on the steady-state performance of hydrodynamic … · deformation of bearing surface on the...

32

Int. J. Mech. Eng. & Rob. Res. 2012 S K Guha, 2012

ON THE STEADY-STATE PERFORMANCE OFHYDRODYNAMIC FLEXIBLE JOURNAL BEARINGS

OF FINITE WIDTH LUBRICATED BY FERROFLUIDS WITH MICRO-POLAR EFFECT

S K Guha1*

*Corresponding Author: S K Guha, [email protected]

The present work deals with the micro-polar ferro fluid lubrication theory to the problem of thestatic characteristics of finite flexible journal bearings. Based on a magnetic field distributionwith a gradient in both circumferential and axial directions, a modified Reynolds equation hasbeen derived by incorporating the characteristics of micro-polar fluid and using the magneticforces on magnetic particles as the body-external forces. The numerical solution of the discretizedReynolds equation by the finite difference technique with an appropriate iterative procedure hasresulted in the film pressure distribution. With the help of the film pressure, load capacity, sideleakage flow and frictional parameter are obtained for various parameters. The results indicatethat the influences of micro-polar and magnetic effects on the bearing performance characteristicsare signicantly apparent.

Keywords: Flexible journal bearings, Hydrodynamic lubrication, Static characteristics,Magnetic fluids, Micro-polar effect

INTRODUCTIONFerro fluids are suspensions of single domainmagnetic particles, stabilized by surfactantsin the standard lubricating fluids. When amagnetic field is applied to the ferro fluid, eachparticle experiences a force that depends onthe magnetization of the magnetic material ofthe particles and on the strength and positionof the applied field (Rosensweig, 1985).

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Int. J. Mech. Eng. & Rob. Res. 2012

1 Mechanical Engineering Department, Bengal Engineering & Science University, Shibpur, Howrah 711103, India.

As a controllable fluid in terms of its position,orientation and movement, the ferro fluids havedifferent applications in a variety ofengineering devices and systems such as inlubrication and sealing of lubricated bearings.Therefore, this invokes interest in carrying outmore research for the improvement of self-acting characteristics of journal bearingslubricated with ferro fluid.

Research Paper

33


Several studies (Tipei, 1982 and 1983;Sorge, 1987; Cheng et al., 1987; and Zhang,1991) have been reported in the field of ferrofluid – lubricated journal bearings. All the aboveinvestigations assume that the ferro fluidbehaves as a Newtonian fluid. The researchwork by Osman (1999) is the first one thatconsidered the non-Newtonian behavior of theferro fluid, using a parabolic distribution ofmagnetic field model in the axial direction.Osman et al. (2001) analyzed the static anddynamic characteristics of finite journalbearings lubricated with ferro fluid, consideringthe perturbation technique. Later on, Nada andOsman (2007) obtained the staticcharacteristics of ferro fluid – lubricated journalbearings with coupled stress effects.

As the ferro magnetic lubricant contains themagnetic particles together with the chemicaladditives for better lubricating effectiveness,constituting special substructure, the conceptof Newtonian fluid approximation is not asuitable approach. A number of micro-continuum theories has been developed fordescribing the peculiar behavior of fluidscontaining substructure, which can translate,rotate or even deform independently (Arimanand Turk, 1973 and 1974). The theory of micro-polar fluids and the field equations were firstdeveloped by Eringen (1966) from the basictheories of micro-continua. Based on this work,extensive studies (Prakash and Sinha, 1975a,1975b and 1976; Zaheeruddin and Isa, 1978;Tipei, 1979; Singh and Singh, 1982; Huang etal., 1988; Khonsari and Brewe, 1989; and Daset al., 2002) on the performances of journalbearings lubricated with micro-polar fluid havebeen reported in the literature. All these workshave not included the elastic deformation ofbearing surface.

