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Linear stability analysis of hydrodynamic journalbearings with a flexible liner and micropolarlubricationPikesh Bansala, Ajit K Chattopadhayaya & Vishnu P Agrawalba Mechanical Engineering Department, Faculty of Engineering and Technology, Mody Instituteof Technology and Science, Lakshmangarh332311, Rajasthan, Indiab Mechanical Engineering Department, Thapar University, Patiala147001, Punjab, IndiaAccepted author version posted online: 06 Oct 2014.

To cite this article: Pikesh Bansal, Ajit K Chattopadhayay & Vishnu P Agrawal (2014): Linear stability analysis of hydrodynamicjournal bearings with a flexible liner and micropolar lubrication, Tribology Transactions, DOI: 10.1080/10402004.2014.969817

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Linear stability analysis of hydrodynamic journal bearings with a flexible liner and

micropolar lubrication

Pikesh Bansala, Ajit K Chattopadhayay

a, Vishnu P Agrawal

b

a Mechanical Engineering Department, Faculty of Engineering and Technology, Mody Institute

of Technology and Science,Lakshmangarh - 332311,Rajasthan, India

b Mechanical Engineering Department, Thapar University, Patiala-147001, Punjab, India

Abstract – A linear stability analysis of hydrodynamic journal bearings is presented, including

the effects of elastic distortion of the liner and micropolar lubrication. Hydrodynamic equations

of the lubricant and equations of motion of the journal are solved simultaneously with the

deformation equations for the bearing surface to predict the fluid film pressure distributions

theoretically. The components of stiffness and damping coefficients, critical mass parameter and

whirl ratio, which reflect the dynamic characteristic of the journal bearing, are calculated for

varying eccentricity ratio taking into account the flexibility of the liner and the micropolar

properties of the lubricant. The results presented show that stability decreases with an increase in

the value of the elasticity parameter of the bearing liner and micropolar fluids exhibit better

stability in comparison to Newtonian fluids.

Keywords

Micropolar Fluid; Hydrodynamic Lubrication; Elasticity of Liner; Non-Newtonian Behavior

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INRODUCTION

Hydrodynamic journal bearings are used for medium to large sized machinery such as steam and

gas turbines, generators, pumps and compressors due to their relatively long life. The best

operating conditions for hydrodynamic journal bearings are reached when they operate above a

certain minimum speed required to generate a fluid film of sufficient thickness such that metal to

metal contact can be avoided. Hydrodynamic journal bearings are subjected to excessive friction

and wear operated at lower speeds and especially during starting and stopping. As the cost of

replacing a bearing is high, these bearings are often provided with flexible liners so that only the

liners need to be replaced when damaged. There is a need to understand how the flexibility of the

bearing liner affects the maximum steady state pressure and the cavitation zone especially at high

values of eccentricity ratio.

Higginson (1) was the first to conduct research in this area in 1965. Since then many researchers

(2-17) have studied the performance characteristics of bearings with flexible liners. Further, all

of these studies were carried out assuming Newtonian behavior of the lubricant. The open

literature lacks sufficient design data on the effect of flexible bearing liners on the performance

of journal bearings operating with non-Newtonian lubricants.

Additives are used to enhance the performance of lubricants. The additives are polymeric in

nature while the bearing lubricants are often mineral based. Under normal working conditions,

lubricants become contaminated with wear debris. Henniker (18) found a tenfold increase in

viscosity within 5000 Å of the surface in his experiment of Couette type flow using leuben oil

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mixed with aluminium napthenate. Similar observations were made by Needs (19) in his

boundary lubrication experiments. These experimental results showed that classical continuum

theory was unable to predict the behavior of fluids in such conditions and generated a need for

new theory which could predict the behavior of such fluids.

The theory of micropolar fluids as discussed by Eringen (20) is based on microcontinua where

each material point of a continuum has six degrees of freedom, three rotational and three

translational. Nonsymmetrical stress tensors and couple stress are brought due to rotational

degree of freedom which are missing from the classical theory. Eringen (20) presented the

definition of micropolar fluids and a study of flow behavior of micropolar fluids in a circular

pipe. Micropolar fluids are fluids which consist of rigid, randomly oriented particles in a viscous

medium, where deformation of the fluid particles is ignored. An excellent review of various

theories of microcontinua is given by Ariman et al. (21). As evidenced by various investigators

(22-23), the theory of micropolar fluid may physically serve as a satisfactory model to predict the

behavior of polymeric fluids, real fluid suspensions, and flow through narrow channels like

animal blood.

Micropolar theory is applied to the problem of lubrication for two main reasons. First, the case

when lubricants are contaminated by residual metal particles and dirt under general operating

conditions falls in the domain of micropolar fluids as represented by fluids containing certain

suspensions and mixtures. Second, the clearance in bearings is of the order of an average grain

size of a non-Newtonian fluid. A practical application of the theory of micropolar fluids was

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given by Balaram (24) in the design of journal bearings in the area of nuclear power plants where

the heat transfer agent sodium is used as a lubricant and by Allen and Kline (25) in the case of

lubricant contamination with metal particles and dirt.

Early researchers (Shukla and Isa (26); Prakash and Sinha (27); Zaheeruddin and Isa (28); Tipei

(29)) discussed the effect of micropolar fluids on a one-dimensional bearing or mainly limited

their studies to infinitely long and short bearings. It was found that micropolar fluid theory was

capable of explaining certain phenomena encountered in lubrication problems like increased

effective viscosity as observed by Needs (19). Micropolar lubricants were observed to improve

bearing performance as evidenced by higher pressures, greater load carrying capacity and

increased side flow for the same bearing kinematics and geometry. Later, Singh and Sinha (30)

presented three-dimensional equations for bearings lubricated with micropolar fluids. A rigorous

mathematical analysis of micropolar fluid lubrication theory was presented by Bayada and

Lukaszewicz (31).

