Non-Newtonian couple-stress squeeze film behaviour between oscillating anisotropic porous circular discs with sealed boundary

. The thrust of this paper is to investigate theoretically the non-Newtonian couple stress squeeze ﬁ lm behaviour between oscillating circular discs based on V. K. Stokes micro-continuum theory. The lubricant squeezed out between parallel porous and rigid facings is supposed to be a concentrated suspension which consists of small particles dispersed in a Newtonian base ﬂ uid (solvent). The effective viscosity of the suspension is determined by using the Krieger-Dougherty viscosity model for a given volume fraction of particles in the base ﬂ uid. For low frequency and amplitude of sinusoidal squeezing where cavitation as well as turbulence are unlikely, the governing equations including the modi ﬁ ed Reynolds equation coupled with the modi ﬁ ed Darcy ’ s equation are derived and solved numerically using the ﬁ nite difference method and a sub-relaxed iterative procedure. The slip velocity at the porous- ﬂ uid interface is directly evaluated by means of the modi ﬁ ed Darcy ’ s law considering laminar and isothermal squeezing ﬂ ow. For a given volume fraction, the couple stress effects on the squeeze ﬁ lm characteristics are analyzed through the dimensionless couple stress parameter ~ ℓ considering sealed and unsealed boundary of the porous disc. The obtained relevant results reveal that the use of couple stress suspending ﬂ uids as lubricants and the effect of sealing the boundary of the porous matrix improves substantially the squeeze ﬁ lm behaviour by increasing the squeeze ﬁ lm force. On the other hand, side leakage ﬂ ow calculated in the sealed case remains constant in comparison to that of open end (unsealed) porous disc for all values couple


Introduction
The squeeze film lubrication mechanism is ubiquitous in many engineering applications such as gears, bearings in rotating and reciprocating machinery, and so on. In such cases, a positive and a negative load can act by turns, and positive and negative squeeze occurs alternatively.
The squeeze effect in lubrication occurs when positive pressure arises in a fluid film between two mutually approaching surfaces. This phenomenon is also called the positive squeeze effect and the lubricating film is called the squeeze film. However, when negative pressure arises in the fluid film between two mutually receding surfaces, this phenomenon is then called negative squeeze. In this case, fluid is sucked into the gap between the two surfaces contiguous to the film.
In his famous paper on lubrication, Osborne Reynolds [1] stated that the squeeze effect is an important mechanism, together with the wedge and stretching effects, for the self-generation of pressure in a lubricating film, especially when the contribution due to the wedge effect is small compared to the squeeze effect. This is due from the fact that the viscous fluid film cannot be squeezed out instantaneously, so it resists to extrusion which results in build-up of pressure.
It should be emphasized that there is a huge number of investigations into porous and non-porous squeeze films. For instance, Jurczak [12] proposed bio-bearings models for the squeeze film with one porous layer to study the inertia effect on the pressure field through the use of the Morgan-Cameron [13] approximation for coupling the Navier-Stokes and Poisson equations. He concluded that the pressure generated in the couple stress fluid is higher than that of the Newtonian fluid. He also found that the pressure decreases when increasing the permeability.
Using the Barus law, Lin et al. [14] analysed the combined effects of viscosity-pressure variations and non-Newtonian couple stresses on the wide parallel rectangular plate squeeze film characteristics. The semi-analytical solution, obtained by means of the small perturbation method, was compared by the authors with that derived in the iso-viscous Newtonian case. They found that the piezo-viscous and couple stresses effects can improve the squeeze film performance characteristics because of the greater carrying capacity obtained for a lower film thickness.
Nabhani et al. [15] presented a new model for combining the effects of couple stresses and porous matrix deformations considering fluid inertia. The authors used the Darcy-Brinkman-Forchheimer generalized model for describing the fluid flow in the porous disc and the simplified thin elastic liner approach to determine the elastic deformation at the porous-fluid interface. They found that the couple stresses and the poro-elasticity of the disc increased the friction coefficient and decreased the load capacity.
Ponnuswamy and Govindaraj [16] analysed squeeze film performances for parallel porous annular disc, with a sinusoidal motion of the upper disc. They derived an analytical expression for pressure by using the Fourier-Bessel series. The obtained results showed that the load capacity increased with increasing the couple stress or the sinusoidal movement amplitude.