It has long been realized that the assumptionof perfectly rigid bearing shell in the analysismay be unrealistic especially for bearingsoperating with small film thickness, i.e., athigher values of eccentricity ratios. Because,the higher eccentricity ratios result in higherfilm pressure distribution in the bearingclearance, which cause the deformation of thebearing surfaces and consequently thechanges in the configuration of film profile.Eventually, the performance characteristics oflubricated journal bearing are changed ascompared with the case of a rigid bearing shell.Conway and Lee (1977) carried out theanalysis of short flexible journal bearings. Theinvestigation reveals that the average filmpressure deteriorates with increase in flexibleparameter. In several studies (Donoghue et al.,1967; Brighton et al., 1968; Conway and Lee,1975; and Majumdar et al., 1988), the effectof elastic deformation of bearing surface hasbeen considered in the analysis ofperformance of finite hydrodynamic journalbearings. Later on, a few investigations havedealt with the elastohydrodynamic effect (Ohand Huebner, 1973; and Jain et al., 1981) inthe field of lubricated journal bearings.

However, so far there is no investigationavailable that addresses the effect of elasticdeformation of bearing surface on the steady-state characteristics of finite hydrodynamicjournal bearing under the micro-polarlubrication with ferro fluid. The objective of thispaper is to reveal the micro-polar effect on thesteady-state performance of flexible journalbearings lubricated with magnetic fluids. In thepresent analysis, a more generalizedReynolds equation is derived taking intoaccount of the non-Newtonian behavior of the

34


ferro fluid. More realistic magnetic fieldmodels are used. Influence of the micro-polarand magnetic parameters on the staticcharacteristics of magnetized bearings isinvestigated.

MODIFIED REYNOLDSEQUATIONThe induced magnetic force per unit volumeof a ferro fluid under the effect of magnetic fieldis given by Cowley and Rosensweig (1967)and Zelazo and Melcher (1969).

mgmm hMBhF 0 …(1)

Where, B is the magnetic field densityvector and mh represents the inducedfree current. Since the electrical properties ofthe ferro fluid are similar to those of the basefluid, they are non-conductive and no freecurrents are induced. The first term, then, canbe cancelled and the equation reduces to

mgm hMF 0 ...(2)

Considering isothermal conditions andlinear behavior of the ferro fluid, themagnetization M

g of the magnetic material is:

Mg = X

mh

m

The induced magnetic force can bedefined as:

mmmm hhXF 0 ...(3)

This force will be considered as a bodyexternal force in the governing equations asderived as follows:

The governing equations for the steady-state flow of the micro-polar fluid, written in thetensor notation as mentioned in the reference(Khonsari and Brewe, 1989) can be simplified

considerably for the incompressible fluid( = constant) in the absence of inertial forcesand in the presence of the magnetic forces asthe external body forces. Moreover, thethermodynamic pressure can be replaced bythe film pressures (Khonsari and Brewe, 1989).Having made the afore-mentionedassumptions, the so called governingequations for the micro-polar lubricationreduces to as follows:

0

z

V

y

V

x

V zyx...(4)

0

2

2 32

2

mx

x Fx

p

y

v

y

V

...(5)

0

y

p...(6)

0

2

2 12

2

m z

z Fz

p

y

v

y

V

...(7)

0 2 121

2

vy

V

y

v z ...(8)

0 2 222

2

vy

v ...(9)

02 323

2

vy

V

y

v x ...(10)

Now, by solving Equations (7) and (8) withthe following boundary conditions

at y = 0, Vz = 0, v

1 = 0 …(11)

at y = h, Vz = 0, v

1 = 0 …(12)

Vz and v

1 are obtained.

35


Similarly, by solving Equations (5) and (10)with the following boundary conditions

at y = 0, Vx = 0, v

3 = 0 …(13)

at y = h, Vx = U, v

3 = 0 …(14)

Vx and v

3 are obtained.

So, in presence of magnetic forces as theexternal body forces, the velocitycomponents, V

x and V

z of the micro-polar

fluid are derived as:

m

NhF

x

phyyVx mx

2

22

)(

mh

hmhhmyhmy

sinh

)1(cos)1(cossin

hmhhmhh

mN

UF

x

pmx

sin)1cos(2

22

hmy

m

N

hmh

hmhysin

2

sin

1cos2 2

hmh

hmh

hmy

hmy

sin

1cos

sin

1cos...(15)

m

NhF

z

phyyV mzz

2

22

hmh

hmyhmy

sin

1cos1coshmy sin

mzFz

p...(16)

where,

2Nm

Substituting Vx and V

z in the continuity

Equation (1) and integrating across the filmthickness using the following boundaryconditions

