Table of Contents
ISRN Tribology
Volume 2013, Article ID 431508, 9 pages
http://dx.doi.org/10.5402/2013/431508
Research Article

Squeeze Film Lubrication between Rough Poroelastic Rectangular Plates with Micropolar Fluid: A Special Reference to the Study of Synovial Joint Lubrication

Department of Mathematics, Gulbarga University, Gulbarga 585106, India

Received 24 November 2012; Accepted 12 December 2012

Academic Editors: L. Bourithis and H. Zhu

Copyright © 2013 N. B. Naduvinamani and G. K. Savitramma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The effects of surface roughness and poroelasticity on the micropolar squeeze film behavior between rectangular plates in general and that of synovial joints in particular are presented in this paper. The modified Reynolds equation, which incorporates the randomized surface roughness structure as well as elastic nature of articular cartilage with micropolar fluid as lubricant, is derived. The load-carrying capacity and time of approach as functions of film thickness during normal articulation of joints are obtained by using Christensen stochastic theory for rough surfaces with the assumption that the roughness asperity heights are to be small compared to the film thickness. It is observed that the effect of surface roughness has considerable effects on lubrication mechanism of synovial joints.

1. Introduction

The study of mechanism of synovial joints has recently become an active area of scientific research. The human joint is a dynamically loaded bearing which employs articular cartilage as the bearing and synovial fluid as the lubricant. Once a fluid film is generated, squeeze film action is capable of providing considerable protection to the cartilage surface. The loaded bearing synovial joints of the human body are the shoulder, hip, knee, and ankle joints; such joints have a lower friction coefficient and negligible wear. Synovial fluid is a clear viscous fluid, a dialysate of plasma containing mucopolysaccharides. Synovial fluid usually exhibits a non-Newtonian shear thinning behavior. However, under high shear rates, the viscosity of synovial fluid approaches a constant value not much higher than that of water [1]. Therefore a Newtonian lubricant model has often been used for synovial fluid in lubrication modeling [2]. In this study the synovial fluid is modeled as non-Newtonian micropolar fluid.

Articular cartilage is poroelastic or biphasic consisting of both fluid and solid phases. The importance of the unique biphasic load-carrying characteristics of articular cartilage and fluid flow inside has been recognized in the lubrication of synovial joints such as weeping and boosted lubrication theories. A more general biphasic lubrication theory was subsequently proposed by Mow and Lai [3]. However, it was not until in the 1990s that the relation between friction and interstitial fluid pressurization was comprehensively studied [47]. A number of friction studies have been carried out under a wide range of tribological conditions to investigate the biphasic lubrication of articular cartilage. Under both start-up and reciprocating motions of a cartilage plug against a metallic counterface friction was found to increase with loading time [4]. The transient friction behavior observed was a direct result of the interstitial fluid pressurization and fluid load support, directly measured experimentally [6, 8]. However, for a similar configuration but under cyclic loading, friction was found to be similar or even at a higher level [9]. The importance of the biphasic lubrication has also been studied by enzymatic treatment of articular cartilage altering the biphasic properties and fluid pressurization such as chondroitinase [1013]. However, the results obtained have been contradictory. Pickard et al. [10] found no major differences in friction levels following chondroitinase treatment, while Kumar et al. [11] and Basalo et al. [12] showed a significant reduction.

All these studies were confined to the smooth cartilage surfaces of human knee. But Sayles et al. [14] revealed experimentally that cartilage surfaces are rough, and roughness height distribution is Gaussian in nature. This has motivated us to investigate the influence of roughness of cartilage surfaces in lubrication aspects of synovial joint. Christensen [15] developed the stochastic theory to understand the influence of surface roughness in hydrodynamic lubrication of bearings. Many researchers have used this theory to analyze the effect of surface roughness of various types of bearings. Naduvinamani et al. [16] have studied the problem of squeeze film lubrication between rough rectangular plates with couple stress fluid as lubricant. These investigations have not incorporated the poroelasticity of the bearing surface.

The squeeze film lubrication characteristics of micropolar fluid have been extensively studied in the literature. Agrawal et al. [17] studied the squeeze film and externally pressurized bearings lubricated with micropolar fluids and found that the time of approach is more for the micropolar fluids as compared to the corresponding Newtonian fluids. The analytical solution of the problem of squeeze film lubrication of micropolar fluid between two parallel plates (one-dimensional) has been given by Bujurke et al. [18].

In the present paper, a theoretical study of combined effects of surface roughness and micropolar fluid in squeeze film lubrication between poroelastic rectangular plates is presented. For mathematical simplicity, the average of three layers of the cartilage is modeled as a single poroelastic layer.