The performance characteristic of finite width journal bearing was investigated by Huang et al.

(32) and the performance analysis of a finite journal bearing was studied by Khonsari and Brewe

(33). However, Lin (34) investigated the effect of three dimensional irregularities on a

hydrodynamic journal bearing lubricated with micropolar fluids. Das, et al. (35-36) studied

conical whirl instability of a hydrodynamic journal and the steady state characteristics of

misaligned journal bearings lubricated with micropolar lubricants. Wang and Zhu (37) presented

a study on lubricating effectiveness of micropolar lubricants for a dynamically loaded journal

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bearing. Das et al. (38) studied the stability characteristics of a hydrodynamic journal bearing

under micropolar lubrication. Later, Das, et al. (39) presented a linear stability analysis for a

hydrodynamic journal bearing using micropolar lubricants. Wang and Zhu (40) discussed journal

bearing characteristics including thermal and cavitation effects with micropolar fluids as the

lubricant. Nair, et al. (41) investigated the static and dynamic performance characteristics of an

elastohydrodynamic journal bearing with micropolar fluids as the lubricant. Verma, et al. (42)

analyzed a multirecess hydrostatic journal bearing with micropolar lubricants. Nicodemus and

Sharma (43) presented the influence of wear on the performance of a multirecess hydrostatic

journal bearing with micropolar lubricants. Rahmatabadi, et al. (44) studied the effect of

micropolar lubricants on the performance characteristics of noncircular lobed bearings.

Naduvinamani and Santosh (45) studied squeeze film lubrication of finite porous journal

bearings with micropolar lubricants. Nicodemus and Sharma (46) studied the performance

characteristics of a micropolar lubricated membrane compensated worn hybrid journal bearing.

Lin, et al. (47) studied the dynamic characteristics of parabolic film slider bearings with

micropolar fluids as lubricants. Quite recently, Sharma and Rajput (48) investigated the effect of

geometric imperfections of a journal on a 4-pocket hybrid journal bearing using micropolar

fluids as a lubricant.

There is a lack of information in the literature addressing the stability analysis of journal

bearings with flexible liner under micropolar lubrication. The aim of the present study is to

explore analytically the effect of micropolar lubricants on the linear stability of journal bearings

with flexible liners as shown in Fig 1. First, deformation of the flexible liner is calculated by

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solving displacement components in the elastic medium as discussed by Oh Huebner (6),

Conway and Lee (7), and Majumdar, et al. (10). Deformation of the flexible liner is used to

calculate fluid film thickness which is then used to calculate film pressure by solving the

modified Reynolds equation for micropolar lubricants as discussed in the literature survey (32-

48). Using a first order perturbation, the pressure is converted to steady-state and dynamic

pressures. Stiffness and damping components are calculated from these pressures which are used

in the equation of motion to obtain the mass parameter and whirl ratio. A parametric study is

conducted for linear stability analysis of a journal bearing with a flexible liner lubricated by a

micropolar fluid. However, due to a dearth of literature regarding actual values of micropolar

parameters, the values assumed by previous researchers (32-48) were used in present study.

Kolpashchikov, et al. (49) attempted to calculate the values of micropolar parameters by taking

data for water flow in capillaries from Deryagin, et al. (50). In the absence of experimental data,

the theoretical mass parameter obtained by this analysis has been compared with available

results. The results presented herein are expected to be useful for practicing engineers in the field

of bearing design and as well as for academicians.

ANALYSIS

Figure 1 shows a schematic diagram of a journal bearing with a flexible liner. The elliptical path

of the journal has a major axis Cε0 1 and a minor axis Cε1.

Modified Reynolds Equation

The modified Reynolds equation (Prakash and Sinha (27), Singh and Sinha (30)) for journal

bearings lubricated with micropolar fluids with assumptions to generalize the micropolar effect

(Prakash and Sinha (27), Singh and Sinha (30)) for linear stability analysis is

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3 3p ph h 2 h h

,N,h ,N,h 6U 1 12x x z z t x t

[1]

where

1/ 2 1/ 22

2

N Nh( , N,h) 1 12 6 coth , , N

h 2 4 2h

The parameter η represents the viscosity of the Newtonian fluid. γ and χ are the material

coefficient and spin viscosity, respectively for the micropolar fluid. The film thickness is h and

the micropolar film pressure for the bearing is p. Λ and N are two parameters which are

associated with the micropolar fluid and are used to distinguish the micropolar fluid from a

Newtonian fluid. Λ is termed the characteristic length of the micropolar fluid, representing the

interaction between the film gap and the micropolar fluid. As Λ tends to zero the modified

Reynolds equation is reduced to the usual Reynolds equation for a Newtonian fluid. N is termed

the coupling number and is a dimensionless number which couples the linear and angular

momentum equations which arise as a result of the microrotational effect of the suspended

particles such as additives, dirt, and metal particles in the fluid.

Equation [1] will be reduced to its non-dimensional form as

2

'

m m

p D p 1 h hg l , N,h g(l , N,h) 1 2

L z z 2 t

[2]

with the following substitutions

2

m2

x 2z h pC C, z ,h , p , l , t t

R L C R

where,

3 l Nhh h Nh mg l , N, h coth

m 212 2l 2l mm

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and

'

t

In the case of a flexible bearing, the steady state film thickness h is given by (O’Donoghue, et al.

(3) and Majumdar, et al. (10))

h 1 cos [3]

h eh , ,

C C C

δ, represents the radial deformation of the bearing liner surface.

