Effect of Surface Roughness on the Squeeze Film Lubrication of Finite Poroelastic Partial Journal Bearings with Couple Stress Fluids: A Special Reference to Hip Joint Lubrication
A simplified mathematical model has been developed for understanding the combined effects of surface roughness and couple stresses on the squeeze film behavior of poroelastic bearings in general and that of hip joints in particular. The cartilage is modeled as biphasic poroelastic matrix and synovial fluid is modeled as couple stress fluid. The modified form of averaged Reynolds equation which incorporates the randomized roughness structure as well as elastic nature of articular cartilage with couple stress fluid as lubricant is derived. For the study of rough surfaces, Christensen's stochastic theory is used to study the effect of two types of one-dimensional random roughness, namely, longitudinal roughness pattern and the transverse roughness pattern. The averaged film pressure distribution equations are solved numerically by using the conjugate gradient method. It is observed that the surface roughness effect is dominant and pattern dependent and the influence of couple stresses is to improve the joint performance. 1. Introduction Synovial joints which are usually globular in appearance are covered with sponge like material called articular cartilage. The joint cavity is filled with sponge like material called synovial fluid. In the recent years considerable attention is paid by researchers to the studies of human locomotion such as knee joints and hip joints. The surfaces of synovial joints have a high degree of geometrical conformity. Their behavior is governed by articular cartilage which is soft glistening tissue with porous properties and synovial fluid which is dialysate of plasma with concentration of hyaluronic molecules, which do not normally pass through cartilage pores. In hydrodynamic lubrication the thickness of the fluid film is much larger than the height of surface roughness asperities. The pressure in the fluid film is generated because of the relative motion of surfaces and wedge action. The squeeze film lubrication phenomenon is observed in several engineering applications such as gears, bearings, machines tools, rolling elements and automotive engines [1], dampers, and human joints [2]. The hip joint is a spherical joint between the femoral head and the acetabulum in the pelvis; it is a diarthrosis or synovial joint, since it is wrapped in a capsule that contains the synovial fluid, a biological lubricant that acts also like a shock absorber [3]. The hip joint can transmit high dynamic loads (7-8 times body weight) and accommodate a wide range of movements. A number of lubrication theories have been proposed in the literature to
References
[1]
N. B. Naduvinamani and S. Santosh, “Micropolar fluid squeeze film lubrication of finite porous journal bearing,” in Proceedings of the 13th Asian Congress of Fluid Mechanics, pp. 970–973, Dhaka, Bangladesh, December 2010.
[2]
J.-R. Lin, W.-H. Liao, and C.-R. Hung, “The effects of couple stresses in the squeeze film characteristics between A cylinder and a plane surface,” Journal of Marine Science and Technology, vol. 12, no. 2, pp. 119–123, 2004.
[3]
M. Nordin and V. Frankel, Basic Biomechanics of the Musculoskeletal System, Lippincott Williams & Wilkins, 3rd edition, 2001.
[4]
N. Tepei, “Lubrication with micropolar fluids and its application to short bearings,” Journal of Tribology, vol. 101, no. 3, pp. 356–364, 1979.
[5]
Y. Tsukamoto, M. Yamamoto, and K. Mabuchi, “Boundary lubricating property of synovial fluid on artificial materials and lubrication of artifical joints,” Journal of the Japanese Orthopaedic Association, vol. 57, no. 1, pp. 91–99, 1983.
[6]
G. A. Ateshian and C. T. Hung, “The natural synovial joint: properties of cartilage,” Proceedings of the Institution of Mechanical Engineers J: Journal of Engineering Tribology, vol. 220, no. 8, pp. 657–670, 2006.
[7]
V. C. Mow and H. Rik, Basic Orthopaedic Biomechanics and Mechano-BiologyEdition, Lippincott Williams & Wilkins, Philadelphia, Pa, USA, 3rd edition, 2005.
[8]
N. M. Bujurke, S. G. Bhavi, and N. B. Naduvinamani, “The effect of couple stresses in squeeze film poro-elastic bearings with special reference to synovial joints,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 7, no. 4, pp. 231–243, 1990.
[9]
V. K. Stokes, “Couple stresses in fluids,” Physics of Fluids, vol. 9, no. 9, pp. 1709–1715, 1966.
[10]
E. Walicki and A. Walicka, “Inertia and couple-stress effects on squeeze-film characteristics with reference to biological bearings,” TriboTest, vol. 8, no. 3, pp. 195–203, 2001.
