Lubrication of Isotropic Permeable Porous Inclined Slider Bearing with Slip Velocity and Squeeze Velocity

This paper theoretically discusses about the inclined slider bearing with porous layer ( plates or mat rix ) attached to slider as well as stator including effects of slip velocity, and squeeze velocity. General Reynolds’s equation which is useful for calculat ing load capacity is obtained. The load capacity is calculated for isotropic case for various porosities of the above and below porous matrix. Various sizes of the porous matrix at both the ends are also discussed for the possible optimization of bearing performance. From the results we conclude that better load capacity is obtained when the thicknesses of both the porous plates are small and porosity is less. It can also be concluded from results that for smaller thickness of the porous layer and lesser porosity the load capacity increased 1.5 %.


Introduction
In an innovative analysis Wu [1], dealt with the case of squeeze film behavior for porous annular disks in which he showed that owing to the fact that fluid can flow through the porous material as well as through the space between the bounding surfaces, the performance of a porous walled squeeze film can differ substantially fro m that of a solid walled squeeze film. The above analysis of Wu [1] extended by Sparrow et. al. [2] by introducing the effect of velocity slip to porous walled squeeze film with porous matrix appeared in the above plate. They found that the load capacity decreases due to the effect of porosity and slip. Prakash and Vij [3] investigated a porous inclined slider bearing and found that porosity caused decrease in the load capacity and friction, while it increased the coefficient of frict ion. Recently, Shah et. al. [4] studied mathematical modelling of slider bearing of various shapes with comb ined effects of porosity at both the ends, anisotropic permeab ility, slip velocity, and squeeze velocity. Many other authors have also worked in this direction, for example Kulkarn i et. al. [5], Patel et. al. [6], Gupta et. al. [7], Naduvinaman i et. al. [8], Guo et. al. [9] , H. Urreta et. al. [10].
In all above investigations, none of the authors in their study considered effects of isotropic permeab ility with the different porosities in both porous plates bearing. The porous layer in the bearing is considered because of its advantageous property of self lubrication. W ith this motivation the aim of the present work is to study the behavior of a inclined slider bearing with porous slider as well as stator including slip velocity, and squeeze velocity with effects of isotropic permeability with different porosities. The squeeze velocity is because when the upper plate approaches to lower one. A lubrication Reynolds's equation is derived for the above system and the load capacity is calculated for isotropic permeab ility and different porosities at both the ends.

Derivation of the Mathematical Model
Porous matrix of thickness l 2 and l 1 metres have attached with the slider and stator respectively. Both the porous matrix are backed by a solid wall. The slider moves with a uniform velocity U in the x-d irect ion. Also, stator moves normally towards the slider with a uniform velocity , where t is time in seconds. The basic flow equations of the above phenomenon are as follows: (1) The one dimensional flo w equation governing the lubricant flo w in the film region for the above phenomenon follows form Navier-Stokes's equation under the usual assumption of lubrication, neglecting inertia terms and that the derivatives of velocities across the film p redominate, yields where u is the film flu id velocity in the x-d irect ion, p is film pressure there and η is fluid viscosity.
(2)The integral form of continuity equation in the film region is given by where w is the axial co mponent of the fluid velocity in the film.
(3) Using Darcy's law, the velocity co mponents of the flu id in the porous matrix are g iven as follo w: For upper porous region: For lower porous region: where k is fluid permeab ility in both the porous region in x and z directions respectively, and P is the fluid pressure in the porous region.
(4) The continuity equation in the porous region are given as follows: For upper porous region: For lower porous region: Solving equation (2) under the slip boundary conditions given by Sparrow et.al. [2] and modified by Shah et.al. [11] with the addition of slider velocity U to [2] Substituting equations (4) and (5) in the continuity equation for upper porous region (8) and then integrating the result with respect to z across the upper porous matrix ( h, h+l 1 ), we obtain using Morgan-Cameron approximat ion [8] and the fact that the surface 1 z h l = + is non-porous. Again, substituting equations (6) and (7) in the continuity equation for lower porous region (9) and then integrating the result with respect to z across the lower porous matrix ( − l 2 , 0), we obtain using Morgan-Cameron approximat ion [8] and the fact that the surface z = − l 2 is non-porous. Considering the continuity of the normal co mponent of flu id velocity across the film porous interface, so that 1 , Since the pressure is neglig ible on the boundaries of the slider bearing co mpared to inside pressure, solving equation The load carrying capacity W in dimensionless forms as

Results and Discussion
Both porous plates inclined slider bearing with slip velocity and squeeze velocity lubricated by conventional flu id are examined to explo re the possible effects on the bearing characteristic like load capacity.
The dimensionless load capacity W for various values of 1 l and 2 l are displayed in the Table 1 Table 1 shows that the dimensionless load capacity increases when porous layer at both the ends are small. Other way we can say that the dimensionless load capacity increases when the porous layer at both the plates decreases.
It can be seen form Tab le 1 that for 0.81  Table 1.   Table 2, it is observed that, the dimensionless load capacity remains same if we interchange the size of the porous layers at both the ends.

Conclusions
Based upon the above formulat ion and results & discussion, the conclusions can be drawn as follows for designing porous inclined slider bearing: In order to have better load capacity it is suggested that (1) The porous layer attached at both the ends should be thin and also the thickness remains same at both the ends.
(2) The values of porosity at both the ends should be small and equal.
(3) It should also be noted that interchange of the size of porous layers at both the ends will not effect on the load capacity.