-ellipsoidal dimple. Furthermore, in order to be consistent with the previous case study, the viscosity ratio ( $\alpha$ ) is also fixed at 2 for all simulations performed in this sub-section.

Table 2.

The geometric and operational conditions used to investigate dimple size effect.

Number of dimples	25
$R_y$ and $R_z$ of semi-ellipsoidal dimples	0.7
Velocity of the lower surface in x direction ( $U_1$ )	1
Velocity of the upper surface in x direction ( $U_2$ )	0
Velocity of the lower surface in y direction ( $V_1$ )	0
Velocity of the upper surface in y direction ( $V_2$ )	0
Thickness of porous layer ( $\Delta$ )	0.1
Length of channel	50

Table 3.

The porous layer parameters used to investigate dimple size effect.

Case	β	K	Shape of dimples
1	Nonporous bearing		Semi-spherical
2	Nonporous bearing		Semi-ellipsoidal
3	0.99	$2\times {10}^{-4}$	Semi-spherical
4	0.99	$2\times {10}^{-4}$	Semi-ellipsoidal
5	0.97	$3\times {10}^{-4}$	Semi-spherical
6	0.97	$3\times {10}^{-4}$	Semi-ellipsoidal

Here, Cases 1 and 2 are associated with the nonporous bearing, whereas Cases 3–6 are associated with the porous bearing with either $\beta =0.99$ or $\beta =0.97$ and with K to be the value corresponding to the smallest LC values for a given parameter set of $\alpha$ and $\beta$ .

Figure 7(a) and (b) represents the LC and EFC variations, respectively, as a function of the dimple radius. The LC values shown in Figure 7(a) tend to increase almost linearly with dimple size within the present computation range regardless of dimple shape. However, the increasing rate of LC with semi-spherical dimples is shown to be greater than that with semi-ellipsoidal dimples.

Figure 7.

Variations in (a) load capacity and (b) equivalent friction coefficient with respect to dimple radius.

For computations for the cases associated with semi-ellipsoidal dimples, only $R_x$ is changed, while $R_y$ and $R_z$ are kept fixed at 0.7. Therefore, when the $R_x$ is smaller than 0.7, the LC values for Cases 1, 3, and 5 (the cases associated with semi-spherical dimples) become smaller than those for Cases 2, 4, and 6 (the cases associated with semi-ellipsoidal dimples), respectively. After the dimple radius exceeds 0.7, the LC values for the cases associated with semi-spherical dimples become larger than those associated with semi-ellipsoidal dimples.

In this situation, the LC values for Cases 1 and 2, which are the nonporous bearing cases patterned with semi-spherical dimples and semi-ellipsoidal dimples, respectively, are higher than those for the porous bearing counterparts. On the other hand, for the porous bearing cases associated with semi-ellipsoidal dimples, the LC values for Case 4 are smaller than those for Case 6.

Figure 7(b) shows the trend in EFC variations with respect to the dimple size. Unlike the trend in LC variations, the EFC value decreases with dimple radius. Furthermore, it can be observed that when the dimple radius is smaller than 0.7, the EFC values for Cases 1, 3, and 5 (the cases associated with semi-spherical dimples) are larger than those for their respective semi-ellipsoidal counterparts, that is, Cases 2, 4, and 6, respectively. After passing through the cross-over point of the radius (0.7), the EFC values for the cases associated with semi-spherical dimples are smaller than those with their semi-ellipsoidal counterparts. In addition, for Cases 3–6, in which a porous layer exists, the same trend as examined in section “Case studies of dimple size on the lubrication characteristics of a porous bearing textured with either semi-ellipsoidal or semi-spherical dimples” can be observed. Like the trend that the EFC value for $\beta =0.99$ is greater than that for $\beta =0.97$ , the EFC values for Cases 3 and 4 are greater than those for Cases 5 and 6, respectively. Conversely, the EFC values for Cases 1 and 2 (associated with nonporous bearing) are the smallest among all the cases (Cases 1, 3, and 5 (associated with semi-spherical dimples) and Cases 2, 4, and 6 (associated with semi-ellipsoidal dimples)), respectively.