Available technical literature reveals that most of the researches done basically consider porous circular or annular discs with unsealed boundary. The Morgan-Cameron approach [13] which ignores any slip effect at the porous-fluid interface has been often adopted to take into account the porosity effects.
Moreover, the slip phenomenon at the porous-fluid interface has been incorporated by means of some more complex models such as Beavers et al. [17] and Brinkman [18]. An elaborate discussion on the choice of slip parameters and the validity of the different slip-flow boundary conditions is given in [19].
The aim of this paper is to investigate more particularly the non-Newtonian couple-stress squeeze film behaviour between two oscillating circular discs with either sealed or unsealed boundary with a view to application for polluted mechanisms. The present investigation is based on the V. K. Stokes micro-continuum theory which accounts for couple-stresses due to the presence of small particles in the suspending fluid.
The upper disc is impermeable and mobile, while the lower one is assumed to be a stationary anisotropic porous medium. The effective viscosity of the suspension is determined by using the Krieger-Dougherty viscosity model for a given volume fraction of particles in the base fluid.
It should be noted that under dynamic loading conditions (i.e., for intense sinusoidal motion), cavitation may occur as the upper and lower surfaces move away from each other during part of the harmonic squeeze motion (i.e., for negative squeeze phase) executed by the upper disc. As the surfaces contiguous to the film separate, this creates sub-cavity pressures within the film gap which are less than the absolute atmospheric pressure. If the sub-cavity pressures fall below a threshold cavitation pressure, the fluid film will break, i.e., cavitate. The cavitation may be gaseous, vaporous, or some combination thereof.
Considering low amplitude and frequency sinusoidal squeezing where cavitation as well as turbulence is unlikely, the governing equations including the modified Reynolds equation coupled with the modified Darcy's equation are derived for a non-Newtonian suspending couple stress fluid model with variable viscosity considering slip effects at the porous-fluid interface. The slip velocity is directly computed by means of the modified Darcy's law for low squeeze Reynolds numbers, i.e. for laminar squeezing flows. The squeeze film performance characteristics are then obtained from the numerical solution of the two governing equations using the finite difference method and a sub-relaxed iterative procedure. For a given volume fraction, the couple stress effects on the squeeze film characteristics will be analyzed for a range of the dimensionless couple stress parameter ℓ considering sealed and unsealed boundary of the porous disc.

Theoretical formulation
The study of lubrication of such mechanism is schematically shown in Figure 1 by two horizontal discs, separated by a polluted film. The effect of squeezing between two discs is analyzed when one rigid disc which approaches with uniform velocity to the other disc has a porous facing, without neglecting the sliding motion on them. The lubricant fluid in the film region is supposed to be a couple stress fluid [20,21]. When the body forces, and the body moments are absent, the momentum equation and the continuity equation for inertialess incompressible couple stress fluid take the forms [22,23]: whereṼ ¼ uẽ r þ vẽ u þ wẽ z is the flow velocity vectorp, the pressure, m s is the effective viscosity of suspension fluid, and h is the material constant responsible for the couple stress property. The dimensional equation of the new material constant is that of momentum, i.e. h = MLT À1 . Some experiments to determine this material constant were given in reference [21].
To calculate the effective viscosity, Krieger-Dougherty [24] in 1959 derived an equation for the dynamic viscosity covering the full range of volume fraction in the base fluid (solvent): where m bf is the viscosity of base fluid, f is the volume fraction of particle in base fluid, f m is the maximum particle packing fraction, which varies from 0.495 to 0.54 under quiescent conditions, and is approximately 0.605 at high shear rates [25,26]and h ½ is the intrinsic viscosity, whose typical value for mono disperse suspensions of hard spheres is 2.5 [27].
Typical variations of the relative viscosity m s m bf with the volume fraction f are presented in Figure 2 using the Krieger-Dougherty relationship (3) and the well-known classical Einstein model [28].
