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Feature

On the squeeze film lubrication of long porous

journal bearings with couple stress fluidsN.B. Naduvinamani

Department of Mathematics, Gulbarga University, Gulbarga, India

P.S. Hiremath

Department of Computer Science, Gulbarga University, Gulbarga, India, and

Syeda Tasneem Fathima

Department of Mathematics, Bellary Engineering College, Bellary, India

AbstractPurpose This paper aims to advance the squeeze film characteristics of long partial journal bearings with couple stress fluid studied by Lin to includethe effect of permeability on the squeeze film lubrication of long partial porous journal bearings with couple stress fluids.Design/methodology/approach A semi-analytical and semi-numerical solution for the squeeze film lubrication of long porous partial journalbearings lubricated with couple stress fluid is presented in the paper. The modified Reynolds equation governing the fluid film pressure is derived. Themodified Reynolds equation is solved analytically and closed form expressions for the squeeze film pressure and load carrying capacity are presented.The first-order non-linear equation for the time-height relation is solved numerically with the given initial condition. The effect of couple stresses andpermeability on the squeeze film characteristics are discussed.Findings It is found that the effect of couple stresses is to increase the load carrying capacity and to lengthen the squeeze film time as compared tothe corresponding Newtonian case. The effect of permeability is to reduce the load carrying capacity and to decrease the squeeze film time as comparedto the corresponding solid case.Originality/value In the design of porous partial journal bearings, the reduction in the load carrying capacity and the response time can becompensated by the use of lubricants with proper microstructures by which the bearing life can be increased.

Keywords Lubrication, Fluids

Paper type Research paper

Nomenclature

C radial clearance

e eccentricity

h fluid film thickness, h C2 e cosuh non-dimensional film thickness,

h=C 1 2 1 cosuh0 non-dimensional minimum film thickness,

h0=C 1 2 1H0 thickness of the porous layer

l characteristic material length of the suspended

particles (h/m)1/2

l couple stress parameter, l/C

p pressure in the film region

p pressure in the porous regionp non-dimensional pressure pC2=mR2d1=dtR radius of the journal

t response time taken by journal centre to move from

1 0 to 11u,v fluid velocity components in the x, and y directions,

respectively

u,v fluid velocity components in the x and y directions,respectively, in the porous region.

W load carrying capacity per unit length of the

bearingW non-dimensional load capacity WC2=mR3d1=dt

x,y local Cartesian co-ordinates

b ratio of microstructure size to pore size h=m=kh material constant responsible for couple stress

property

u circumferential co-ordinate1 eccentricity ratio, e/C

m lubricant viscosityt dimensionless response time WC2t=mR3c permeability parameter kH0=C

3

The Emerald Research Register for this journal is available at

www.emeraldinsight.com/researchregister

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0036-8792.htm

Industrial Lubrication and Tribology

57/1 (2005) 1220

q Emerald Group Publishing Limited [ISSN 0036-8792]

[DOI 10.1108/00368790510575941]

The authors acknowledge the financial support under the Special

Assistance Program DRS project by the University Grants Commission,

New Delhi, India.

12
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Introduction

Self-lubricating porous bearings have the advantage of

reducing the need for certain lubricating equipments (oil

pipes, pumps, etc.) as well as reducing other problems related

to lubrication mechanism. One of the advantageous feature of

the porous bearings is that, no external supply of lubricant is

required for running-in. It was Morgan and Cameron (1957)who first gave an analytical survey of study of porous bearings

with the aid of hydrodynamic conditions. There have been

numerous studies of various types of porous bearings, such as

slider bearings (Uma, 1977), journal bearings (Prakash and

Vij, 1974), squeeze films (Wu, 1971) and thrust bearings

(Gupta and Kapur, 1979).

Many of the studies on porous bearings are confined to

Newtonian lubricants. But the use of non-Newtonian fluids as

lubricants is of growing interest in recent times. In particular,

the addition of long chain polymer solutions to the lubricant

enhances the bearing performance. These lubricants are fluids

with microstructures. The failure of the classical continuum

theory in representing the flow behaviour of such fluids

adequately has led to the development of the microcontinuum

theories (Ariman et al., 1973, 1974). One of these theories iscouple stress theory proposed by Stokes (1966), which

account for the polar effects due to the presence of

microstructures in the fluid. The Stokes couple stress

model describes adequately the rheological behaviour of the

lubricants with polymer additives. Many of the investigations

for the study of performance characteristics of various bearing

systems with couple stress fluid are found in the literature.

The studies by Ramaniah (1979), Ramaniah and Sarkar

(1978), Bujurke and Jayaraman (1982), Bujurke and

Naduvinamani (1990), Lin (1997a b) have shown that the

effect of couple stresses is to increase the load-carrying

capacity and the response time of approach in the squeeze

films.

