The most prominent theory of lubrication regarding human joints was regulated by hydrodynamic. Well-built investigations were carried out to find the similarities between physical wedges and articulating surfaces in human joints shown by Reynolds which is necessary for hydrodynamic or fluid film lubrication [10]. The intraarticular cartilage (synovial fluid in knee joint) behaves as a slight inclination of the coupled surfaces to cater wedge shaped lubricant films, which is suggested by MacConail [10] [11]. After his research, there was conclusion about joints to view as hydrodynamically lubricated systems by performing the first detailed experiments on friction in naturally acting joints, observing a horse stifle mounted as the fulcrum in a pendulum machine. A very low coefficient of friction ( f = 0.02 ) in case of sliding was observed when the joint was lubricated by synovial fluid whereas an unlubricated joint revealed creaking noise after 8000 cycles when applied a constant load of 445 N [12] [13]. After 4 hours, observations were the joint heating, debris being thrown, bone granting, stream rising, etc. [12] [13]. After such experiment he highlighted the essential role of lubrication in natural joints. So hydrodynamic theory at that time played a fundamental role in the study of natural human joint lubrication [12] [13] [14]. Even if this topic remains under investigation, to summarize and highlight the scientific progress coupled with mathematical equations made through the years in the understanding and modelling of lubrication mechanisms in natural human synovial joints, covering more popular and relevant theories from earlier research in this field up to the latest ones, can be viewed by introducing briefly some milestones from the 1930’s to today [12] [13] [14].

The conclusion was drown which was based on several pendulum experiments (in which a linear decay in amplitude was observed) that hydrodynamic action could not exist due to low sliding velocities under the heavy loads acting in the human joints. An alternative to MacConail theory, a boundary lubrication model was proposed by Charnley in 1959 by his experiment which recorded friction coefficient values [13] [14]. Such lubrication exists between unlubricated sliding surfaces and fluid-film lubrication. It is also characterized as condition in which the friction between surfaces is determined by features of the surfaces and properties of the lubricant other than viscosity [13] [14]. Barnett and Cobbold made critical observations on Charnley’s theory, in which linear decay in amplitude was attributable to the dissection of the joint [15]. Such decay showed an essential linear relationship with time when replaced the joint in the pendulum fulcrum with a hydrostatic bearing. In 1968, Linn and Radin conducted experiments on animal joints at a constant load attracted a lot of interests in which extraneous dynamic force component give rise to friction force [16]. The most important conclusion was the animal joints operate within mixed film region of lubrication.

McCutchen considered the porosity and elasticity of articular cartilage of joint lubrication problem for the first time in 1966 [17]. According to his views, the pressured synovial fluid travels through the porous cartilage which acts as a sponge like material [17]. Weeping lubrication is a form of fluid lubrication in which the load bearing surfaces are held apart by a film of lubricant that is maintained under pressure [16] [17] [18]. Weeping implied that the lubricant film was sweated into the high-pressure zone between opposing cartilages, while the boundary lubrication effect between the contact surfaces remained [16] [17] [18]. McCutchen came up with the term ‘‘weeping lubrication” because of bearing materials that accomplish it weep fluids when pressurized. An external pump is frequently used to supply pressure in engineering aspects while the pump action in the human body is provided by muscle contractions around the joint or by weight bearing compression of articular cartilage, which causes the cartilage to distort and “weep” fluid, forming a fluid film across the articular surfaces [16] [17] [18]. Because the impervious layer of calcified cartilage prevents the fluid from being driven into the subchondral bone, it can only move into the joint. When the stress is removed, osmotic pressure causes the fluid to flow back into the articular cartilage [16] [17] [18]. When cartilage weeps under force, the weight is absorbed by the hydrostatic pressure of the fluid, even though the liquid content of the porous cartilage would be evacuated and the friction coefficient would grow if the squeeze is sustained indefinitely. It also appears that wringing water out of the porous matrix takes longer and is more difficult, whereas resoaking is simple [16] [17] [18]. This sort of lubrication is most effective when the load is high, but it can be used in a variety of situations.

The protective fluid film is maintained at an adequate thickness in the elastohydrodynamic lubrication model by the elastic deformation of the articular surfaces [19]. Elastohydrodynamic lubrication is a hydrodynamic process in which the fluid film pressure produces elastic deformation of the bearing surfaces, altering the pressure in the film region. To put it another way, the elastic cartilage deforms slightly to keep a layer of fluid between the opposing joint surfaces. Under strong loading, the elastohydrodynamic motion can keep a fluid film intact [19]. The fluid film thickness varies between 10⁻⁵ and 10⁻⁴ cm [19]. In general, elastic distortion offers greater geometrical conformity in the contact zone than is ordinarily present, resulting in a substantially thicker lubricating film for a given load [19]. Furthermore, the high pressure may produce a significant rise in the viscosity of the lubricant. The thickness of the lubricating film is further increased by this viscosity effect [19]. With the addition of the effective elastic modulus, the minimum film thickness is a function of the same factors as in hydrodynamic lubrication [19]. In 1963, Dintenfass firstly, showed failure of the hydrodynamic lubrication mode in which he completely neglected deformations while he took into account the deformability of articular cartilage and led to Elastrohydrodynamic lubricayion theory (EHD) [20]. The main finding was that the operating film thickness in highly loaded lubricated contacts could be up to 100 times greater than predicted by conventional lubrication theory, because the human joint requires the correct operating conditions for the cartilage to be separated by a synovial film and thus boundary lubrication does not exist [20]. In 1966, Tanner theoretical values suggested the possibility of lubricating film in the human joints when surfaces are subject to relative motion [21]. Later on, in 1986 confirmed EHD lubrication as prominent mode of lubrication of soft bearing or highly deformable systems, such as synovial joints [21].

