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

e effects of surface roughness and poroelasticity on the micropolar squeeze �lm behavior between rectangular plates in general and that of synovial joints in particular are presented in this paper. e modi�ed Reynolds equation, which incorporates the randomized surface roughness structure as well as elastic nature of articular cartilage with micropolar �uid as lubricant, is derived. e load-carrying capacity and time of approach as functions of �lm 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 �lm thickness. It is observed that the effect of surface roughness has considerable effects on lubricationmechanism of synovial joints.


Introduction
e study of mechanism of synovial joints has recently become an active area of scienti�c research.e human joint is a dynamically loaded bearing which employs articular cartilage as the bearing and synovial �uid as the lubricant.�nce a �uid �lm is generated, squeeze �lm action is capable of providing considerable protection to the cartilage surface.e loaded bearing synovial joints of the human body are the shoulder, hip, knee, and ankle joints; such joints have a lower friction coe�cient and negligible wear.Synovial �uid is a clear viscous �uid, a dialysate of plasma containing mucopolysaccharides. Synovial �uid usually exhibits a non-Newtonian shear thinning behavior.However, under high shear rates, the viscosity of synovial �uid approaches a constant value not much higher than that of water [1].erefore a Newtonian lubricant model has oen been used for synovial �uid in lubrication modeling [2].In this study the synovial �uid is modeled as non-Newtonian micropolar �uid.
Articular cartilage is poroelastic or biphasic consisting of both �uid and solid phases.e importance of the unique biphasic load-carrying characteristics of articular cartilage and �uid �ow 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 �uid pressurization was comprehensively studied [4][5][6][7].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].e transient friction behavior observed was a direct result of the interstitial �uid pressurization and �uid load support, directly measured experimentally [6,8].However, for a similar con�guration but under cyclic loading, friction was found to be similar or even at a higher level [9].e importance of the biphasic lubrication has also been studied by enzymatic treatment of articular cartilage altering the biphasic properties and �uid pressurization such as chondroitinase [10][11][12][13].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 signi�cant reduction.
�ll these studies were con�ned 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.is has motivated us to investigate the in�uence of roughness of cartilage surfaces in lubrication aspects of synovial joint.Christensen [15] developed the stochastic theory to understand the in�uence 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 �lm lubrication between rough rectangular plates with couple stress �uid as lubricant.ese investigations have not incorporated the poroelasticity of the bearing surface.
e squeeze �lm lubrication characteristics of micropolar �uid have been extensively studied in the literature.�grawal et al. [17] studied the squeeze �lm and externally pressurized bearings lubricated with micropolar �uids and found that the time of approach is more for the micropolar �uids as compared to the corresponding Newtonian �uids.e analytical solution of the problem of squeeze �lm lubrication of micropolar �uid between two parallel plates (onedimensional) has been given by Bujurke et al. [18].
In the present paper, a theoretical study of combined effects of surface roughness and micropolar �uid in squeeze �lm 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.

Mathematical Formulation of the Problem
e geometry and coordinates of the problem are as shown in Figure 1.e squeezing �ow of micropolar �uid between two rectangular surfaces is considered.e upper rough articular surface is approaching the lower smooth poroelastic matrix normally with a constant velocity  .e lubricant in the joint cavity is taken to be Eringen's [19] micropolar �uid.�s the load-bearing area of the synovial knee joint is small, the two surfaces may be considered to be parallel under high loading conditions.e moving boundary is characterized by where  represents the nominal smooth part of the �lm 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 de�nite roughness parameter.
where , , and  are velocity components along , , and  axes, respectively  1 and  2 are the micropolar velocity components in the  and  directions, respectively, and  is the pressure in the �lm region.

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 �uid contained in it can be written in the following form: Matrix: Fluid: where   and   denote the densities of solid matrix and �uid, respectively,   is the corresponding displacement vector,   is the �uid 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 �uid components, of the cartilage, respectively.In these equations, le hand terms denote the local forces (mass × acceleration) which are counterbalanced by right porous media driving force, respectively.
In fact these two equations may be viewed simply as a generalized form of �arcy�s law for unsteady �ow in a deformable porous medium in terms of the relative velocity ((   −   between the moving cartilage and the �uid contained in its pores.
e classical stress tensor  for a continuous homogeneous medium may be expressed for the matrix and �uid, respectively, as where  1 , , and  are the elastic parameters of the cartilage.Aer neglecting the inertia terms, addition of ( 6) and (7) eliminates the pressure and �uid velocity, and, thereaer, taking the divergence of the results yields the following Laplace equation: where (= div   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 e relevant boundary conditions for the velocity �eld are

Solution of the Problem
Solving ( 4) and ( 5 By neglecting inertia terms, (7) may be arranged in terms of relative velocity in the form and elimination of  through (10) and (18) gives e normal component of the relative �uid velocity at the cartilage surface is Integrating (3) across the �uid �lm region and using the boundary conditions for  given in (12) and also using the expressions ( 13) and ( 16) for  and , the modi�ed Reynolds equation is obtained in the form where (  ) =  3 + 12 2   6 2 cot(2).