Recently, Naduvinamani et al. (2001) have studied theeffect of couple stresses on the squeeze film lubrication of

short porous journal bearings and confirmed the earlier

findings of increased load-carrying capacity and delayed time

of approach. In the present study, the squeeze film

characteristics of long partial journal bearings with

couplestress fluid studied by Lin (1997a b) has been

advanced to include the effect of permeability on the

squeeze film lubrication of long partial porous journal

bearings with couple stress fluids.

Analysis

A schematic diagram of the problem under study is shown

in Figure 1. The journal of radius R approaches the

porous bearing surface at a circumferential section u withvelocity Vu.

The film thickness h is a function of u and is given by

h C2 e cosu; 1

where C is the radial clearance and e is the eccentricity of the

journal centre. The lubricant in the film region and also in the

porous region is assumed to be a Stokes (1966) couple stress

fluid.

Under the usual assumptions of fluid film lubrication

applicable to thin films (Cameron, 1987), the equation of

motion of an incompressible couple stress fluid within the film

region, when the body forces and body couples are absent, are

given by

p

x m

2u

y22 h

4u

y4; 2

p

y 0; 3

u

x vy

0: 4

The flow of couplestress fluid in a porous matrix is

governed by the modified Darcy law, which account for the

polar effects

~q 2k

m1 2 b7p; 5

where ~q u; v; and b h=m=k:The ratio (h/m)1/2 is of dimensional length and hence

characterizes the chain length of the polymer additives. Hence

the parameter brepresents the ratio of the microstructure size

to the pore size. The p is the pressure in the porous region.Owing to continuity, it satisfies the Laplace equation

2p

x2

2p

y2 0: 6

The relevant boundary conditions for the velocity

components are:

(1) at the boundary surface y h :

u 0; 7a

2u

y2 0; 7b

v 2Vu: 7c

(2) at the porous journal surface y 0 :

u 0; 8a

2u

y2 0; 8b

v 2v: 8c

The solution of equation (2) subject to the boundary

conditions (7(a) and 7(b)) and (8(a) and 8(b)) is

ux;y 1

2m

p

xyy 2 h 2l2 1 2

cosh2y 2 h=2l

coshh=2l

;

9

where l h=m1=2 is the couple stress parameter.Integrating equation (6) with respect to y over the porous

layer thickness H0 and using the boundary conditions of solid

bearing

p

y 0

at y 2H0

we obtain,

p

y

y0

2

Z02H0

2p

x2

dy: 10

On the squeeze film lubrication of long porous journal bearings

N.B. Naduvinamani, P.S. Hiremath and Syeda Tasneem Fathima


Volume 57 Number 1 2005 1220

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Assuming that the porous layer thickness H0 is very small and

using the pressure continuity condition p p at theinterface y 0 of porous matrix and fluid film, equation (10)reduces to

p

yy0

2H0

2p

x2 : 11

Then, the vertical component of the modified Darcy velocity,

v at the interface y 0 is given by

vjy0kH0

1 2 b

p

x2

: 12

Integrating equation (4) across the fluid film and utilizing

boundary conditions (7(c)) and (8(c)) and expressions in (9)

and (12) in the modified Reynolds type equation is obtained

in the form

xfh; l

12kH0

12b

p

x 212mVu; 13where fh; l h3 2 12l2h 24l3 tanhh=2l and

Vu 2dh

dt C

d1

dtcosu:

Introducing the non-dimensional quantities:

p pC2

mR2d1=dt; u

x

R; h

h

C 1 2 1 cosu; l

l

C

and ckH0

C3:

Equation (13) takes the form

ufh; l

12c

1 2 b

p

u

212 cos u 14

where fh; l h3 2 12l2 h 24l3 tanhh=2l:As the permeability parameter c! 0; equation (14) reduces

to the corresponding solid case studied by Lin (1997a b).

Squeeze film pressure

For the 1808 partial journal bearing, the boundary conditions

for the fluid film pressure are

p 0 at u ^p

215a

dp

du 0 at u 0: 15b

Integrating equation (14) with respect to u and the use of

boundary conditions (15(a) and (b)) yield the non-

dimensional fluid film pressure as

p 212

Zuuu2p=2

sin2 u

fh; l 12c12b

h i du: 16

Load carrying capacity

The load carrying capacity of the 1808 porous partial journal

bearing is evaluated by integrating the fluid film pressure field

acting on the journal:

W

Zup=2u2p=2

p cosuR du; 17

Figure 1 Physical configuration of porous partial journal bearing





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where W represents the load carrying capacity per unit

length of the porous bearing generated by the squeeze film

pressure.

The non-dimensional form of equation (17) is

WWC2

mR3d1=dt 12

Zup=2u2p=2

sin2 u

fh; l 12c12bh idu

g1; l;c: 18

The closed form integration of the integrals appearing in

equations (16) and (18) is not possible and hence they are

obtained numerically.