Using the classical assumption of thin film lubrication [29,30] and when the problem is considered axisymmetric (i.e., ∂ ∂u ¼ 0 equations (1) and (2) in cylindrical coordinates (r, u, and z) reduce to where u, w are the components of the flow velocity vector in the radial and axial directions, respectively. The velocity components in the porous matrix are given for an isothermal laminar flow by the modified Darcy's law, which accounts for the couple stress effects [31]: where u* and w* are the radial and axial components of the flow velocity in the porous layer, respectively, p* is the pressure field in the porous layer, Þ are the permeability parameters of the anisotropic porous matrix in the r and z directions, respectively. b r and b z [32,33] appearing in the relationships of a r and a z which represent the microstructure size to the pore size ratios are defined by the following relationships : where h m s has a dimension of the length squared, and k r and k z are the permeability coefficients in the r and z directions, respectively. The pressure field in the porous layer must satisfy the following modified Laplace equation written for a couple stress fluid: If we assume that the film thickness as a function of time, t, in the case of positive and negative squeeze changes in a sinusoidal manner, in this case, the film thickness h (t) and the squeeze velocity W sq ¼ _ h t ð Þ will be given by: where h 0 is an initial film thickness, h a is the amplitude, and f is the sinusoidal motion frequency. Typical variations of the dimensionless film thicknessh and the dimensionless squeeze velocityW sq with dimensionless timet ¼ tf obtained from equations (11) for h a ¼ 0:2 are shown in Figure 3.
The relevant boundary conditions for the velocity components are: -At the fluid-porous interface z = 0 The relevant boundary conditions for the pressure are: where H * is the thickness of the porous layer. The solution of equation (4) subjected to the boundary conditions (12) is: is the couple stress parameter, U s is the slip velocity at the fluid-porous layer interface (z = 0) given for a laminar squeezing flow by: This is a direct incorporation of the slip effect at the porous-fluid interface.
The fluid velocity w in the z-direction can be obtained numerically using the finite difference method from the derivation of the continuity equation (6) with respect to the z Àvariable, i. e.
Integrating the continuity equation (6) through the fluid film thickness and using the modified Darcy's law (14) and the boundary conditions (12) for the axial fluid velocity w we get the modified Reynolds' equation: where Substituting equation (8) into equation (17), a final form of the modified Reynolds equation can then be derived: Recall that U s ¼ Àa r

Numerical formulation
The modified Reynolds equation (19) and the modified Laplace equation (10) are coupled at the porous interface by the continuity of pressure and velocities, and solved numerically by the finite difference method (Fig. 4).
The finite difference equations resulting from the discretization process of the two governing equations are given below -For the modified Reynolds' equation where: -For the modified Darcy's equation where: The obtained system of linear algebraic equations for both governing differential equations is solved numerically by means of the iterative Gauss-Seidel method with overrelaxation factor v GS whose value ranges from 1 to 2 (i.e., 1 v GS < 2) [34]. Figure 5 illustrates the general procedure for the iterative process used at each time step. To obtain a converged solution for the system of coupled partial differential equations an iteration scheme is required. For the first iteration, the modified Reynolds equation (20) is solved for p old i without the influence of the porous matrix. The obtained fluid-film pressure field can then be used as a boundary condition at the fluid-porous interface to solve the modified Darcy's equation (22) for p Ã i;j in the porous medium. The calculated pressure p Ã i;j serves to compute the radial and axial pressure gradients ∂p Ã ∂r j i;1 and ∂p Ã ∂z j i;1 and therefore velocity components u Ã i;1 and w Ã i;1 at the interface. It is now possible to solve a second time the modified Reynolds equation (20) for p new i taking into account the terms related to porosity appearing on the RHS of the fluidfilm equation. With the resulting pressure the process of calculation returns to solving the modified Darcy's equation (22). This procedure is repeated until a converged solution has been found which results in obtaining two successive fluid pressure fields very close.
To avoid numerical oscillations it is important to limit the influence per iteration of the porous matrix into the pressure solution of the Reynolds equation (20). To this end an under-relaxation parameter is introduced, which has to be chosen such that convergence is guaranteed, but the calculation time still remains acceptable.
Once the convergence of process is obtained we can calculate the hydrodynamic performance characteristics such as the lift load, the drag force, the leakage flow, and the power loss.

Hydrodynamic performance characteristics
The instantaneous leakage flow of lubricant is calculated by the following relation: zÞdz Substituting the expression for u from equation (14) into the leakage flow equation (24) and then integrating over the film thickness yields: By integrating the film pressure over the fluid layer, we express the instantaneous load carrying capacity so-called lift force W of the squeeze film as follows: for couple stress fluid, the parietal shear stress t rz given by: By using the expression of u and boundary conditions, we find: Thus, the friction force F f can be calculated by integrating the shear stress : The total power loss can be calculated by: where F 1 and F 2 are the dissipation functions expressed by: After integration with respect to z, we get the final form of total power loss: The obtained results are displayed and compared in Figure 6. The calculations are performed using the data reported in Table 1 for a r = 0. Good agreement is observed between numerical and analytical results and we may conclude that the proposed method of solution used in the present analysis is validated for a small porous layer thickness.