When the bearing surfaces are moving perpendicularly towards one other, squeeze-film lubrication occurs. The movement of articular surfaces that are perpendicular to one another creates pressure in the fluid film. The fluid film is squeezed out of the area of approaching contact as the opposing surfaces draw closer together. The viscosity of the fluid in the gap between the surfaces creates pressure, which causes the lubricant to be forced out. The surface is kept separated by the pressure caused by the fluid’s viscosity [22]. This sort of lubrication is best for heavy loads that are only used for a short time. This device has the ability to carry heavy loads for brief periods of time [22]. The layer of fluid lubricant thins when the fluid is driven out, and the joint surfaces come into contact. Under particular conditions, the film’s height may be measured, and the time it takes for the film to shrink from a given height to zero can be calculated [22]. A squeeze film mechanism of lubrication was suggested by Dowson and Popov in 1966 which based on assumption that the analysis concerning a loaded rigid cylinder approaching a rigid plate [23]. A rigid plane opposes a rigid bone covered by a porous, elastic layer of articular cartilage lubricated by synovial fluid with relative velocities are both sliding and squeezing [23]. Dowson used the primary results of hydrodynamic and EHD theories in a simple model for human joints for the first time in tribology history [22] [23]. Theoretical and experimental values of film thickness were found to be in excellent agreement. Higginson and Norman (1974) and Higginson (1977) looked at a pair of compliant surfaces where the lubricant was squeezed out. However, the non-Newtonian behaviour of the synovial fluid was not taken into account in the computations [24]. An asymptotic analysis of a lubrication problem for a model of articular cartilage and synovial fluid under squeeze-film conditions, was proposed an original analytical approximate model for the synovial pressure field determination in the ankle joint in a pure squeeze motion, accounting for the non-Newtonian behaviour of synovial fluid and cartilage porosity [23] [24].

● Steady, two dimensional and incompressible.

● e I is x component of the seepage velocity in fluid film region.

● e ¯ I is average velocity.

● S M x and S M y are source terms include contribution due to body forces only and ℙ is pressure. We consider S M x = S M y = 0 and linear deformation rate.

● Neglect the variation of pressure normal to very thin film of lubrication.

● n is power law index with n = 1 .

Above assumptions led to the following expression,

0 = − ∂ ℙ ∂ x + μ ∂ ∂ y [ | ∂ e I ∂ y | n − 1 ∂ e I ∂ y ] . (6)

0 = − ∂ ℙ ∂ y . (7)

Let us consider a special slip boundary conditions at lower and upper permeable surfaces.

e I = − K α ∂ e I ∂ y ,     at     y = h , (8)

where K and α are permeability of porous medium and slip parameter respectively [26] [27]. The boundary of porous medium with a free viscous fluid is the continuity of pressure and the normal mass flux which give rise to wipe up the tangential velocity of the synovial fluid [28] [29]. Such boundary condition was proposed by Beavers and Joseph (BJ) in 1967 as a empirical formula. The analysis showed that BJ condition is macroscopic law which describes a phenomena at a macroscopic scale whereas characteristic length is much larger than the pore characteristic size [28] [29]. Similarity transformation is useful to scale out the problem.

e I = 0 ,     at     y = 0. (9)

Convection diffusion phenomena has great potential for applications in mathematical biology as well as pattern formation. The hyaluronic acid (HA) molecules in synovial fluid linked by convection diffusion phenomena is simple mass balance between changes in concentration ℂ of solutes [28] [29].

∂ ℂ ∂ t + ( e I − e ¯ I ) ∂ ℂ ∂ x + ( e J − e ¯ J ) ∂ ℂ ∂ y = D D i f f ( ∂ 2 ℂ ∂ x 2 + ∂ 2 ℂ ∂ y 2 ) − ℝ 1 ℂ , (10)

where ℂ , ℝ 1 and D D i f f are concentration of solutes, chemical reaction with first order rate (such reaction occurs in fully developed flow after diffusion) and diffusion constant respectively.

● Transverse diffusion is very much higher than longitudinal diffusion.

⇒ ∂ 2 ℂ ∂ x 2 ≪ ∂ 2 ℂ ∂ y 2 ,

● Steady convection diffusion phenomena.

We apply the following boundary conditions on Equation (10),

ℂ = c 0 ,     at     y = h , (11)

ℂ = c 1 ,     at     y = − h . (12)

Computational methods like finite difference, finite volume, finite element or their variants, that are most frequently used in computational fluid dynamics. The present work emphasises mainly the finite volume method for computational solution to aforementioned problems (13) & (14). Implementation steps are as follows:

● Divide the domain into the finite sized subdomains (finite control volumes) and each subdomain is represented by a finite number of grid points (like Nodes) [33],

● integrate the governing differential Equation (GDE) over each subdomain [33],

● consider a profile assumption for the dependent variable (like, interpolation function) to evaluate the above integral which express the result as an algebraic quantity at the grid points [33].

∫ Δ V     u η η d V + ∫ Δ V     S o u r c e η d V = 0, (24)

above Equation (24) can be written as,

∫ Δ V     d η ( u η ) d V + ∫ Δ V     S o u r c e η d V = 0. (25)

which implies,

[ n u η ] n − [ A s u η ] s + S o u r c e η A δ η = 0, (26)

profile assumptions for u η at control volume faces n and s are as follows,

S o u t h   F a c e   F l u x η = A s ( u P − u S ) δ η S P , (27)

N o r t h   F a c e   F l u x η = A n ( u N − u P ) δ η P N , (28)

Combining Equations (26) → (28) to get the following system,

( A s δ η S P + A n δ η P N − S P ) u P = A s δ η S P u S + A n δ η P N u N + S u (29)