World Tribology Congress 2013 Torino, Italy, September 8 – 13, 2013
Couple stress and poro-elasticity effects on squeeze film behaviour Mohamed NABHANI1)*, Mohamed EL KHLIFI1) and Benyebka BOU-SAÏD2) 1)
University Hassan II Mohammedia – Casablanca, Faculty of Sciences and Techniques, LMCM P.O. Box 146, 20650 Mohammedia, Morocco 2) University of Lyon, CNRS INSA-Lyon, LaMCoS, UMR CNRS 5259, 18-20 Avenue Albert Einstein 69621 Villeurbanne, France * Corresponding author:
[email protected]
1. Introduction The investigation of squeeze-film characteristics between porous surfaces draws attention of many researchers because of a wide use in industry and biomechanics. Typical studies are the squeeze film behaviour of Newtonian fluid between parallel circular discs by Lin 1, Megat et al.2 and Nabhani et al.3,4. In practice, polymer chains are added to modern lubricants to improve their properties under various operating conditions. This resulting non-Newtonian behaviour significantly influences the fluid characteristics and thus must be taken into account. Based on the Reynolds equation and the Darcy-Brinkman model, Nabhani et al.5 investigated the porous squeeze film of a couple stress lubricant between a rigid fixed porous disc and another mobile one. However, considering the deformation of the porous disc is necessary for predicting more correctly the porous squeeze film performances. Considering couple stress and elastic deformation effects, a numerical study of squeeze-film behaviour between porous elastic discs is presented in the present work. The Reduced Navier-Stokes (RNSP) equations for couple stress fluids are derived in the thin film and the Darcy-Brinkman-Forchheimer equations are used for the flow model in the porous disc. This porous disc, considered homogeneous and isotropic, is saturated with a Newtonian fluid with a viscosity equals to that in the fluid film. A simple approach based on the elastic thin layer model (Wrinkler model)6 to investigate the elastic deformation effect of the porous disc is proposed. The governing equations are discretized using an implicit finite difference scheme. The flow in the fluid film is coupled to that in the porous disc using a sequential iterative algorithm. The generalized Dracy-Brinkman-Forchheimer equations are solved with the over-relaxation Gauss-Seidel method. The RNSP equations are solved using an inverse method. The present numerical predictions show that the combined couple stress and elastic deformation have significant effects on the porous squeeze film characteristics.
lubricant. The upper rigid disc approaches the lower porous one with a normal velocity dg/dt, where g is the upper disc position and t is the time. Using the assumptions of thin fluid films7 and in the absence of body forces and body couples, the momentum and continuity equations of laminar axisymmetric flow of an incompressible non-Newtonian couple stress lubricant in the film region are given in cylindrical coordinates by: u u p 2u 4u u u w 2 4 r z r z z t p 0 z
1 (ru) w 0 r r z
(1) (2) (3)
where is the fluid density, p is the pressure, is the fluid dynamic viscosity, is a material constant responsible for couple stress effect, and u and w are the fluid film radial and axial velocities. We refer to this set of equations as the Reduced Navier-Stokes equations of Prandtl (RNSP). The porous disc, considered homogeneous and isotropic, is saturated with a Newtonian fluid having the same viscosity as the fluid film. The generalized momentum equations of the fluid in the porous disc are given in the case of an axisymmetric flow8 by:
u * u * u * w* u * p * * u t r z r k 1 ru * 2u * C f * * u u r r r z 2 k
w* u * w* w* w* p * * w t r z z k
(4)
(5)
2. Governing equations and numerical solution
1 w* 2 w* C f * * r u w r r r z 2 k
The geometry used consists of two parallel circular discs of the same radius R, one of which has a poroelastic face of thickness H, separated by a fluid film
Applying the divergence operator to equations (4) and (5) in r and z directions and considering the fluid incompressibility, one obtains:
C f * 1 p * 2 p * r u .grad u * r r r z 2 k 2 2 2 u * u * w* u * w* 2 2 z r r r z
p p* 0 at r R
(6)
where p * is the pressure, u * is the velocity vector,
u * and w* are the radial and axial velocities in the porous disc, k is the permeability and is the porosity. The axial component of the motion equation for the upper disc of mass m is written as: m
d 2g t W t F0 dt 2
(7)
R
where W t 2 prdr is the applied load on the upper
(13)
At the interface z g h between the fluid and the porous medium, the continuity equations of velocity, tangential stress, normal stress and pressure are applied: (14) u u * and w w*
u 3u u* 3 z z z
(15)
w* 0 z
(16)
p p*
(17)
At the lower wall, non-permeability condition is: p * 0 at z 0 z
(18)
0
disc by the fluid film forces and F0 is the upper disc weight. The squeeze velocity dg/dt and the position g of the upper disc are therefore deduced using an Euler explicit scheme of acceleration in equation 7. The film thickness, considering elastic deformation of the porous interface under pressure effect, is given by: h(r, t) = g(t) - H + (r, t)
(8)
The fluid flow is assumed to be continually smooth through the porous disc edge
u * w* 0 at r R r r
(19)
No local fluid particle rotation at the interface between the fluid and solid boundaries leads to:
The deformation of the porous interface is given in the case of the elastic thin layer model by the expression 6:
2u z 2
(r , t )
H 1 1 2 p(r , t ) (9) 1 E where E is the Young’s modulus and is the Poisson ratio.