Parametric study
The numerical model developed is used to investigate the effects of couple stresses for given values of volume fraction on hydrodynamic performance characteristics over one cycle of oscillating squeeze motion of sealed or unsealed porous circular discs. The couple stress effects are controlled by the dimensionless parameterl varying from 0. (Newtonian case) to 0.4. The data used in this investigation are given in Table 1. Figure 7 shows the axisymmetric positive squeeze film pressure p as a function of non-dimensional radial coordinater at dimensionless timet ¼ 0:3 for various values of the couple stress parameterl. It is observed that the effect of couple stresses increases the squeeze film pressure as compared to the Newtonian casel ¼ 0 À Á and this increase is more pronounced for larger values ofl. This effect has also been confirmed by the research authors V. Ponnuswamy and S. Govindaraj [16].
In addition, for a same value of the couple stress parameter the pressure calculated for sealed configuration is greater than that obtained for unsealed case especially for higher values ofl. Indeed, the highest rate of increase of central-pressure is about 3% in Newtonian case and 10% in non-Newtonian case whenl ¼ 0:4. Furthermore, the distributions of pressure within the fluid film and porous disc for sealed and unsealed configurations obtained at t ¼ 0:3 andl ¼ 0:2 have the same patterns except in the region near of the porous disc outer edge as clearly illustrated in Figure 8.
The influence of the couple stress parameterl in the axisymmetric radial fluid velocity profile u at the edge (r = R) ort ¼ 0:3 is depicted in Figure 9. The increase of the couple stress parameterl results in the reduction of the radial velocity in the fluid layer compared to the Newtonian case. This reduction is due to the non-Newtonian couple stress behaviour which provides more resistance to fluid flow. The obtained results are qualitatively in good accordance with those obtained by Nabhani et al. [15] who used a CFD approach, and by Nabhani et al. [35] who used Darcy-Brinkman equations for describing the fluid flow within the porous layer. It is also observed that there are no significant differences of couple stress parameter effect on the radial velocity profile for the sealed configuration. Figure 10 represents axisymmetric velocity fields and streamlines att ¼ 0:3 and forl ¼ 0:2 obtained in the fluid film and porous disc. We can see that the fluid is accelerated in the low pressure zone (Fig. 8) and the velocity profiles in the film gap have a parabolic shape whereas the axisymmetric velocity profiles are nearly uniform across the porous disc for both sealed and unsealed cases. In the same figure, we can find the same flow patterns (streamlines) in the porous disc for both sealed and unsealed configurations except in the outlet region.
The effects of the couple stress parameterl in the shear stress distributions across the film thickness, t rz , are shown in Figure 11 att ¼ 0:3 and for r = R. It is observed that the greater the couple stress parameterl the greater is the shear stress which results in an increased resistance to the flow. As expected, the shear stresses calculated for both Newtonian and non-Newtonian cases are maximum at the walls and vanish at mid film gap. On the other hand, the parietal shear stresses for sealed configuration are greater than those obtained for the unsealed one. Therefore, the sealed case leads to the greater drag force. Figure 12 presents the axisymmetric distribution of the squeeze film pressure p during one cycle of oscillating squeeze motion obtained forl ¼ 0:2 (couple stress fluid case) and different values of volume fraction f. The positive and negative pressures are obtained during positive and negative squeeze motions, respectively. It can be seen that the pressure p generated in the fluid layer is parabolic and is symmetric about the horizontal axis because the fluid inertia forces are ignored as stated previously. Similar trends have been found experimentally by Y. Hori [11] (Chap. 7) for the Newtonian case as well as by J. R. Lin [36] for oscillating circular squeeze films considering non-Newtonian couple stresses through a theoretical investigation.