For a flexible bearing, the non-dimensional radial deformation is first obtained before trying to

find the solution of equation [2]. The method of calculating elastic deformation is as suggested

by Brighton, et al. (4) and Majumdar, et al. (10). Satisfying the boundary conditions, the three

displacement components u, v and w are calculated. The method is briefly explained below.

The non-dimensional radial deformation of the bearing liner surface can be calculated as

suggested by O’Donoghue, et al. (3) and Majumdar, et al. (10) using:

m,n m,n m,n

m 0 n 0

2m z2 1 F p d cos cos n

L

[4]

3

3

RF

EC

F is defined as the deformation factor which gives a measure of the relative magnitude of viscous

shear stress compared with the normal stress per unit normal strain for the bearing liner material

in the radial direction for a particular bearing geometry. The Poisson’s ratio for the bearing liner

material is ν.

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To calculate the values of pm,n and βm,n, the two-dimensional modified Reynolds equation is

solved to obtain pressure distribution assuming the bearing is rigid and assuming steady state

conditions. This pressure distribution is expressed as a double Fourier series. (O’Donoghue (3)

and Majumdar, et al. (10))

2m z| |p p cos cos n

m n m,n m,nL

[5]

where |

indicates that the first term of the series is halved, and m,n and m,n are calculated as

follows

2 22 1 2 1

m,,n

0 0 0 0

2p cos m z cos n dzd p cos m z sin n dzd

Lp

[6]

2 1

1 0 0

m,n 2 1

0 0

p cos m z sin n dzd

tan

p cos m z cos n dzd

and 2 1

0,0

0 0

1p pd dz

where 2 2

m,n

m,n 2 2

p C pC zp , p , z

L / 2R R

To calculate dm,n, three displacement components in r, θ, z can be assumed as (Mazumder (52))

r m,n

m,n

z m,n

2m zu u a cos(n )cos

L2m z

u v a sin(n )cosL

2m zu w a cos(n )sin

L

[7]

where u , v and w are functions of r only and a is the inside radius of the bearing liner.

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The stress displacement relationship, Mazumder (52) and Rekach(53), gives

r z r

rr

r z

r

r z z

zz

r

r

z

z

u(ru ) u u1 12

r r r z ru u(ru ) u1 1

2 ur r r z r

u(ru ) u u1 12

r r r z z

uu1r

r r r

u u1

z r

z r

zr

u u

r z

[8]

The equations of equilibrium from the basic theory of elasticity can be written as (Mazumder

(52) and Rekach (53))

r rrrr rz

r z r

zrz zz rz

10

r r z r1

2 0r r z r

10

r r z r

[9]

Substituting the values of ur, uθ and uz from equation [7] in the stress-displacement equations [8]

then further substituting the stress components in the equilibrium equations [9] gives following

relations:

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22

2 2

2

2

22 2

2 2

2

2

2

d u C du u n dvC (C n ) (C 1)

y dy y dydy yn dw

(C 1) v k u (C 1)k 0dyy

d v 1 dv v n du(1 C n ) k v (C 1)

y dy y dydy yn w

(C 1) u nk(C 1) 0yy

d w

dy

22

2

1 dw n duw C k w (C 1)k

y dy dyyu v

k(C 1) nk(C 1) 0y y

[10]

2 ma r E EC 2 ,k , y , ,

L a (1 )(1 2 ) 2 1

The boundary conditions are (Majumdar(10) and Mazumder (52))

1. The ends of the bearing are prevented from expanding axially, but are free to move

circumferentially or radially.

m,n

du 1 nv uy 1,C p (C 2)( kw )

dy y ydv nu v

dy y y

dw u k

dy

[11]

2. The outer surface of the bearing is rigidly enclosed by the housing, preventing any

displacement of the outer surface.

bat y= , u =v =w =0

a

First u is calculated by solving equation [10], taking into consideration the boundary conditions

in equation [11] with value of pressure as unity. Next, dm,n is calculated using equation given

below.

m,n

ud =

Rp

[12]

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Perturbation Method

Non-dimensional pressure and film thickness are calculated using a first order perturbation

technique with the assumption that the journal whirls about its mean steady position given by the

eccentricity ratio ε0 with amplitudes Re i t1e

and Re i t

0 1e along the line of centers and

perpendicular to the line of centers respectively. The non-dimensional pressure and local film

thickness are

i t i tp p p e p e0 1 1 2 0 1

i t i th h e cos e sin0 1 0 1

[13]

where

i th 1 cos , e0 0 0 1

pi te and 0 1

[14]

The subscript i = 0, 1 and 2 represents pressures for steady state and first order perturbations in

Eqs.[12]-[14], respectively (Das et al. (39)).

Equation for Steady-State and Dynamic Conditions

Expanding Equation [2] with all the substitutions made and by neglecting higher order terms of

(ε1) and (ε0 1), the following three equations are obtained by collecting the zeroth and first order

terms of ε1 and ε0 1

2 22 2

0 0 0 0 0 0 0

1 2 2 2

h p h p p p hD D 1C C

L z z L 2z

[15]

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2 2 22 22 2

0 0 0 01 1 1 12 1 12 2 2 2

2 2

0 0 01 1

3 1

p p h hp p p pD D DC C cos C

L L L z zz z

h h pp pD D pC cos C sin cos

L z z L z

1sin i cos

2

[16]

2 2 22 22 2

0 0 0 02 2 2 2

2 1 12 2 2 2

2 2

0 0 0 02 2

3 1

p p h hp p p pD D DC C sin C

L L L z zz z

h h p pp pD DC sin C cos sin

L z z L z

0

0

h1 icos i sin

2

[17]

where,

2 2 2

20 0 m 0 0 m 0

1 2

mm

h Nh Nl h N h Nl h1C coth cosech

4 l 2 4 2l

3 2

0 0 0 m 0

2 2

mm

h h Nh Nl hC coth

12 2l 2l

3 2

2 2 20 m 0 m 0 m 0 m 0 m 0

3 0

m

h Nl h Nl h N l h Nl h Nl hNC N h cosech coth cosech coth

2 2 l 2 4 2 2

Boundary Conditions

The boundary conditions relevant to the problem are as follows:

1. The steady state pressures and perturbed pressures at the ends of the bearing are zero

p ( ,z) 0, i 0,1,2 at z= 1i

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2. The pressure distribution is symmetrical about the mid-plane for steady state pressures

and perturbed pressures of the bearing

p ,z

i 0, i 0,1,2 at z=0z

3. Cavitation boundary conditions

1

p ( ,z)0 2p ( ,z) 0

0 2 zp ( ,z) 0, i 0,1,2 for

2i

where θ1 and θ2 represent, respectively, the angular coordinates where the film starts and

the film cavitates. The cavitation boundary condition is replaced by the method of

constraints suggested by Christopherson (54) to facilitate computation while economizing

time.

4. Periodic boundary conditions

p ,z p ( 2 ,z), i 0,1,2i i

Steady State Load

The components of the fluid film force in non-dimensional forms along the line of centers and

perpendicular to the line of centers are

2

1

2

1

1

R 000

1

000

F p cos d dz

F p sin d dz

[18]

Utilizing the above non-dimensional fluid film force components, the non-dimensional steady

state load, attitude angle and the Sommerfeld number are obtained.

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1/ 222

0 R 0 0

1 0

0

R 0

0

W F F

Ftan

F

1S

w

[19]

where

2

0

0 3

2W CW

R L

Stiffness and Damping Coefficients

Non-dimensional components of stiffness and damping coefficients can be calculated from the

dynamic film pressures as follows (Majumdar, et al. (10))

2

1

2

1

2

1

2

1

2

1

2

1

1

RR 1

0

1

R 1

0

1

R 2

0

1

2

0

1

RR 1

0

1

R 1

0

S Re 2 p cos d dz

S Re 2 p sin d dz

S Re 2 p cos d dz

S Re 2 p sin d dz

ImD 2 p cos d dz

ImD 2 p sin d dz

2

1

2

1

1

R 2

0

1

2

0

ImD 2 p cos d dz

ImD 2 p sin d dz

[20]

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Equations of Motion

Considering the rotor to be rigid, the equations of motion of the journal along the line joining

centers and in the direction perpendicular to it, can be written as (Majumdar, et al. (10))

22

R2

d dMC F W cos

dtdt

[21]

2

2

d d dMC 2 F Wsin

dt dtdt

[22]

where FR and F are the resultant film forces in the R and directions, respectively.

For the steady state condition (Majumdar et al. (10))

R 0 0 0 00 0F W cos 0 and F W sin 0 [23]

Substituting equations [14] and [23] into equations [21] and [22] and non-dimensionalising with

the following substitutions

3 3

ij ij ij ij3 3

2C 2CS S ;D D ;

R L R L

2 2

0 03

0

2C MCW W ;M ;

WR L

and retaining only the first order terms gives the non-dimensional equations of motion

(Majumdar et al. (10))

2

0 RR RR 1

0 R 0 R 0 0 1

MW S i D

S i D W sin 0

[24]

2S i D MWR R 1 0 0

S i D W cos 00 0 0 0 1

[25]

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For a non-trivial solution, the determinant of the coefficient matrix of equation [24] and equation

[25] must vanish. When the imaginary and real terms of the determinant are equated to zero

separately, the following two equations are generated (Majumdar et al. (10))

0

0 RR RR R R R R R 0 RR 02

0RR

W1MW S D S D S D S D (D sin D cos )

(D D )

[26]

2 4 20 0

0 0 RR RR R R RR R R

0

0

RR 0 R 0

0

W cosMW MW S S D D D D S S S S

WS cos S sin 0

[27]

Whirl ratio and mass parameter can be calculated from the above two equations.

METHOD OF SOLUTION

The dimensionless stability parameter for a journal bearing with a flexible liner with micropolar

lubrication is obtained via the following procedure.

1. Initialize the values of the input parameters i.e. L/D, H/R, ν, F, ε0, lm, N2, over relaxation

factors, convergence criterion and number of maximum iterations.

2. Initialize iteration number n to 0.

3. Initialize dimensionless steady state pressureijp 0 .

4. Calculate dimensionless film thickness assuming non-dimensional radial deformation of

bearing liner surface =0.

5. Solve equation [15] in finite difference form using a successive over relaxation scheme

satisfying boundary conditions and taking film thickness as calculated in step 4. This step is

repeated iteratively until convergence is achieved or number of maximum iteration is

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exceeded. If the maximum iteration number is exceeded, change the over relaxation factor

and go to step 2 and repeat until convergence is achieved.

6. The pressure profile generated in step 5 is expressed as a double Fourier series as given in

equation [5]. Then m,,np and m,n are calculated by Simpson’s 1/3

rd rule.

7. Values of dm,n are calculated by solving equation [10] for unit pressure, and μ and R by the

finite difference method (Gauss-Seidel iteration) with a successive over relaxation scheme.

These values of dm,n are stored in computer memory for further use.

8. is calculated by substituting the values of dm,n, m,,np and m,n in equation [4].

9. Using these values of , calculate a new dimensionless film thickness from step 4. Repeat

the procedure until convergence is achieved to obtain final steady state pressure profile.

10. Similarly solve equations [16] and [17] to get perturbed pressure along the line joining the

centers and perpendicular to the line joining the centers.