Vy(x, 0, z) = V

y(x, h, z) = 0 (non-squeezing

case) the modified Reynolds equation isobtained as:

z

phNg

zx

phNg

x,,,,

hNgFxx

hU mx ,,6

hNgFz mz ,,

...(17)

where,

2

cot612,, 223 NhhNhhhhNg

2

1

2

1

4,

2

N

is the viscosity of the base fluid as in thecase of the Newtonian fluids and and aretwo additional viscosity coefficients of themicro-polar ferro fluids. These viscosityparameters are grouped in the form of twoparameters viz. N2 and . N is a dimensionlessparameter called the coupling number as itsignifies the coupling of the linear and angularmomentum equations. is called thecharacteristics length as it characterizes theinteraction between the micro-polar fluid andthe film gap.

Fmx

and Fmz

are the magnetic forcecomponents which are defined as perEquation (3) as follows.

36


x

hhXF m

mmmx

0 ,

z

hhXF m

mmmz

0 ...(18)

With the following substitutions

2

2

, ,2

,R

pCp

C

hh

L

zz

R

x

0

, ,m

mmm h

hhRU

C

Equation (17) reduces to its non-dimensional form as:

z

phNg

zL

DphNg m ,,,, m

2

mmm

hhhNg

D

Lh,,46

2

z

hhhNg

z

mmm ,,4 ...(19)

where,

2cot

612,, m

2

2

hNh

hNhhhNg

mm

3

m

,

2

20

20

L

CXh mm

MAGNETIC FIELD MODELThe applied magnetic field is that producedby displaced finite current carrying wire. It hasa field gradient in both axial and circumferentialdirections of the bearing. In the non-dimensional form, it is defined by Nada andOsman (2007).

tansinz,1- zhm

tansin, 1- zzhm ...(20)

where,

5.02 cos21

KK

D

L

R

L

R

Ih

h

hh m

m

mm

2,

4, 0

0

0

ParameterRatioDistanceR

RK w

FILM THICKNESS OFFLEXIBLE BEARINGIn case of a flexible bearing shell, the steady-state film thickness, h is given by

zC

hh ,cos1 0

Where, z, is the non-dimensional

deformation of bearing surface. Following amethod similar to that as mentioned in

Donoghue et al. (1967), z, is obtained

as follows.

0,00,012, dpFσz

0 0

,/

,, coscosk n

nk nknk zkndp

(k, n) (0, 0) ...(21)

where as per the Majumdar et al. (1988), thefollowings are defined.

2

0

1

0

0,0

1zddpp

37


c

sscnk A

AAAp tan ,

2 1-/2

122

,

2

0

1

0coscos zddnzkpAc

2

0

1

0sincos zddnzkpAs

F = Flexibility parameter = 3

3

EC

R,

12

EG

Rp

Gud r

nk tCoefficienDistortion, ,

ur= Radial displacement of bearing liner

NUMERICAL PROCEDUREEquation (19) is discretized into finitedifference form and solved by the Gauss-Seidel iterative technique using the over-relaxation factor, satisfying the appropriateboundary conditions. The boundary conditionsare as follows:

0

0,0,1

z

pp

0

,, 2

2

zp

zp

Where, 2 is the angular location of the film

rupture.

Prior to starting the iteration process, thesteady-state film pressure is initiallydetermined considering the bearing liner asrigid one. With the help of this pressureexpressed in double Fourier series, nkp , and

k, n/ are obtained by numerical integration

method. From the known values nkp , , k, n/ of

and the distortion coefficient, dk,n

, the non-dimensional deformation, z, is computed.With this new deformation, z, , Equation(19) is solved again to get new film pressuredistribution. This process is repeated throughiteration technique till convergence isachieved. With the help of this film pressureafter convergence, the steady-stateperformance characteristics of flexible bearingare obtained.