Time-height relationship

The time taken by the journal centre to move from 1 0 to 11can be obtained from equation (18) as

d1

dt

1

g1; l;c; 19

where

tWC2

mR3t

is the non- dimensional response time.

The first-order non-linear differential equation (19) is

solved numerically using the fourth-order Runge-Kutta

method (Steven and Reymond, 1998) with the initial

condition

1 0 at t 0: 20

Results and discussion

The effect of permeability on the squeeze film characteristicsof a long porous journal bearing is observed through the non-

dimensional permeability parameter c and effect of couple

stresses through the non-dimensional couple stress parameterl: The parameter l

ffiffiffiffiffiffiffiffiffih=m

p=C is the ratio of microstructure

size to the radial clearance. Hence l gives a mechanism of

interaction of fluid with the bearing geometry. It is to be noted

that as c! 0 the problem reduces to the corresponding solidcase and as l;b! 0 it reduces to the correspondingNewtonian case.

Squeeze film pressure

The variation of non-dimensional squeeze film pressure p as a

function of circumferential co-ordinate u for various values of

lis shown in Figure 2 with the parametric values c 0:01 and1 0:1:

It is observed that the effect of couple stresses is to increase

the squeeze film pressure p as compared to the corresponding

Newtonian case. Increase in p is more pronounced for larger

values of l: The effect of eccentricity ratio parameter 1 onthe variation of p with u is shown in Figure 3 for c 0:01

Figure 2 Non-dimensional film pressure p as a function ofu for different values of lwith c 0.01 and 1 0.1





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and l 0:2: It is observed that p increases for increasingvalues of 1.

Figure 4 shows the variation of p with u for different

values of the permeability parameter c with 1 0:5 andl 0:2:The dotted curve in the figure corresponds to the solid case.

The effect of permeability parameter c is to decrease p ascompared to the corresponding solid case. The variation of

maximum pressure pmaxi:e: p at u 0 with c fordifferent values of l is shown in Figure 5 with 1 0:5: It isobserved that pmax increases as c decreases. This increase inpmax is more accentuated for larger values of l:

Load-carrying capacity

Figure 6 shows the variation of non-dimensional load-

carrying capacity W as a function of the eccentricity ratio

parameter 1 for the different values of l:It is found that W increases for increasing values of1. The

increase in W for couplestress fluids as compared to the

corresponding Newtonian case is observed and is more

pronounced for larger values ofl: The variation in Wwith lfordifferent values ofc is shown in Figure 7.

It is observed that W decreases for increasing values of

cand this decrease in W is more accentuated for larger valuesof l:

Time-height relationship

The response time of squeeze film is an important factor in

the design of squeeze film bearings. This is the time elapsed to

reduce the initial film thickness to the minimum permissible

squeeze film height. The variation of the non-dimensional

minimum squeeze film height h0 1 2 1 with the non-dimensional response time t for the different values of l is

shown in Figure 8. It is observed that the bearing withcouplestress fluid as lubricant have longer response time as

compared to the Newtonian case.

Figure 9 shows the variation of h0 with tfor different valuesof c with l 0:2: The response time of the squeeze film tdecreases for increasing values of c. This is due to thereduction in W with increasing c.

Conclusions

The squeeze film lubrication of long porous partial

journal bearings with couplestress fluid is presented on the

basis of Stokes microcontinuum theory for couple stress

fluids. As the permeability parameter c! 0; the squeezefilm characteristics presented in this paper agrees with

those of solid case studied by Lin (1997a b). It is found

that the effect of permeability is to reduce the non-

dimensional load-carrying capacity and to decrease the

response time as compared to the corresponding solid

case. However, the effect of couple stresses is to enhance

the load carrying capacity and to lengthen the response

time. Hence, in the design of porous partial journal

bearings, the reduction in the load carrying capacity and the

response time can be compensated by the use of lubricants

with proper microstructures by which the bearing life can

be increased.

Figure 3 Non-dimensional film pressure p as a function ofu for different values of1 with c 0.01 and l 0.2





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Figure 4 Non-dimensional film pressure p as a function ofu for different values ofcwith 1 0.5 and l 0.2

Figure 5 Maximum film pressure pmax vs c for different values of lwith 10.5





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Figure 6 Non-dimensional load-carrying capacity W vs 1 for different values of lwith c 0.01 and 1 0.1

Figure 7 Non-dimensional load-carrying capacity W vs l for different values ofcwith 1 0.3





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Figure 8 Non-dimensional minimum film height h0 vs t for different values of lwith 10.1 and c 0.01

Figure 9 Non-dimensional minimum film height vs t for different values ofcwith l 0.2





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