11. Calculate steady state load from equation [16] using the components force in the direction of

the line of centers and perpendicular to the line of centers which are calculated by Simpson’s

1/3rd

rule. Calculate the stiffness and damping coefficients by Simpson’s 1/3rd

rule using

equation [20].

12. Use the these values calculated in steps 10 and 11 in equations [26] and [27] to obtain the

mass parameter and whirl ratio. Equations [26] and [27] are solved to get whirl ratio. If the

roots of the whirl ratio are real, the mass parameter is obtained using equation [26]. If the

roots are imaginary, then the equilibrium position is stable.

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RESULT AND DISCUSSIONS

In this paper, the dynamic characteristics for a journal bearing in terms of critical mass parameter

and whirl ratio are computed considering the flexibility of the bearing liner and micropolar

lubrication. The critical mass parameter (a measure of stability) is the threshold value of M above

which the bearing will be unstable. The journal speed corresponding to the critical mass

parameter is the threshold speed. Whirl ratio gives the ratio of frequency of the center trajectory

with respect to spin of the journal. Whirl ratio helps diagnose whether self excited vibration is

due to oil or synchronous whirl. Since the critical mass parameter and whirl ratio are dependent

on steady state and perturbed pressure and thus, in turn are functions of lm, N2 and F, a

parametric study is done to show the effect of these parameters on critical mass parameter and

whirl ratio. The components of stiffness and damping coefficients are used to calculate the

critical mass parameter and whirl ratio, so their variation is not shown.

In order to validate the computer program developed and the solution algorithm the results were

compared with published results. The mass parameter for journal bearings with a flexible liner

using a Newtonian fluid as a lubricant is compared with that obtained by Mazumdar, et al. (10)

as shown in Figure 2. Also the results for the mass parameter for the case of a rigid bearing

lubricated by a micropolar fluid are compared with Das, et al. (39) as shown in Figure 3. The

results shown in Figure 2 and Figure 3 compare well, agreeing with in 4-7%.

The critical mass parameter and whirl ratio are calculated for various values of ε0 (0.6-0.85),

Sommerfeld Number (0-0.35), lm (10-80), N2 (0.1-0.9), F (0.0-0.5), H/R = 0.3, ν = 0.4, L/D =1.0.

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Constant values of H/R, L/D, ν are used in order to compare the results with Majumdar et al.

(10). Values of ε0 above 0.6 are considered since the effect of the flexibility of the liner is

negligible below ε0 = 0.6, as predicted by Majumdar, et al. (10) and Brighton, et al. (4). The

values of lm and N2 can be determined experimentally as shown by Kolpashchikov et al. (49).

Table 1 (55) gives the values for lm and N2 as calculated by Pietal (55) for exemplifying fluids

whose chemical composition is given in detail in (56, 57). Due to limited data available for

values of micropolar parameters lm and N2 for real fluids, the values used herein are the same as

those taken by other researchers in the field of micropolar lubricants (32-48), considering that all

the real fluids working as lubricants will have values in the range provided.

Figure 4 shows the variation of the critical mass parameter versus Sommerfeld number. The

graph shows how the mass parameter is affected by the elasticity parameter F of the bearing liner

when considering a micropolar fluid as the lubricant, a bearing geometry of L/D = 1.0, ν=0.4,

H/R=0.3, and micropolar parameters of lm=40.0 and N2=0.5. The graph clearly shows that as the

value of the elasticity parameter increases from 0.0 to 0.4 the stability of the bearing tends to

decrease. The elasticity parameter is inversely proportional to the Young’s modulus. So, the

stability of the bearing will increase when a bearing liner with a higher value of Young’s

modulus is chosen.

Figure 5 shows a parametric study of the effect of the characteristic length of the micropolar

fluid on the critical mass parameter for L/D=1.0, ν=0.4, F=0.2, H/R=0.3 and N2=0.5. The

variation of the plot exhibits typical trends for critical mass parameter versus Sommerfeld

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number curves for a Newtonian fluid but the micropolar fluid shows a higher stability threshold.

As lm→∞ the stability threshold shifts towards that of the Newtonian fluids. This is due to the

fact that lm represents the characteristic length of the substructure and as lm→∞, the fluid loses its

micropolar characteristics and is reduced to a Newtonian fluid.

Figure 6 shows the combined effect of the elasticity parameter F and the characteristic length. In

this graph the critical mass parameter is plotted against lm for various values of F. There are two

important observations. First, the value of the critical mass parameter is maximum as the value

of lm→10.0. Second, the value of the critical mass parameter decreases as the value of F

increases.

The variation of the critical mass parameter versus the elasticity parameter is shown in Figure 7

for various values of eccentricity ratio at L/D=1.0, ν=0.4, H/R=0.3, lm=40.0, N2=0.5. The

variation in the graph depicts that the value of critical mass parameter decreases as the value of

elasticity parameter is increased from 0.0 to 0.5. The value of the critical mass parameter

decreases rapidly for ε0=0.85 and goes below values of the critical mass parameter for ε0=0.8.

This is due to the fact that as F increases also increases, which in turn increases minimum film

thickness. Due to an increase in minimum film thickness there is a reduction in pressure and

hence lower values of the critical mass parameter. Similar observations were made by Conway

and Lee (7) using the short bearing approximation and by Majumdar, et al. (10) using a finite

bearing. The value of the critical mass parameter decreases sharply as the value of the elasticity

parameter increases from 0.0 to 0.1.

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The variation of whirl ratio versus non-dimensional characteristic length lm for various values of

the elasticity parameter F for L/D=1.0, ν=0.4, H/R=0.3, ε0=0.5, N2=0.5 is shown in Figure 8. The

value of whirl ratio increases as the value of F is increased up to F = 0.2, but is found to decrease

with further increase in value of F for a given value of lm. However, the values of whirl ratio are

taken at an accuracy of 10-4

in order to show the variation and the values in all cases lie in a

range between 0.52 and 0.56.