2. Mathematical Formulation of the Problem

The geometry and coordinates of the problem are as shown in Figure 1. The squeezing flow of micropolar fluid between two rectangular surfaces is considered. The upper rough articular surface is approaching the lower smooth poroelastic matrix normally with a constant velocity . The lubricant in the joint cavity is taken to be Eringen’s [19] micropolar fluid. As the load-bearing area of the synovial knee joint is small, the two surfaces may be considered to be parallel under high loading conditions. The moving boundary is characterized by where represents the nominal smooth part of the film geometry and is part due to the surface asperities measured from the nominal level and is a randomly varying quantity of zero mean and is an index parameter determining a definite roughness parameter.

431508.fig.001
Figure 1: A geometry of simplified model for knee joint.
2.1. Basic Equations for Micropolar Fluids (Region-I)

The basic equations for the flow of micropolar fluid in the film region in vectorial form are [19] For the three-dimensional steady motion of an incompressible micropolar fluid under the usual assumption of hydrodynamic lubrication with negligible body forces and body couples, the field equations (2) reduce to the following

2.1.1. Conservation of Mass

Consider

2.1.2. Conservation of Linear Momentum

Consider

2.1.3. Conservation of Angular Momentum

Consider where , , and are velocity components along , , and axes, respectively and are the micropolar velocity components in the and directions, respectively, and is the pressure in the film region.

2.2. Basic Equations for Poroelastic Region (Region-II)

Following Torzilli and Mow [20] and Collins [21] the coupled equations of motion for deformable cartilage matrix and the mobile portion of the fluid contained in it can be written in the following form:

Matrix: Fluid: where and denote the densities of solid matrix and fluid, respectively, is the corresponding displacement vector, is the fluid velocity vector, is the permeability of the cartilage, and denotes the material derivative. Equations (6) and (7) represent the force balance for the linear elastic solid and viscous fluid components, of the cartilage, respectively. In these equations, left hand terms denote the local forces (mass × acceleration) which are counter-balanced by right porous media driving force, respectively.

In fact these two equations may be viewed simply as a generalized form of Darcy’s law for unsteady flow in a deformable porous medium in terms of the relative velocity between the moving cartilage and the fluid contained in its pores.

The classical stress tensor for a continuous homogeneous medium may be expressed for the matrix and fluid, respectively, as where , , and are the elastic parameters of the cartilage. After neglecting the inertia terms, addition of (6) and (7) eliminates the pressure and fluid velocity, and, thereafter, taking the divergence of the results yields the following Laplace equation: where is known as the cartilage dilatation. Following Hori and Mockros [22] we characterize the cartilage dilatation by a sample similar linear equation in terms of corresponding average bulk modulus , in the following form: From (9) and (10) we get The relevant boundary conditions for the velocity field are

3. Solution of the Problem

Solving (4) and (5) for the velocity components , , and and microrotation velocity components and with the respective boundary conditions given in (12) we get where for , By neglecting inertia terms, (7) may be arranged in terms of relative velocity in the form and elimination of through (10) and (18) gives The normal component of the relative fluid velocity at the cartilage surface is

Integrating (3) across the fluid film region and using the boundary conditions for given in (12) and also using the expressions (13) and (16) for and , the modified Reynolds equation is obtained in the form where .

3.1. Stochastic Reynolds Equation

In the context of rough surfaces, there are two types of roughness patterns which are of special interest. The one-dimensional longitudinal structure where the roughness has the form of long narrow ridges and valleys running in the -direction and the one-dimensional transverse structure where roughness striations are running in the -direction in the form of long narrow ridges and valleys. However, the present study is restricted to one-dimensional longitudinal roughness since the one roughness structure can be obtained from others by just rotation of coordinate axes. For the one-dimensional longitudinal roughness pattern, the film thickness assumes the form

Taking the expectation on both sides of (21) and simplifying them, the stochastic modified Reynolds type equation is obtained in the form where denotes the expectancy operator defined by is the probability density function of the stochastic film thickness variable . According to the Sayles et al. [14], the cartilage surfaces are rough and roughness height distribution is Gaussian in nature. Therefore, polynomial form which approximates the Gaussian distribution is chosen in the present study. Such a probability density function is where and is the standard deviation.

The relevant boundary conditions for and are(i)for the fluid film region (ii)for the poroelastic region (iii)at the interface where is the cartilage layer thickness.

The solution of Laplace equation (11) with boundary conditions (28) and (29) is where and the constants are the Fourier coefficients to be determined.

Using the mean pressure continuity condition (30) in (31), we get On substituting (31) and (33) into (23) and using the orthogonality of eigen functions and , the Fourier coefficients are obtained in the form where .