BEARING PERFORMANCECHARACTERISTICSLoad Capacity

The non-dimensional load components whichare shown in Figure 1 are as follows:

1

0 0

2

cos

zddpW r ,

1

0 0

2

sin

zddpW ...(22)

The dimensionless load carrying capacityis given by

WWW r

2 ...(23)

where LR

WCW

2

,

Attitude angle,

rW

W 1tan …(24)

End Leakage Flow

The end leakage flow can be obtained byintegrating the axial velocity component,V

z as

given by Equation (16) across the end sectionof bearing. The dimensionless end leakageflow is calculated by

38


2

06

1h, N, gQ m

dz

hh

z

p

z

mm

z

11

22

2

1...(25)

where D

L

LRC

QQ

,

2

Frictional ParameterThe shear stress at the journal surface isgiven by:

hy

xs y

V

...(26)

Based on Equation (15), the aboveexpression for

s reduces to:

hhNh

UF

x

phmxs

sin1cos

2

1

22

...(27)

where

Nh

The frictional force at the journal surface is:

2

0

2

02

L

ss dzdRF

...(28)

After substituting the expression for s from

Equation (27) into Equation (28), the non-dimensional frictional force becomes asfollows:

Figure 1: Schematic Diagram of Hydrodynamic Journal Bearing with Flexible LinearUnder the Displaced Finite Wire Magnetic Field

e

c

O

b

Rh

Oj

W sin

W cos

WLine of Centres

Solid Housing

Flexible Liner

Journal

IWire

39


Figure 2: W vs F for Various Values of 0 and Magnetic Force Coefficient,

Variation of W with F for Various Values of 0 for = 0 and = 0.1

= 0, 0 = 0.4

= 0, 0 = 0.8

= 0.1, 0 = 0.7

α =

α =

α =

= 0, 0 = 0.5

= 0.1, 0 = 0.4

= 0.1, 0 = 0.8

= 0, 0 = 0.6

= 0.1, 0 = 0.5

= 0, 0 = 0.7

= 0.1, 0 = 0.6

--------------------- α = 0____________ α = 0.1

F

= 0.1 = 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

6

5

4

3

2

1

0

L/D = 1.0, H/R = 0.3, = 0.4, lm = 20, N2 = 0.5

1

0

2

0

222

m

mcsh

hhKph

F

zddhN

hN

h

K

m

c

2tan

2

m

...(29)

where D

L

LR

CFF s

s

,2

As per the Ariman and Turk (1974), Kc is acontrol number. Inside the full film region, Kc isequal to 1.0 and outside this region (partial

film region), hhKc /min which is less than 1.0.

minh is the non-dimensional minimum film

thickness.

Frictional parameter, f(R/C) is defined as:

W

F

C

Rf

s

RESULTS AND DISCUSSIONResults for the steady-state characteristics areexhibited in Figures 2-10 for flexible journalbearings lubricated with micro-polar ferro-fluids at various values of non-dimensionalcharacteristics length, l

m, coupling number, N,

flexibility parameter, F, eccentricity ratio, 0 and

magnetic force coefficient, . The numericalcomputations pertain to a finite journal bearingof L/D = 1.0 with H/R = 0.3, = 0.4, K = 1.2and = /2 for parametric study. Theconditions for magnetic and non-magneticlubricants are characterized with = 0.1 and = 0 respectively.

Load Capacity

Figure 2 shows the dimensionless loadcapacity, W as a function of bearing flexibilityparameter, F at various values of eccentricityratios for both magnetic and non-magneticlubricant cases. A scrutiny of the figure reveals

40


that as eccentricity ratio increases, thedimensionless load increases for all valuesof flexibility parameter. Further, the family ofcurves shows the declining trend withincrease in F for all values of

0. It is also

observed by comparing the non-magneticcurves with the magnetic ones that the non-magnetic case has an adverse effect on theload capacity.

Effect of lm on the load capacity can be

studied from Figure 3. Here, too, it is observedthat the other factors remaining same, anincrease in l

m deteriorates the load capacity

and tends to Newtonian load value as lm is

increased more and more. For a value of lm,

load capacity decreases with increase in F butat lower values of F, the change in loadparameter with F is significant.