[11]
N. M. Bujurke and R. B. Kudenatti, “An analysis of rough poroelastic bearings with reference to lubrication mechanism of synovial joints,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 309–320, 2006.
[12]
M. Nabhani, M. El Khilfi, and B. Bou-Said, “Non-Newtonian couplestresses poroelastic squeeze film,” Tribology International, vol. 64, pp. 116–127, 2013.
[13]
A. E. Yousif and A. A. Al-allaq, “The hydrodynamic squeeze film lubrication of the ankle joint,” International Journal of Mechanical Engineering and Applications, vol. 1, no. 2, pp. 34–42, 2013.
[14]
G. R. Higginson and R. Norman, “The lubrication of porous elastic solids with reference to the functioning of human joints,” Journal of Mechanical Engineering Science, vol. 16, no. 4, pp. 250–257, 1974.
[15]
M. Hlavá？ek, “Lubrication of the human ankle joint in walking with the synovial fluid filtrated by the cartilage with the surface zone worn out: steady pure sliding motion,” Journal of Biomechanics, vol. 32, no. 10, pp. 1059–1069, 1999.
[16]
M. Hlavá？ek, “The influence of the acetabular labrum seal, intact articular superficial zone and synovial fluid thixotropy on squeeze-film lubrication of a spherical synovial joint,” Journal of Biomechanics, vol. 35, no. 10, pp. 1325–1335, 2002.
[17]
Z. M. Jin, D. Dowson, and J. Fisher, “Effect of porosity of articular cartilage on the lubrication of a normal human hip joint,” Proceedings of the Institution of Mechanical Engineers H: Journal of Engineering in Medicine, vol. 206, no. 3, pp. 117–124, 1992.
[18]
P. N. Tandon, J. K. Misra, R. S. Gupta, and S. Satyanand, “Role of ultrafiltration of synovial fluid in lubrication of human joints,” International Journal of Mechanical Sciences, vol. 27, no. 1-2, pp. 29–37, 1985.
[19]
R. S. Sayles, T. R. Thomas, and J. Anderson, “Measurement of the surface microgeometry of articular cartilage,” Journal of Biomechanics, vol. 12, no. 4, pp. 257–267, 1979.
[20]
H. Christensen, “Stochastic models of hydrodynamic lubrication of rough surfaces,” Proceedings of the Institution of Mechanical Engineers, vol. 184, no. 55, pp. 1013–1026, 1970.
[21]
A. Walicka, “Inertia effects in porous squeeze film biobearing with rough surfaces lubricated by a power-law fluid,” Special Topics & Reviews in Porous Media-An International Journal, vol. 3, no. 3, pp. 247–256, 2012.
[22]
N. M. Bujurke, R. B. Kudenatti, and V. B. Awati, “Effect of surface roughness on squeeze film poroelastic bearings with special reference to synovial joints,” Mathematical Biosciences, vol. 209, no. 1, pp. 76–89, 2007.
[23]
J. Prakash and K. Tiwari, “Effect of surface roughness on the squeeze film between rotating porous annular discs with arbitrary porous wall thickness,” International Journal of Mechanical Sciences, vol. 27, no. 3, pp. 135–144, 1985.
[24]
N. B. Naduvinamani and G. K. Savitramma, “Micropolar fluid squeeze film lubrication between rough anisotropic poroelastic rectangular plates: special reference to synovial joint lubrication, Tribology,” Materials, Surfaces and Interfaces, vol. 6, no. 4, pp. 174–181, 2012.
[25]
P. A. Torzilli and V. C. Mow, “On the fundamental fluid transport mechanisms through normal and pathological articular cartilage during function. II. The analysis, solution and conclusions,” Journal of Biomechanics, vol. 9, no. 9, pp. 587–606, 1976.
[26]
V. C. Mow and W. M. Lai, “Recent development in synovial joint biomechanics,” SIAM Review, vol. 22, no. 3, pp. 275–317, 1980.
[27]
R. Y. Hori and L. F. Mockros, “Indentation tests of human articular cartilage,” Journal of Biomechanics, vol. 9, no. 4, pp. 259–268, 1976.
