Rayleigh step bearings are capable of carrying the highest load carrying capacity as compared to many other slider bearings. Due to this, the step bearings found extensive applications in industry to improve the performance of automotive machines. The first study on step bearing was conceived in 1918 by Lord Rayleigh

The use of different fluids with additives of high molecular weight polymers to improve the viscosity index of lubricants is considered. Mouda et.al

Bujurke et.al.

Uma Srinivasan

A double-layered porous facing would be useful as it would not only increase the load capacity of the bearing because of reduced oil seepage into its wall but would also help to bring oil between the surfaces, thereby improving the performance of the bearing when it is not completely saturated with oil. Hence in this paper an attempt has been made to analyze the double-layered porous step-slider bearing lubricated with couple stress fluid which has not been studied so far. Results are compared with that of single-layered porous step-slider bearing analyzed by Naduvinamani and Siddangouda^{ 11}. The numerical results are presented in the graphical form. The presented results show that the introduction of the double-layer porous facing increases the load carrying capacity and frictional force however decreases the co-efficient of friction which are the desired attributes of efficient lubrication system. This investigation bridges the gap of study between the single layered porous Rayleigh step bearing and double layered bearings.

h film thickness

h_{1} thickness of inlet film

h_{2} thickness of outlet film

k porous matrix permeability

L bearing length (=L_{1}+L_{2})

L_{1}_{, }L_{2} Bearing lengths in the entry and exit regions respectively

l couple stress parameter =

p pressure in film region

P_{1}, P_{2} Non-dimensional film pressure in the entry and exit region respectively.

u^{*} , v^{*}^{ }modified Darcy velocity in the x and y direction.

w load

k_{1}, k_{2} permeabilities in the layer-1 and porous layer-2 respectively

The governing equations for the Stokes couple stress fluid are given by

As shown in

Bearing length (L) is equal to the sum of the lengths of entry and exit regions (=L_{1} +L_{2})_{. }The velocity boundary conditions are

1. at

2. at

Assuming that v^{*}is the Darcy’s velocity component carried along y axis in the porous region. Couple stress fluid flows within the porous region according to Darcy's modified law

where k_{1}, k_{2} are the permeability parameters of porous layer-1 and layer-2 respectively

Due to continuity of fluid flow in the porous regions, the pressure ^{ 18 }^{ }

The related pressure boundary conditions are

Integrating Eq. (8a) with respect to y over the wall thickness

Integrating Eq. (8b) with respect to y over the wall thicknessuse of Eq. (13) gives

The wall thicknesses

Equation (3) indicates that the pressure p in the film region is independent of y. Solving Eqn.(2) with relevant boundary conditions for u in equations (4a), (4b) and (5a), (5b), the fluid velocity in the film region is obtained in the form

where

Using the expression for u given in Eq. (17) into the continuity equation (1) and integrating over the film thickness and using boundary conditions (4a), (4b) and (5a), (5b), we get the modified Reynolds equation

where

Assuming that the double layered porous thickness to be very small, the Morgan-Cameron approximation gives

Put Equation (19) in Eq.(18). Then the modified Reynolds –type equation is acquired in the form of

Introducing the following non- dimensional quantities

Eq.(20) takes the form given below

where

The fluid film pressure boundary conditions are

Here _{ }is the non–dimensional pressure at the step. Integrate Eq. (22) twice with respect

From Equations (25) and (26), we get

Now the pressure for the entry region (0

For the exit region (

Using Eqs.(28) and (29) we get the non-dimensional load carrying capacity.

The frictional force f per unit width on the bearing surface y = 0 is defined by

Put Eq. (18) into Eq.(33) and substituting it in Eq.(32) gives dimensionless frictional force

where,

The Coefficient of friction is computed as follows

Variations of

Variations of

_{1}= 0.3, β_{2}= 0.6, KR=0.5, δ_{1}=200, δ_{2}=200. It is observed that, non-dimensional load carrying capacity increases for increasing the value of couple stress parameter

The variation of non-dimensional frictional force

The key parameter to assess the performance of slider bearings is the coefficient of friction. The variation of the coefficient of friction

The double-layered porous Rayleigh step-slider bearings lubricated with couplestress fluid is analyzed on the basis of Stokes couplestress fluid theory. Following conclusions are drawn on the basis of the numerical results presented in the above section:

1. The enhanced load carrying capacity of the double-layered porous Rayleigh step –slider bearing is observed as compared to that of single-layered porous bearings.

2. Even though the non-dimensional frictional force increases for the double layered porous Rayleigh step slider bearings, the coefficient of friction decreases.

3. The adverse effect of reduced load carrying capacity of the single layered porous Rayleigh step-slider bearing can be well compensated by the presence of double-layered porous facing with appropriable permeabilities.