Figure 3: W vs F for Various Values of lm and Magnetic Force Coefficient,

Variation of W with F for Various Values of lm for = 0 and = 0.1

F

= 0.1 = 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

5

4

3

2

L/D = 1.0, H/R = 0.3, = 0.4, 0 = 0.6, N2 = 0.5

---------------------____________

= 0, lm = 5

= 0.1, lm = 5

= 0, lm = 10

= 0.1, lm = 10

= 0, lm = 20

= 0.1, lm = 20

= 0, lm = 40

= 0.1, lm = 40

Variation of the load capacity as a functionof flexibility parameter, F is shown in Figure 4for various values of N2, keeping the non-dimensional characteristics length, l

m = 20 and

eccentricity ratio, 0 = 0.6. It is found that at

any F value, the load capacity improves withN2. This is because of the fact that an increasein N2 means a strong coupling effect betweenlinear and angular momentum. This eventuallygives enhanced effective viscosity and hencenon-dimensional load capacity improves. Thishas also been reported by Khonsari and Brewe

(1989) while analyzing the performance ofplane journal bearings lubricated with micro-polar fluids. Moreover, the effect of themagnetic field coefficient is to significantlyincrease the load capacity for all values of N2

at any value of F.

End Flow Rate

Variation of end flow rate (from clearancespace) of the bearing with F is presented inFigure 5 at various eccentricity ratios. Forboth magnetic and non-magnetic lubrication

41


Figure 5: Q vs F for Various Values of 0 and Magnetic Force Coefficient,

Q vs. F for Various 0 Values of = 0 and = 0.1 respectively

F

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

1.2

1.0

0.6

0.2

0.0

= 0

L/D = 1.0, H/R = 0.3, = 0.4, lm = 20.0, N2 = 0.5

---------------------____________

0.8

0.4

0 = 0.4

0 = 0.4

0 = 0.5

0 = 0.5

0 = 0.6

0 = 0.6

0 = 0.7

0 = 0.7

0 = 0.8

0 = 0.8

= 0.1

cases, the effect of eccentricity ratio is toincrease the end flow rate at any value offlexibility parameter. It is further observed that

end flow rate improves with increase in F forany value of eccentricity ratio and the increasein the end flow is more conspicuous at higher

Figure 4: W vs F for Various Values of N2 and Magnetic Force Coefficient,

W vs. F for Various Values of at = 0 and = 0.1 respectively

F

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

5.5

5.0

4.5

3.5

3.0

2.0

1.5

= 0.1 = 0

L/D = 1.0, H/R = 0.3, = 0.4, lm = 20.0,

0 = 0.6

---------------------____________

4.0

2.5

NewtonianNewtonian

Nsq = 0.3Nsq = 0.3

Nsq = 0.5Nsq = 0.5

Nsq = 0.7Nsq = 0.7

Nsq = 0.9Nsq = 0.9

42


values of F. Moreover, there is a considerabledecrease in the side leakage for ferro-magnetic lubrication. It is found that themagnetic lubrication has two opposing effects.The first effect is an increase in pressure andthen an increase in the pressure gradient atthe end section. This effect causes an increasein the side leakage. The second effect is thesealing effect by the magnetic force (F

mz) at

the end section, which causes a decrease in

the side leakage. The net effect is the resultantof the above two opposite effects. The resultsindicate that F

mz has more predominant effect.

The effect of lm on end flow rate is shown in

Figure 6. For a particular value of F, the endflow rate decreases with increase in l

m. This

decrease in end flow is more conspicuous athigher values of F. Here, too, it is observedthat the effect of á is to decrease the end flowrate significantly at any value of F.

Figure 6: Q vs F for Various Values of lm and Magnetic Force Coefficient,

Variation of Q with F for Various Values of lm for = 0 and = 0.1

F

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

1.2

1.0

0.6

0.2

0.0

= 0L/D = 1.0, H/R = 0.3, = 0.4,

0 = 0.6, N2 = 0.5

---------------------____________

0.8

0.4

= 0.1

lm = 5

lm = 5

lm = 10

lm = 10

lm = 20

lm = 20

lm = 40

lm = 40

Figure 7 shows the variation of the end flowrate with respect of F for various values of N2.It is observed from the figure that the end flowincreases with F at any value of N2. In case ofmagnetic lubrication, the rate of increase in

Q is more predominant. Moreover, the effectof N2 is to decrease the end flow rate at anyvalue of F for both types of lubrication. Theeffect of magnetic force coefficient, on Qfollows the similar trend when N2 is taken as aparameter.