These governing equations and boundary conditions are transformed into dimensionless form using cylindrical coordinates and discretized using an implicit finite difference scheme.
The following initial and boundary conditions are used. In the fluid film at t=0: u, w and p are given by the analytical solution of the Reynolds equation for the non porous case. In the porous region: u * and w* are taken to be zero, but p * is identical to the pressure p for each radial section. The beginning of the squeezing occurs at zero acceleration and the porous matrix is assumed to be initially undeformed.
Velocity and pressure conditions are initially specified. At each time step, the squeezing velocity and position of the upper disc are calculated from its equation of motion written in the previous time step by an explicit time scheme. The flow in the fluid film is coupled to that in the porous medium using a sequential iterative algorithm. This iterative process begins by fixing a pressure field within the fluid film. A good estimate is to consider that of the previous time step. This pressure field permits the calculation of the fluid film thickness. The porous disc geometry is then known, which therefore enables solving the generalized Dracy-Brinkman-Forchheimer and Poisson equations by Gauss-Seidel method. A new pressure field within the fluid film is then calculated by solving the RNSP equations by an inverse method9.
The no slip and the non-permeability conditions on the lower wall z 0 and on the upper disc z = g(t) are: dg at z g (t ) (10) u 0 and w dt (11) u* w* 0 at z 0 On the symmetry axis, we have w w (12) 0 at r 0 r r At the edge of the fluid film and the porous disc, a zero ambient pressure is assumed:
u u* 0 and
*
z g h
2u z 2
0
(20)
zg
The convergence of the global iterative process is monitored by the fluid film pressure. The convergence is considered reached when the root mean square change of the difference between the obtained pressures for two successive iterations is less than 10-3. Five to 2
seven iterations were sufficient to obtain the solution convergence. This calculation process is repeated for each iteration squeeze time step until achieving a possible minimum fluid film thickness.
coefficient of the lubricant at the fluid film – porous disc interface. Friction increases rapidly at the initial stage of the squeezing, and then begins to decrease very slowly.
A grid experiment was performed before computation. It was ensured that the results were grid independent. A 41×21 grid configuration for both the fluid film and the porous medium was found to be adequate.
A Newtonian lubricant has a lower friction coefficient when compared to a corresponding couple stress fluid (see Figure 3). The more the couple stress parameter is high, the more the friction coefficient increases; this could limit the contact life. By contrast, the porous disc deformation decreases the friction coefficient. Compared to the impermeable case, the presence of the deformable porous disc reduces the friction coefficient (see Figure 4), since the porosity reduces the viscous resistance to the radial flow of the fluid at the porous interface. This desirable result is favorable to enhance the mechanism time life.
