INTRODUCTION
The performance of hydrodynamic lubricated contacts would be improved by additives
which form a thin porous layer adhered to bearing surfaces due to lubricant
microstructure (Oliver, 1988). Tichy
(1995) developed models applicable for fluid flow through porous medium
considering the effects of lubricant additives. Li (1999)
derived a modified form of Reynolds equation using Brinkmanextended Darcy model
which takes into account the viscous shear and viscous damping effects. Lina
et al. (1996) applied the Brinkman model to predict the load capacity
and friction parameter for flexible long porous journal bearings. Their results
showed that Brinkman model which includes viscous shear effects, predicts an
increase in load capacity and reduction in coefficient of friction. Li
and Chu (2004) and Elsharkawy (2005) utilized porous
media model and the couple stress model to study the effects of lubricant additives
on the performance of hydrodynamic contacts. By modeling the microstructure
of lubricating surfaces as thin porous film press fitted on bearing surfaces,
the theoretical approach on the steady state performance of hydrodynamic contacts
is presented to investigate the effects of lubricant additives.
Naduvinamani (1997) presented a theoretical study
of doublelayered porous Rayleighstep bearings using Darcy’s model and
BeaversJoseph velocity slip at the porous media/fluid film interface. Saha
and Majumdar (2004) investigated steady state and stability characteristics
of hydrostatic twolayered porous oil journal bearings. Twolayered porous bearing
using highly permeable structural support topped by a thin layer with restriction
to fluid flow gives better stability. The characteristics of flow through three
layered porous media are investigated by Allan et al.
(2009). Attia (2007) analyzed steady flow between
two parallel plates in a porous medium. Amiri (2001)
presented the analysis of flow through porous medium using a capillary model.
This study presents onedimensional analysis of twolayered long porous journal
bearing using Newtonian fluid as lubricant. A modified Reynolds equation is
derived using Brinkman model. The nondimensional pressure and shear stress expressions
are obtained. Reynolds boundary conditions are used to solve the pressure distribution.
In this work, the effects of dimensionless permeability parameter and porous
layer thickness on the steady state performance characteristics such as load
capacity and coefficient of friction are analyzed.
MATHEMATICAL FORMULATION
The schematic of a twolayered porous journal bearing is shown in Fig.
1. The porous layer 1 of thickness δ_{1} is adjacent to the
stationery bearing surface while the porous layer 2 of thickness δ_{2}
is adjacent to the porous layer 1. In the present analysis, the variation of
pressure across the porous layer and fluid film is assumed to be negligible.

Fig. 1: 
Geometry of twolayered porous journal bearing 
The fluid motion in the twolayered porous regions (region I: 0≤y≤δ
and region II: δ_{1}≤y≤δ_{1}+δ_{2})
is governed by Brinkman equation considering viscous shear and viscous damping
effects as:
The fluid motion in the film region obeys the conventional governing Newtonian
flow equations:
The boundary conditions are: (1) noslip boundary conditions at the bearing
and journal surface and (2) velocities and viscous shear stresses match at the
porous layer 1/porous layer 2 and porous layer 2/fluid film interface:
Integrating the Eq. 12 using the boundary
conditions in Eq. 36, the dimensionless
velocity distribution is expressed as:
Where:
The equation of continuity across the film is:
Simplifying the equation of continuity across the film, yields the nondimensional
pressure gradient term for a twolayered porous journal bearing as:
Where:
For Δ_{2} = 0, G_{1} and G_{2} in Eq.
1920 reduce to:
Where:
The Reynolds boundary conditions are:
Integrating the Eq. 18 and substituting the first boundary
condition given in Eq. 25, yields the nondimensional pressure
profile as:
Substitution of the Reynolds boundary conditions for nondimensional pressure
at film rupture in Eq. 26 and simplifying results in Q as:
Substituting the pressure gradient boundary condition given in Eq.
23 in the expression for nondimensional pressure gradient in Eq.
18, results in:
The NewtonRaphson iterative procedure is used to solve simultaneously both
θ_{r} and Q using Eq. 27 and 28.
The radial and tangential nondimensional load capacity obtained by integration
of nondimensional pressure along and perpendicular to line of centers are expressed
as:
The nondimensional load capacity is expressed as:
The nondimensional shear stress in the journal bearing at Y = H is obtained
as:
The nondimensional friction force on the journal surface is obtained by integrating
the shear stress along the journal surface as:
The nondimensional friction coefficient is calculated as:
NUMERICAL STUDIES AND DISCUSSION
A twolayered porous journal bearing is considered in the analysis. The influence
of permeability and thickness of porous layers on the nondimensional load capacity
and coefficient of friction are presented.