The pressure vanishes att ¼ 0:5 corresponding to t = 0.5 s for which the squeeze velocity becomes zero as depicted in Figure 3. Furthermore, the squeeze film pressure att ¼ 0:3 i.e., during the positive squeezing motion, is greater than the squeeze film pressure predicted att ¼ 0:2 since the squeeze velocity W sq is higher and the film thickness is lower. Furthermore, it is found that the squeeze film pressure increases significantly when increasing the volume fraction of particles in the base fluid. The rate of increase att ¼ 0:3 is about 5% or f = 0 and 9% or f = 0.3. Accordingly, the suspending fluid gives an enhancement of the carrying load than the non-suspending fluid, i.e. without particles. On the other hand, the squeeze film pressure calculated for the sealed case is much larger than that obtained in the unsealed case for both positive and negative squeeze. The concentration effect on the axisymmetric radial velocity profile at the edge (r = R) during one cycle of oscillating squeeze motion calculated forl ¼ 0:2 are displayed in Figure 13. It seems that there is no significant influence of the concentration on the velocity distributions for both sealed and unsealed cases. For all values of the concentration, we obviously observe that the radial velocities calculated in the sealed case are more important than those obtained in the unsealed case for both positive and negative squeeze. Figure 14 displays the axisymmetric shear stress profile t rz during one cycle of oscillating squeeze motion computed forl ¼ 0:2 at the outlet of contact r = R for different values of volume fraction f. It is observed that the shear stress increases significantly with increasing of the volume fraction. As depicted in the figure, the maximum shear stress occurs at the walls. This increase can be explained by the viscosity increase of the suspension. As noted earlier for a fixed time (Fig. 11) the shear stress t rz calculated for sealed configuration is greater than that obtained for the unsealed configuration and this for both positive and negative squeezing motions. The time variations of the central-pressure p max , the parietal shear stress t rz at r = R, the friction force F f and the load carrying capacity W for various values of the couple stress parameterl are presented in Figures 15-18, respectively. It is shown that the all aforementioned hydrodynamic characteristics increase significantly with increasing of the couple stress parameterl over the entire cycle. This result agrees with that of J. R. Lin [36]. Furthermore, the graphical results are always antisymmetric with respect tot ¼ 0:5 which follow the same trend as the squeeze velocity (Fig. 3). It should be noted that the peak values of these parameters are more important for the sealed case. Figure 19 shows the influence of the couple stress parameterl, the leakage flow Q calculated for sealed and unsealed cases during one cycle of oscillating squeeze motion.
For the sealed case, the leakage flow is not affected by the couple stress parameter since the flow that enters the porous matrix returns to the fluid layer regardless ofl. In contrast, the leakage flow in the film gap decreases when increasing the couple stress parameter for the unsealed case. In this situation, the leak occurs both at the edge of the fluid layer and the porous matrix.

Conclusion
This research was basically concerned with the theoretical and numerical analysis of lubrication by squeezing film effect in polluted mechanism by modelling the lubricant as a couple stress suspended fluid.
The proposed model concerns practically all systems where the squeeze film effect is the only mechanism for pressure generation, especially when a sufficiently large wedge effect is not expected [11,[36][37][38][39][40], for example: the case of the internal combustion engine connectingrod small end bearing, the case of human hip and knee joints (for certain sporting gestures), the case of the squeeze film damper bearings (SFDB) used in certain rotating machinery. On the basis of V. K. Stokes micro-continuum theory, the governing equations including the modified Reynolds' equation and the modified Darcy's equation were derived for laminar squeezing flow and solved numerically by the finite difference method. The nonlinear fluid-solid interaction problem for both sealed and unsealed configurations was solved numerically using a sub-relaxed iterative procedure for given values of the couple stress parameter, the permeability parameters, and the volume fraction of particles. The numerical results were graphically presented for low amplitude and frequency sinusoidal motion of the upper rigid disc taking into account directly the slip phenomenon at the porous-fluid interface. Enhancements in the squeeze film pressure and load carrying capacity (lift force) were observed for larger values of the couple stress parameter. Further for a given value of the couple stress parameter, enhancements in squeeze film pressure and load carrying capacity were obtained by increasing reasonably the volume fraction of particles in base fluid especially for the sealed porous disc configuration.
It was also found that for a given couple stress parameter the parietal shear stresses increased with increase in volume fraction of particles suspended in the base lubricant and this increase was more pronounced for the porous disc with sealed boundary. On the other hand, the side leakage flow obtained for the unsealed case decreased over the entire cycle of oscillation, i.e. fort varying from 0. to 1, when increasing the couple stress parameter. However, the side leakage flow calculated in the sealed case remains unchanged for the different values of couple stress parameter.
Finally, it was shown that the power losses for both sealed and unsealed cases remain weak for all the values of couple stress parameter considered.