The load-carrying capacity of the squeeze film is obtained by integrating the averaged pressure field over the surface of the top plate

The nondimensional mean instantaneous load-carrying capacity of the squeeze film is given by where

The film thickness at any time can be obtained by solving for the load as a function of time: where The nondimensional squeeze film time is obtained as As (i.e., ) and (36) and (40) reduce to the corresponding solid case studied by Sinha and Singh [23], the results of Wu [24] can be recovered in the limiting case of .

4. Results and Discussion

A simplified mathematical model has been developed for understanding combined effects of surface roughness, poroelasticity, and micropolar fluid on lubrication aspects of synovial joints. The governing equations along with the appropriate constitutive relationships and boundary conditions have been formulated for modeling the roughness structure of cartilage with viscous fluid in the lubricant region in synovial joint lubrication. The load capacity and time height relation are functions of non-dimensional parameters , , . The values of parameters , , are taken from Torzilli, which are associated with healthy human articular cartilage during normal functioning.

4.1. Load Carrying Capacity

The variations of non-dimensional load carrying capacity with aspect ratio are depicted in Figure 2 for different values of the coupling number and two values of permeability parameter . The curves corresponding to represent the Newtonian case. It is observed that the non-dimensional load carrying capacity increases for the increasing value of a coupling number . Figure 3 shows the variations of with for different values of with two values of permeability parameter . It is observed that increases for increasing values of further; it is also observed that the maximum load carrying capacity is attained for the square plates ().

431508.fig.002
Figure 2: Variation of non-dimensional load with for different values of with ,  , .
431508.fig.003
Figure 3: Variation of non-dimensional load with for different values of with ,  , .

The effect of elastic parameter on variations of with is shown in Figure 4 for two values of permeability parameter . It is observed that increases with and decreases for increasing values of .

431508.fig.004
Figure 4: Variation of non-dimensional load with for different values of with ,  , .

The effect of roughness parameter on the variations of with is depicted in Figure 5. for two values of permeability parameter . It is observed that increases for the increasing values of . Further, it is interesting to note that the maximum load carrying capacity is a function of and is obtained for the rectangular plates.

431508.fig.005
Figure 5: Variation of non dimensional load with for different values of with ,  , .
4.2. Squeeze Film Time

The variations of non-dimensional squeeze film time with the aspect ratio for different values of coupling number are depicted in the Figure 6 for two values of permeability parameter . The curves corresponding to represent the Newtonian case. It is observed that increases for increasing value of .

431508.fig.006
Figure 6: Variation of non-dimensional squeeze film time with for different values of with ,  .

Figure 7 shows the variation of squeeze film time with for different values of with two values of permeability parameter . The curves corresponding to represent the Newtonian case. It is observed that increases for the increases in values of . Hence the squeeze film bearings lubricated with micropolar fluid carry larger load for a longer time as compared to the corresponding Newtonian fluids, by which the performance of the bearings is improved.

431508.fig.007
Figure 7: Variation of non-dimensional squeeze film time with for different values of with , , .

The effect of elastic parameter on variations of with is shown in Figure 8 for two values of permeability parameter . It is observed that increases with and decreases for increasing values of . The effect of roughness parameter on the variations of with is depicted in Figure 9. It is observed that increases for the increasing value of .

431508.fig.008
Figure 8: Variation of non-dimensional squeeze film time with for different values of with ,  , .
431508.fig.009
Figure 9: Variation of non dimensional squeeze film time with for different values of with ,  ,  .

5. Conclusion

The effect of surface roughness on the squeeze film characteristics of rough poroelastic rectangular plates is presented. On the basis of Eringen’s micropolar fluid theory and the Christensen stochastic theory for the study of rough surfaces, the modified form of stochastic Reynolds equation is derived for one-dimensional longitudinal roughness pattern. As the micropolar fluid parameter and , the squeeze film characteristics reduce to corresponding Newtonian case and as these characteristics reduce to the smooth case. On the basis of the results presented, the following conclusions are drawn.(1)The effect of micropolar fluid provides an increased load carrying capacity and squeeze film time as compared to the corresponding Newtonian case.(2)The effect of surface roughness on the cartilage surface increases the load carrying capacity and squeeze film time as compared to smooth case.

Hence in a practical situation the required shape of the bearing may be rectangular, in which case a specific choice of , , , and will yield larger load carrying capacity and longer squeeze film time as compared to the corresponding Newtonian fluids, by which the performance of the bearings is improved.

Acknowledgments

The authors are grateful to the referees for their valuable comments on the earlier version of the paper. G. Savitramma also acknowledges the financial support under the Research Fellowship in Science for Meritorious Students (RFSMS) by the University Grants Commission, New Delhi, India.

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