Figure 9 shows the variation of whirl ratio versus the elasticity parameter F for various values of

ε0 at L/D=1.0, ν=0.4, H/R=0.3, lm=40.0, N2=0.5. The graph indicates that the variation of whirl

ratio for ε0=0.6 and 0.8 (0.43 < <0.53) is not significant as the value of F is increased but the

value of whirl ratio increases drastically (0.31 < < 0.53) for ε0 = 0.85 as the value of F is

increased. This increase is also due to the fact that the pressure profile reduces as F is increased

due to an increase in minimum film thickness.

CONCLUSION

From the above study it can be concluded that

1) The performance characteristics of a journal bearing with a flexible liner are significantly

affected by the flexibility of the liner and the micropolar effect of the lubricant. The effect of

the flexibility of the liner should be taken into account especially at higher eccentricity ratio

(i.e. ε0≥0.8) in order to accurately predict the performance characteristics.

2) When a bearing liner is used, the designer should choose a liner material with a high Young’s

modulus to achieve better stability, especially when the bearing is operating at high

eccentricity ratio (i.e. ε0≥0.8).

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3) The stability parameter improves for any value of elasticity parameter for micropolar

lubricated bearings in comparison to bearings lubricated with Newtonian fluids. The

maximum effect of micropolar fluids is observed when lm→10.0.

4) An increase in the micropolar characteristics of a fluid increases stability while an increase in

the deformation factor of the liner reduces stability. An optimized solution may be created

when designing a journal bearing with a flexible liner.

5) The whirl ratio for journal bearings using a flexible liner remains near 0.5. However, the value

of whirl ratio may vary from 0.3 to 0.5 at higher eccentricity ratios (i.e. ε0≥0.8).

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REFERENCES

(1) Hinggison, G.R. (1965-66), “The Theoretical Effects of Elastic Deformation of the Bearing

Liner on Journal Bearing Performance,” Proc. of the symp. on Elastohydrodynamic

Lubrication, IMechE, London (UK) 180, Part 3B, pp. 31-37.

(2) Holmes, R. (1966), “Instability Phenomena due to Circular Bearing Oil Films,” J. of Mech.

Engg. Sci. V8, pp. 419-425.

(3) O’Donoghue, J., Brighton, D.K. and Hooke, C.J.K. (1967), “The Effect of Elastic

Distortions on Journal Bearing Performance,” J. Lubr. Tech., Series F 89, n4, pp. 409-417.

(4) Brighton, D.K., Hooke, C.J.K. and O’Donoghue, J. (1967-68), “Theoretical and

Experimental Investigation of the Effect of Elastic Distortions on the Performance of Journal

Bearing,” Tribology Convention, Proc of IMechE 182, Part 3N, pp. 192-200.

(5) Benjamin, M.K., Castelli, V. (1971), “A Theoretical Investigation of Complaint Journal

Bearing,” ASME J. Lubr. Tech. 9, n1, pp. 191-201.

(6) Oh, K.P. and Huebner, K.H. (1973), “Solution of the Elastohydrodynamic Finite Journal

Bearing Problem,” ASME J. Lubr. Tech. 95, n3, pp. 342-352.

(7) Conway, H.D. and Lee, H.C. (1975), “The Analysis of the Lubrication of a Flexible Journal

Bearing,” ASME J. Lubr. Tech. 97, pp. 599-604.

(8) Jain, S.C., Sinhasan, R. and Singh, D.V. (1982), “Elastohydrodynamic Analysis of a

Cylindrical Journal Bearing with a Flexible Bearing Shell,” Wear 78, pp. 325-335.

(9) Jain, S.C., Sinhasan, R. and Singh, D.V. (1982), “The Performance Characteristics of Thin

Compliant Shell Journal Bearings,” Wear 81, pp. 251-261.

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(10) Majumdar, B.C., Brewe, D.E. and Khonsari, M.M. (1988), “Stability of a Rigid Rotor

Supported on Flexible Oil Journal Bearing,” Tribol. Trans. 110, pp. 181-187.

(11) Chandrawat, H.N. and Sinhasan, R.A. (1988), “Study of Steady state and Transient

Performance Characteristics of a Flexible Shell Journal Bearing,” Tribol. Int. 21, pp. 137-

148.

(12) Braun, M. J. and Dougherty, J. D. (1989), “Hydrodynamic Analysis and Fluid-Solid

Interaction Effects on the Behavior of a Compliant Wall (Thick) Journal Bearing. Part 1:

Theory,” Trans. of ASME, Jan. 1989 111, pp. 70-79.

(13) Braun, M. J. and Dougherty, J. D. (1989), “Hydrodynamic Analysis and Fluid-Solid

Interaction Effects on the Behavior of a Compliant Wall (Thick) Journal Bearing. Part 2:

Results,” Trans. of ASME, Jan. 1989 111, pp. 80-86.

(14) Mokhiamer, U.M., Crosby, W.A. and El-Gamal, H.A. (1999), “A Study of a Journal Bearing

Lubricated by Fluids with Couple-Stress Considering the Elasticity of the Liner,” Wear 224,

pp. 194-201.

(15) Lahmar, M. (2005), “Elastohydrodynamic Analysis of Double-Layered Journal Bearing

Lubricated With Couple-Stress Fluids,” Proc. IMechE, Part J: Journal of Engg. Tribol. 219,

pp. 145-171.

(16) Boucherit, H., Lahmar, M. and Bou-Saïd, B. (2008), “Misalignment Effects on Steady-State

and Dynamic Behavior of Complaint Journal Bearings Lubricated With Couple Stress

Fluids,” Lubr. Sci., Wiley Inter-Science 20, pp. 241-268.