Frictional Parameter

Frictional parameter, f(R/C) is shown as afunction of bearing flexibility parameter, F forvarious values of eccentricity ratio in Figure 8.It is observed that the frictional parameterincreases with F at any value of eccentricityratio,

0 for both magnetic and non-magnetic

lubricating conditions. Moreover, the effect ofeccentricity ratio is to decrease the frictionalparameter at any value of F. It is clearly shownthat the magnetic lubricant has a significant

43


Figure 7: Q vs F for Various Values of N2 and Magnetic Force Coefficient,

Q vs. F for Various Values of N2 at = 0 and = 0.1 respectively

F

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

1.2

1.0

0.6

0.2

0.0

= 0L/D = 1.0, H/R = 0.3, = 0.4, l

m = 20.0,

0 = 0.6

---------------------____________

0.8

0.4

NewtonianNewtonian

Nsq = 0.3Nsq = 0.3

Nsq = 0.5Nsq = 0.5

Nsq = 0.7Nsq = 0.7

Nsq = 0.9Nsq = 0.9

= 0.1

Figure 8: f(R/C) vs F for Various Values of 0 and Magnetic Force Coefficient,

Variation of f(R/C) with F for Various Values of 0 for = 0 and = 0.1

= 0, 0 = 0.4

= 0, 0 = 0.8

= 0.1, 0 = 0.7

α =

α =

α =

= 0, 0 = 0.5

= 0.1, 0 = 0.4

= 0.1, 0 = 0.8

= 0, 0 = 0.6

= 0.1, 0 = 0.5

= 0, 0 = 0.7

= 0.1, 0 = 0.6

F

= 0.1 = 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

7

6

5

4

3

2

0

f(R

/C)

L/D = 1.0, H/R = 0.3, = 0.4, lm = 20, N2 = 0.5

---------------------____________

1

decreasing effect on the frictional parameterat all values of F. This is due to higher load

capacity obtained by the magnetic effect at allvalues of F, when

0 is taken as a parameter.

44


Variation of frictional parameter with F forvarious values of l

m is shown in Figure 9. The

effect of lm is to deteriorate the frictional parameter

at any value of F. As lm tends to indicating the

loss of individuality of the micro-structure in themicro-polar lubricant, the frictional parameter

Figure 9: f(R/C) vs F for Various Values of lm and Magnetic Force Coefficient,

Variation of f(R/C) with F for Various Values of lm for = 0 and = 0.1

F

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

6

5

3

1

0

f(R

/C)

= 0L/D = 1.0, H/R = 0.3, = 0.4,

0 = 0.6, N2 = 0.5

---------------------____________

4

2

= 0.1

= 0, lm = 5

= 0.1, lm = 5

= 0, lm = 10

= 0.1, lm = 10

= 0, lm = 20

= 0.1, lm = 20

= 0, lm = 40

= 0.1, lm = 40

Figure 10: f(R/C) vs F for Various Values of N2 and Magnetic Force Coefficient,

f(R/C) vs. F for Various Values of N2 at = 0 and = 0.1 respectively

F

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

6

5

3

1

0

f(R

/C)

= 0L/D = 1.0, H/R = 0.3, = 0.4, l

m = 20.0,

0 = 0.6

---------------------____________

4

2

NewtonianNewtonian

Nsq = 0.3Nsq = 0.3

Nsq = 0.5Nsq = 0.5

Nsq = 0.7Nsq = 0.7

Nsq = 0.9Nsq = 0.9

= 0.1

45


approaches the value of that in the classicallubrication. At any value of l

m, the frictional

parameter improves with increase in F. Here,also the magnetic lubrication condition results inthe lower value of frictional parameter when l

m is

taken as a parameter.

Figure 10 shows the variation of thefrictional parameter as a function of F for anumber of N2. It can be discerned from thefigure that the effect of N2 is to increase thefrictional parameter for both types of lubricationat any value of F. Moreover, the frictionalparameter tends to increase with F at any valueof N2. Here, too, the magnetic lubrication haspotential in lowering the frictional parameter,when N2 is taken as a parameter.

CONCLUSIONThe conclusions that may be drawn from theabove analysis are summarized as follows:

• At any value of flexibility parameter for bothtypes of lubricant, the effect of eccentricityratio is to:

– Improve load carrying capacity.

– Improve end flow rate.

– Decrease frictional parameter.

• At any value of flexibility parameter, anenhancement in the non-dimensionalcharacteristics length of micro-polar ferro-fluid decreases the load capacity, end flowrate and frictional parameter for both typeof lubrication.