3. Results and conclusion The non-Newtonian couple stress and elastic deformation effects on porous squeeze film between two circular discs are here investigated. The calculation was performed using the data as follows: =900 kg/m3, 0.1 Pa.s, and10-9Pa.s.m2, H = 1.6 mm, R = 40 mm, ho = 1.2 mm, k = 1.0 10-15 m2, = 0.4, E = 1.5 MPa, 0.4, m = 30 kg and F0 = 300 N. Figure 1 shows the evolution of the upper disc position with respect to the squeeze time. At the initial stage of squeezing, the decrease in the position of the upper disc is very rapid, but then becomes less significant. At a given time step, the disc reaches a lower position and therefore travels a longer distance in the case of Newtonian lubricant. The greater the couple stress parameter is, the slower the descent of the disc becomes. This shows the high resistance of the non-Newtonian lubricant to the motion of the disc. It should also be noted that the porous disc deformation induces a lower position compared to the case of a rigid disc. The disc experiences less resistance to its movement when the porous disc is deformable. At a given time step, the fluid film has a smaller thickness when the porous disc is deformable, due to the lower position reached by the upper disc. The couple stresses effect on the film thickness and comparison with the impermeable case are shown in Figure 2. At a given squeezing time, the Newtonian lubricant exhibits the smallest film thickness. The increase in the couple stress parameter significantly increases the non-Newtonian lubricant film thickness at every squeezing time step. The last result is very favorable for the contact life. Finally, it must be noted that the presence of the porous disc reduces the fluid film thickness regardless of the lubricant used. Note that the deformation is maximum on the symmetry axis of the disc, and it decreases to zero at the end of the contact. This is a direct consequence of pressure distribution in the fluid film and therefore the application of thin elastic model.
Acknowledgements The authors would like to thank the French-Moroccan Mixed Inter-University Committee for supporting this work under Grant number SPI 06/12. 4. References [1]
[2]
[3]
[4]
[5]
[6] [7] [8]
[9]
Lin, JR. “Viscous shear effects on the squeeze film behaviour in porous circular discs”, Int. J. Mech. Sc. 38, 4, 1996, 373-384. Megat Ahmed, M.M.H., Gethin, D.T., Claypole, T.C., Roylance, B.J., “Numerical and experimental investigation into porous squeeze films”, Tribology International, 31, 4, 1998, 189-189. Nabhani, M., El Khlifi, M., Bou-SaÏd, B., “A general model for porous medium flow in squeezing film situations”, Lubrication Sciences, 22, 2, 2010, 37-52. Nabhani, M., El Khlifi, M., Bou-SaÏd, B., “Numerical investigation of porous squeeze film using the Darcy-Brinkman-Forchheimer model”, 10th Congress of Mechanics, April 19-22, 2011, Oujda, Morocco. Nabhani, M., El Khlifi, M., Bou-SaÏd, B. “Combined non-Newtonian and viscous shear effects on a porous squeeze film”, Tribology Transactions, 55, 4, 2012, 491-502. Winkler, E., “Die lehre von der elasticitaed und festigkeit”, Prag: Dominicus, 1867. Cameron, A., “Basic Lubrication Theory”, Wiley Eastern Ltd, 1987. Hsu, C.T., Chang P., “Thermal dispersion in porous media”, International Journal of Heat and Mass Transfer, 33, 8, 1990, 1587-1597. Peyret, R. And Taylor, T.D., “Computational methods fluid flow”, Spinger-Verlag, 1983.
Figures 3 and 4 show the variation of the friction
3
2.8
0.6 Rigid Deformable
Porous Non porous
2.7
0.55 2.6 -9 2 = 5.76 10 Pa.s.m Fluid film thickness (mm)
Upper disc position (mm)
2.5
2.4
2.3
2.2
2.1
= 5.76 Pa.s.m
2
= 2.3 Pa.s.m2
1.9
Newtonian
0.5
0.45
-9 2 = 2.3 10 Pa.s.m
0.4
2
0.35
1.8
0
0.005
0.01
0.015 Time (s)
0.02
0.025
Figure 1. Upper disc position for different couple stress parameter values in the porous case.
0.3
0.03
Newtonian
0
5
10
15 20 25 Radial coordinate (mm)
30
35
40
Figure 2. Fluid film thickness for different couple stress parameter values in the deformable case.
-3
13
x 10
Porous Non porous 12
11
Friction coefficient
10
9
8
7
6
5
Figure 3. Friction coefficient for different couple stress parameter values in the porous case.
0
0.005
0.01
0.015 Time (s)
0.02
0.025
0.03
Figure 4. Friction coefficient at Pa.s.m2 in the deformable case.
4