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(17) Lahmar, M., Ellagoune, S. and Bou-Saïd, B. (2010), “Elastohydrodynamic Lubrication

Analysis of a Compliant Journal Bearing Considering Static and Dynamic Deformations of

the Bearing Liner,” Tribol. Trans. 53, 3, pp. 349-368.

(18) Henniker, J.C. (1949), “The Depth of Surface Zone of a Liquid,” Rev. of mod. Phys. 21, 2,

pp. 332-341.

(19) Needs, S.J. (1940), “Boundary Film Investigations,” Trans. of ASME 62, pp. 331-339.

(20) Eringen, A.C. (1966), “Theory of Micropolar Fluids,” J. Math Mech. 16, pp. 1-18.

(21) Ariman, T., Turk, M.A. and Sylvester, N.D. (1973), “Microcontinuum Fluid Mechanics-A

Review”, Int. J. Engng Sci. 11, pp. 905-930.

(22) Ariman, T., Turk, M.A. and Sylvester, N.D. (1973), “A Microcontinuum Model for Blood.”

Proc. Fourth Congress Appl. Mech., Montreal.

(23) Kline, K.A. and Allen, S.J. (1971), “A Thermodynamical Theory of Fluid Suspensions”,

Phys. Fluids 14, pp. 1863-1869.

(24) Balaram, M. (1975), “Micropolar Squeeze Films,” J. Lubr. Tech. 97, pp. 635-637.

(25) Allen, S. and Kline, K. (1971), “Lubrication Theory of Micropolar Fluids,” J. Appl. Mech.

38, pp. 646-650.

(26) Shukla, J.B. and Isa, M. (1975), “Generalised Reynolds Equation for Micropolar Lubricants

and its Application to Optimum One-Dimensional Slider Bearing; Effects of Solid Particle

Additives in Solution,” J. Mech. Eng. Sci. 17, pp. 280-284.

(27) Prakash, J. and Sinha, P. (1975), “Lubrication Theory of Micropolar Fluids and Its

Application to a Journal Bearing,” Int. J. Eng. Sci. 13, pp. 217-232.

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(28) Zaheeruddin,Kh. And Isa, M. (1978), “Micropolar Fluid Lubrication of One-Dimensional

Journal Bearing,” Wear 50, pp. 211-220.

(29) Tipei, N. (1979), “Lubrication with Micropolar Liquids and its Application to Short

Bearings,” J. Lubr. Tech. 101, pp. 356-363.

(30) Singh, C. and Sinha, P. (1982), “The Three Dimensional Reynolds Equation for Micropolar

Fluid Lubricated Bearings,” Wear 76, 2, pp. 199-209.

(31) Bayada, G. and Lukaszewicz, G. (1996), “On Micropolar Fluids in the Theory of

Lubrication. Rigorous Derivation of an Analogue of the Reynolds Equation,” Int. J. Engg.

Sci. 34, pp. 1477-1490.

(32) Huang, T.W., Weng. C.I., Chen C.K. (1988), “Analysis of Finite Width Journal Bearings

with Micropolar Fluids,” Wear 123, pp. 1-2.

(33) Khonsari, M.M., Brewe, D.E. (1989), “On the Performance of Finite Journal Bearings

Lubricated with Micropolar Fluids,” STLE, Tribol. Trans. 32, 2, pp. 155-160.

(34) Lin, T.R. (1996), “Hydrodynamic Lubrication of Journal Bearings Including Micropolar

Lubricants and Three Dimensional Irregularities,” Wear 192, pp. 21-28.

(35) Das, S., Guha, S.K., Chattopadhyay, A.K. (2001), “On the Conical Whirl Instability of

Hydrodynamic Journal Bearings Lubricated with Micropolar Fluids,” Proc. Instn. Mech.

Engrs. J. J. Eng. Tribol. 215, pp. 431-439.

(36) Das, S., Guha, S.K., Chattopadhyay, A.K. (2002), “On Steady State Performance of

Misaligned Hydrodynamic Journal Bearings Lubricated with Micropolar Fluids,” Tribol. Int.

35, pp. 201-210.

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(37) Wang X. and Zhu K.Q. (2004), “A Study of the Lubricating Effectiveness of Micropolar

Fluids in a Dynamically Loaded Journal Bearings,” Tribol. Int. 37, pp. 481-490.

(38) Das, S., Guha, S.K., Chattopadhyay, A.K. (2004), “Theoretical Analysis of Stability

Characteristics of Hydrodynamic Journal Bearings Lubricated with Micropolar Fluids,”

Proc. Inst. Mech. Engrs. J. J. Eng. Tribol. 218, pp. 45-56.

(39) Das, S., Guha, S.K., Chattopadhyay, A.K. (2005), “Linear Stability of Hydrodynamic

Journal Bearings under Micropolar Lubrication,” Tribol. Int. 38, pp. 500-507.

(40) Wang, X.L. and Zhu, K.Q. (2006), “Numerical Analysis of Journal Bearings Lubricated

with Micropolar Fluids including Thermal and Cavitating Effects,” Tribol. Int. 39, pp. 227-

237.

(41) Nair, K.P., Nair, V.P.S. and Jayadas, N.H. (2007), “Static and Dynamic Analysis of

Elastohydrodynamic Elliptical Journal Bearing with Micropolar Lubricant,” Tribol. Int. 40,

pp. 297-305.

(42) Verma, S., Kumar, V., Gupta, K.D. (2009), “Analysis of Multirecess Hydrostatic Journal

Bearings operating with Micropolar Lubricant,” J. Tribol. 131, pp. 021103-1-9.