• At any value of flexibility parameter for bothtypes of lubrication, the effect of couplingnumber is to:

– Improve load capacity.

– Decrease end flow rate.

– Improve frictional parameter.

• The effect of flexibility parameter is todeteriorate load capacity when eccentricityratio or micro-polar characteristics lengthor coupling number is taken as a parameter.Such effect is not observed in cases of endflow rate and frictional parameter.

• The magnetic micro-polar lubricant haspotential for achieving higher load capacity,lower end flow rate and lower frictionalparameter as compared to the non-magnetic micro-polar lubricant.

REFERENCES1. Ariman T, Turk M A and Sylvester N D

(1973), “Micro-Continuum FluidMechanics—A Review”, Int. J. Eng. Sci.,Vol. 11, pp. 905-930.

2. Ariman T, Turk M A and Sylvester N D(1974), “Applications of Micro-ContinuumFluid Mechanics”, Int. J. Eng. Sci.,Vol. 12, pp. 273-293.

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APPENDIX

Nomenclature

B Magnetic field intensity vector.

C Radial clearance.

dk,n

Distortion coefficient at harmonics, k and n.

D Bearing diameter.

e0,

0Steady-state eccentricity,

0 = e

0/C.

E Young’s modulus of flexible lining material.

f Frictional coefficient.

F Flexibility parameter, F = R3/EC3.

Fm

Induced magnetic force per unit volume of ferro-fluid.

Fmx

Component of induced magnetic force per unit volume of ferro-fluid along x-direction.

Fmz

Component of induced magnetic force per unit volume of ferro-fluid along z-direction.

Fs, sF Shear force on the journal surface, sF = F

sC/ R2 L.

G Lame’s constant, G = E/2(1 + ).

h, h Film thickness of ferro-fluid, Chh / .

hm, mh Magnetic field intensity, 0mmm hhh .

48


APPENDIX (CONT.)

Nomenclature

hm0

Characteristics value of magnetic field intensity.

H Thickness of flexible bearing liner.

I Strength of the current passing through the wire.

K Distance ratio parameter, K = R0/R

lm

Non-dimensional characteristics length of micro-polar ferro-fluid, lm = C/.

L Bearing length.

K Axial harmonic.

Mg

Magnetization of the ferro-fluid.

N Circumferential harmonic.

N Coupling number, N = /(2 + ).

p, p Film pressure, 22 RpCp .

pk,n nkp , Film pressure expressed in double Fourier’s series, 22

,, RCpp nknk .

Q, Q Side leakage flow of lubricant, LRCQQ 2

R Radius of journal.

R0

Displaced distance from the wire position to the bearing centre.

ur

Radial displacement of flexible bearing liner.

U Velocity of journal surface, U = R.

Vi

Velocity components of ferro-fluid along i – direction, i = x, y and z.

v1, v

2, v

3Micro-rotational velocity components about the axes.

W, W Steady-state load capacity, LRWCW 2 .

Wr, rW Radial component of the steady-state load, LRCWW rr

2 .

W, W Transverse component of the steady-state load, LRCWW 2 .

X Cartesian coordinate axis along the circumferential direction, = x/R.

Xm

Susceptibility of magnetic fluid.

Y Cartesian coordinate axis along the film thickness direction.

z, z Cartesian coordinate axis along the bearing axis, Lzz /2 .

lnk, Attitude angle at harmonics k and n.

Viscosity coefficient of the micro-polar ferro-fluid.

Deformation of the flexible bearing liner.

Newtonian viscosity coefficient.

Spin viscosity coefficient of the micro-polar ferro-fluid.

Slenderness ratio of journal bearing, = L/D.

49


APPENDIX (CONT.)

Nomenclature

Characteristics length of the micro-polar ferro-fluid, 21

4 .

0

Permeability of free space or air (0 = 4 10–7 AT/m).

Magnetic force coefficient, 2

20

20

L

CXh mm

.

Poisson’s ratio of flexible bearing liner material.

Attitude angle.

Position angle of the displaced wire magnetic model, = /2.

Angular speed of rotation of journal.

Angular coordinate, = x/R.

on the steady-state performance of hydrodynamic … · deformation of bearing surface on the...

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