Fig. 2(ab): 
Nondimensional load capacity, (a) ε = 0.5, Δ_{1}
= 0.2 and (b) ε = 0.5, Δ_{2} = 0.2 
The parameters used in the analysis are: eccentricity ratio (ε) = 0.5;
nondimensional permeability of porous layers 1 and 2, respectively (K_{1},
K_{2}) = 10^{2}, 10^{3} and nondimensional thickness
of porous layers 1 and 2, respectively (Δ_{1}, Δ_{2})=
0.05, 0.1, 0.15, 0.2.
Figure 2a and b show the nondimensional
load capacity (W) of twolayered porous journal bearing with nondimensional
thickness of porous layers 1 and 2, respectively (Δ_{1}, Δ_{2})
for different values of nondimensional permeability of porous layers 1 and 2
(K_{1}, K_{2}). The nondimensional load capacity (W) increases
with (1) decrease in the nondimensional permeability of porous layer and (2)
increase in the nondimensional thickness of porous layer. For all the parameters
of nondimensional thickness of porous layers considered in the study, the higher
nondimensional load capacity (W) is obtained with reduction in the nondimensional
permeability of porous layer 2 (K_{1} = 10^{3}, K_{2}
= 10^{2}10^{4}) compared to reduction in the nondimensional
permeability of porous layer 1 (K_{1} = 10^{3}, K_{2}
= 10^{2}10^{4}).
Figure 3a and b show the coefficient of
friction (C_{f}) of twolayered porous journal bearing as a function
of nondimensional thickness of porous layers 1 and 2, respectively (Δ_{1},
Δ_{2}) for different values of nondimensional permeability of porous
layers 1 and 2 (K_{1}, K_{2}).

Fig. 3(ab): 
Coefficient of friction, (a) ε = 0.5, Δ_{1}
= 0.2 and (b) ε = 0.5, Δ_{2} = 0.2 
The coefficient of friction (C_{f}) decreases with (1) decrease in
the nondimensional permeability of porous layer and (2) increase in the nondimensional
thickness of porous layer. Lower coefficient of friction (C_{f}) is
obtained with reduction in the nondimensional permeability of porous layer 2
(K_{1} = 10^{3}, K_{2} = 10^{2}10^{4})
compared to reduction in the nondimensional permeability of porous layer 1 (K_{1}
= 10^{3}, K_{2} = 10^{2}10^{4}).
CONCLUSION
The present study evaluates the influence of permeability and thickness of
porous layers on improvement in load capacity and reduction in coefficient of
friction for a twolayered long porous journal bearing. A modified Reynolds
equation is derived considering permeability and thickness of twolayered porous
layers using Brinkman model. The conclusions based on the analysis are:
• 
Higher nondimensional load capacity (W) and lower coefficient
of friction (C_{f}) are obtained with (1) decrease in the permeability
and (2) increase in the thickness of porous layer 
• 
For a given thickness of porous layers 1 and 2 (Δ_{1}, Δ_{2}
in the range 0.050.2), a low permeability porous layer 2 with high permeability
porous layer 1 (K_{1} = 10^{3}, K_{2} = 10^{4};
K_{1} = 10^{2}, K_{2} = 10^{3}) would
result in higher nondimensional load capacity (W) and lower coefficient
of friction (C_{f}) compared to a low permeability porous layer
1 with high permeability porous layer 2 (K_{1} = 10^{4},
K_{2} = 10^{3}; K_{1} = 10^{3}, K_{2}
= 10^{2}), respectively 
ACKNOWLEDGMENTS
This research is funded by Exploratory Research Grant Scheme of Ministry of
Higher Education (ERGSMOHE) Malaysia under grant ERGS/1/2011/TK/ UTP/02/40.
The authors greatly appreciate the support provided by Universiti Teknologi
PETRONAS for this research.
NOMENCLATURE
C 
= 
Radial clearance (m) 
f 
= 
Friction force (N) F = fC/μu_{j} RL 
h, H 
= 
Film thickness (m) H = h/C 
ki, i = 1, 2 
= 
Permeability of porous layer 1 and porous layer 2, respectively (m^{2})
K_{i} = k_{i}/C_{2} 
L 
= 
Length of the journal bearing (m) 
p 
= 
Pressure distribution (N m^{2}); P = pC^{2}/μu_{j}R 
q 
= 
Volume flow rate per unit length along film thickness (m^{2} sec^{1})
Q = q/u_{j}C 
R 
= 
Journal radius (m) 
u 
= 
Velocity component along θ direction (m sec^{1}; U = u/u_{j}) 
u_{i}, i = 1, 2, 3 
= 
Local mean velocity components along θ direction in porous layer
1 and 2 and velocity component along θ direction in fluid film, respectively
(m sec^{1}) 
u_{12}, u_{23} 
= 
Velocity component along θ direction at the interface of porous layer
1porous layer 2 and porous layer 2fluid film, respectively (m sec^{1}) 
u_{j} 
= 
Journal velocity along θ direction (m sec^{1}) 
w 
= 
Static load, N (W = wC^{2}/μu_{j} R^{2} L) 
W_{ε}, W_{φ} 
= 
Nondimensional radial and tangential static load for journal bearing 
x 
= 
Coordinate along circumferential (x) direction (m) θ = x/R 
y 
= 
Coordinate along radial (y) direction (m) Y = y/C 
δ_{i}, i = 1, 2 
= 
Thickness of porous layer 1 and porous layer 2, respectively (m) Δ_{i}
= δ_{i}/C 
ε 
= 
Journal bearing eccentricity ratio 
μ 
= 
Fluid viscosity (Nsec m^{2}) 
θ 
= 
Angular coordinate measured from the position of maximum film thickness
in journal bearing 
θ_{r} 
= 
Angular extent of film rupture for journal bearing 
τ 
= 
Shear stress component (N m^{2}; N m^{2}) Π = τC/μu_{j} 
ω 
= 
Angular velocity of journal bearing (rad sec^{1}) 
Subscripts: 
r 
= 
rExtent of outlet film in journal bearing measured 
ε 
= 
long the radial direction 
φ 
= 
Along the tangential direction 