Major axis orientation effects of the semi-ellipsoidal dimple on the bearing’s lubrication characteristics

In this section, a series of case studies are performed to investigate the lubricating effects generated due to the orientation of the major axis of the textured semi-ellipsoidal dimples. The geometric attributes of dimples and operating conditions of the bearings are listed in Table 4. In brief, the dimple number is fixed to be 25 for all cases and the major axes of the patterned semi-ellipsoids are set to be parallel with the x-axis ( $\theta =0$ ) for the baseline configuration of the dimple. Here, the angle between the major axis of a semi-ellipsoid and the x-axis is denoted as $\theta$ . The parameters for the porous layer used in this sub-section are listed in Table 5.

Table 4.

Geometric attributes of dimples and operating conditions of bearing.

Number of dimples	25
x-axis radius of dimples ( $R_x$ )	1.3
y- and z-axes radius of dimples ( $R_y$ and $R_z$ )	0.5
Velocity of the lower surface in x direction ( $U_1$ )	1
Velocity of the upper surface in x direction ( $U_2$ )	0
Velocity of the lower surface in y direction ( $V_1$ )	0
Velocity of the upper surface in y direction ( $V_2$ )	0
Thickness of porous layer ( $\Delta$ )	0.1
Length of channel	50

Table 5.

Porous layer parameters used to investigate the size effect of the textured semi-ellipsoidal dimples.

Case	β	K
1	Nonporous bearing
2	0.99	$2\times {10}^{-4}$
3	0.97	$3\times {10}^{-4}$

Figure 8(a) and (b) shows the variations in LC and EFC with respect to the variations in $\theta$ for the cases associated with semi-ellipsoidal dimples. In Figure 8(a), the LC value decreases as $\theta$ increases. In particular, the steepest changes occur when $\theta$ is around 45° for all cases. The LC values converge to certain limits when $\theta$ approaches 90°, which is equivalent to the case that $R_x=R_z=0.5$ , $R_y=1.3$ , and the x-directional velocity of the lower bearing part is 1 m/s. Figure 8(a) shows that the LC values for Case 1 are the highest and those for Case 2 are the lowest among the three cases depicted.

Figure 8.

Variations in (a) load capacity and (b) equivalent friction coefficient with respect to $\theta$ .

Alternatively, the trend for EFC variations is opposite to that for LC variations; the EFC values increase with $\theta$ . Similar to the previous case study, the steepest changes occur when $\theta$ is around 45° for all cases, and the EFC values converge to certain limits when $\theta$ approaches 90°. The EFC values for Case 1 become the lowest, whereas those for Case 2 become the highest among the three cases treated in this case study.

Conclusion

In this study, the pattern effects on the lubrication characteristics, such as the LC and EFC of a sliding bearing incorporated with a porous layer, are thoroughly examined. Here, the pattern comprised a set of regularly spaced identical dimples either with semi-spherical or semi-ellipsoidal shapes, and the size and orientation of the patterned dimples are independently varied to examine their effects separately. Furthermore, the effects of the porous layer’s material and geometric properties such as the permeability (K), stress jump parameter ( $\beta$ ), and viscosity ratio ( $\alpha$ ) on the LC and EFC are also studied in both (semi-)analytic and numerical ways. In order to take the porous layer into account, the Brinkman-extended-Darcy model, which finally ends up with a modified Reynolds equation, is employed in the present model. Through the examination of the analysis and computation results, the following conclusions can be made:

1. Overall, the LC (or EFC) value for the bearing with a porous layer is lower (or higher) than that without the porous layer under the same parametric and geometric conditions. In addition, with the fixation of other parameters, the LC (or the EFC) value decreases (or increases) with the increase in the stress jump parameter ( $\beta$ ) at the interface between the free flow region and the porous layer. The LC (or the EFC) variations with the permeability of the porous layer (K) show the existence of the minimum (or the maximum) LC (or the EFC) value at the corresponding critical permeability ( $K^{*}$ ).

2. Two representative dimple shapes (one semi-spherical and the other semi-ellipsoidal) are considered for investigating their configurational effects on the lubrication characteristics. The LC (or EFC) values increase (or decrease) almost linearly with the dimple size for both dimple shapes. However, the increasing (or decreasing) rates of the LC (or EFC) with the size of the semi-spherical dimple are much higher than those of the semi-ellipsoidal dimple.

3. The orientation of the major axis of the semi-ellipsoidal dimple ( $\theta$ ) measured from its original configuration (the major axis originally is placed in parallel with the x-axis) decreases (or increases) the LC (or EFC) value. Such a decreasing (or increasing) trend appears more remarkably as $\theta$ approaches 45° and then gradually converges to its limiting value.