Stochastic Reynolds Equation.
In the context of rough surfaces, there are two types of roughness patterns which are of special interest.e one-dimensional longitudinal structure where the roughness has the form of long narrow ridges and valleys running in the -direction and the onedimensional 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 �lm thickness assumes the form  =  () +   ( ) .
Taking the expectation on both sides of (21) and simplifying them, the stochastic modi�ed Reynolds type equation is obtained in the form  2 where where parameter .It is observed that  increases with log 10  and decreases for increasing values of .e effect of roughness parameter  on the variations of  with log 10  is depicted in Figure 5. for two values of permeability .It is observed that  increases for the increasing values of .Further, it is interesting to note that the maximum load carrying capacity  max is a function of  and is obtained for the rectangular plates.

Squeeze Film Time. e variations of non-dimensional
squeeze �lm time  with the aspect ratio  for different values of coupling number  are depicted in the Figure 6 for two values of permeability parameter .e curves corresponding to   0 represent the Newtonian case.It is observed that  increases for increasing value of .
Figure 7 shows the variation of squeeze �lm time  with log 10  for different values of  with two values of permeability parameter .e curves corresponding to   0 represent the Newtonian case.It is observed that  increases for the increases in values of .�ence the squeeze �lm bearings lubricated with micropolar �uid carry larger load for a longer time as compared to the corresponding Newtonian �uids, by which the performance of the bearings is improved.
e effect of elastic parameter  on variations of  with log 10  is shown in Figure 8 for two values of permeability parameter .It is observed that  increases with log 10  and decreases for increasing values of .e effect of roughness parameter  on the variations of  with log 10  is depicted in Figure 9.It is observed that  increases for the increasing value of .

Conclusion
e effect of surface roughness on the squeeze �lm characteristics of rough poroelastic rectangular plates is presented.On the basis of Eringen�s micropolar �uid theory and the Christensen stochastic theory for the study of rough surfaces, the modi�ed form of stochastic Reynolds equation is derived for one-dimensional longitudinal roughness pattern.As the micropolar �uid parameter    and   , the squeeze �lm 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) e effect of micropolar �uid provides an increased load carrying capacity and squeeze �lm time as compared to the corresponding Newtonian case.
(2) e effect of surface roughness on the cartilage surface increases the load carrying capacity and squeeze �lm time as compared to smooth case.
Hence in a practical situation the required shape of the bearing may be rectangular, in which case a speci�c choice of , , , and  will yield larger load carrying capacity and longer squeeze �lm time as compared to the corresponding Newtonian �uids, by which the performance of the bearings is improved.

F 2 :F 3 :F 4 :F 5 :
Variation of non-dimensional load  with log 0 () for different values of  with  = 0.2,  = 0.,  = 0.2.Variation of non-dimensional load  with log 0 () for different values of  with  = 0.2,  = 0.,  = 0.2.e �lm thic�ness at any time  can be obtained by solving for the load as a function of time:  =   0  ( ())  = −768 2  2  √  Variation of non-dimensional load  with log 10 ( for different values of  with   0,   0,   0.Variation of non dimensional load  with log 10 ( for different values of  with   0,   0,   0. (12)r the velocity components , , and  and microrotation velocity components  1 and  2 with the respective boundary conditions given in(12)we get 12.
F 6: Variation of non-dimensional s�uee�e �lm time  with log 10 ( for different values of  with   0   0,   0.Variation of non-dimensional s�uee�e �lm time  with log 10 ( for different values of  with   0,   0,   0.F 8: �ariation of non-dimensional squeeze �lm time  with log 10 ( for different values of  with   0,   0,   0.Load Carrying Capacity.evariations of nondimensional load carrying capacity  with aspect ratio  are depicted in Figure2for different values of the coupling number  and two values of permeability parameter .ecurvescorresponding to   0 represent the Newtonian case.It is observed that the non-dimensional load carrying capacity increases for the increasing value of a coupling number .Figure3shows the variations of  with log 10  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 (  1).e effect of elastic parameter  on variations of  with log 10  is shown in Figure4for two values of permeability F 9: �ariation of non dimensional squeeze �lm time  with log 10 ( for different values of  with   0,   0,   03.