(43) Nicodemus, E.R. and Sharma, S.C. (2010), “Influence of Wear on the Performance of

Multirecess Hydrostatic Journal Bearing operating with Micropolar Lubricant,” J. Tribol.

132, pp. 021703-1-11.

(44) Rahmatabadi, A.D., Nekoeimehr, M. and Rashidi, R. (2010), “Micropolar Lubricant Effects

on the Performance of Noncircular Lobed Bearings,” Tribol. Int. 43, pp. 404-413

(45) Naduvinamani, N.B. and Santosh, S. (2011), “Micropolar Fluid Squeeze Film Lubrication of

Finite Porous Journal Bearing,” Tribol. Int. 44, pp. 409-416.

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(46) Nicodemus, E.R. and Sharma, S.C. (2012), “Performance Characteristics of Micropolar

Lubricated Membrane-Compensated Worn Hybrid Journal Bearings,” Tribol. Trans., 55, pp.

59-70.

(47) Lin, J.R., Chou, T.L., Liang, L.J. and Hung, T.C. (2012), “Non-Newtonian Dynamic

Characteristics of Parabolic-Film Slider Bearings: Micropolar Fluid Model,” Tribol. Int. 48,

pp. 226-231.

(48) Sharma, S.C. and Rajput, A.K. (2013), “Effect of Geometric Imperfections of Journal on the

Performance of Micropolar Lubricated 4-Pocket Hybrid Journal Bearing,” Tribol. Int. 60,

pp. 156-168.

(49) Kolpashchikov, V.L., Migun, N.P. and Prokhorenko, P.P. (1983), “Experimental

Determination of Material Micropolar Fluid Constants,” Int. J. Engg. Sci. 21, pp.405-411.

(50) Deryagin, B.V., Zheleznyl, B.V., Zorin, Z.M., Sobolev, V.D. and Churaev, N.V. (1974),

“Surface Layers in Fine Films and Colloids Stability,” (Edited by M. Nauka), pp. 90-94.

(51) Pinkus, O., Sternlicht B. (1961), “Theory of Hydrodynamic Lubrication,” McGraw Hill NY.

(52) Mazumder, S.K. (2003), “Theoretical Analysis of Steady State and Dynamic Performance

including Stability of Finite Flexible Oil Journal Bearings,” Doctoral Thesis 2003, Bengal

Engineering College (A Deemed University), Department of Mechanical Engineering, West

Bengal, India.

(53) Rekach, V.G. (1979), “Manual of theory of elasticity,” MIR publisher Moscow.

(54) Christopherson, D.G. (1941), “A New Mathematical Method for the Solution of Film

Lubrication Problems,” Proc. Inst. Mech. Engrs., London, 146, pp.126-135.

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(55) Pietal, A.K. (2004), “Microchannels flow modelling with the micropolar fluid theory,”

Bulletin of the polish Acad. of Sci. 52, 3, pp. 209-214.

(56) N. P. Migoun (1996), “Kapillarnaja wiskozimietrija dla mikroobjemow żydkostiej”, in

Fiziczeskaja Mietieorołogija, Nauka, Sankt Pietierburg.

(57) N. P. Migoun and P. P. Prochorenko (1984), “Gidrodinamika i Tiepłoobmien Gradientnych

Tieczenij Mikrostrukturnoj Żydkosti,” Nauka i Tiechnołogija, Minsk.

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Nomenclature

a, b inside and outside radius of bearing

liner (m) t, t

U time (s), t t

velocity of the journal (m/s), U = ωR

C radial clearance (m) Ω whirl ratio

C' 2+λ/μ W0, 0W steady state load (N), 2 3

0 0W 2W C R L

D journal diameter (m) α constant describing the fluid-wall

interaction

Dij, ijD damping coefficient of micropolar

fluid for i = R, and j = R, (Ns/m), 3 3

ij ijD 2D C R L

x, y, z

θ, y , z

circumferential, radial, axial

coordinates (m)

dimensionless coordinates, θ=x/R,

dm,n distortion of m, n harmonic y y a , z z /(L / 2)

E Young’s modulus (N/m2) u', v', w' radial, circumferential and axial

e, ε steady state eccentricity at mid-plane displacements

of bearing (z=0) (m),ε = e/C η viscosity of Newtonian fluid (Pa s)

F deformation factor, 3 3F R C E γ, χ viscosity coefficient for the micropolar

Fi force component along the R and

directions for i = R and (N)

τij

fluid (Pa s)

stress in ith plane in jth direction (N/m2)

H = b-a

h, h

thickness of bearing liner (m)

local film thickness (m), h h C

δ, radial deformation at bearing liner

surface (m), C

Λ

ν

characteristic length of the micropolar

fluid (m)

Poisson’s ratio

ω

ωp

angular velocity of the journal (rad/s)

angular velocity of orbital motion of

the centre (rad/s)

lm non-dimensional characteristic length

of micropolar fluid (m), lm=C/Λ

λ, μ

Lame’s constant

attitude angle (rad)

L bearing length (m) θ angular coordinate of the bearing (rad)

m, n

M, M

CM

axial and circumferential harmonic

mass parameter (kg), 2

0M MC W

critical value of non-dimensional mass

θ1, θ2 angles of end and start of

hydrodynamic film at each axial plane

of bearing (rad)

N

p, p

parameter

coupling number

local film pressure (Pa), 2 2p pC R

¯

S

above a variable represents the non-

dimensional value of parameter

Sommerfeld number

R

Sij, ijS

journal radius (m)

stiffness coefficient of micropolar

0

subscript represents the steady state

value

fluid for i = R, and j = R, (N/m), 3 3

ij ijS 2S C R L

1 and 2 subscript represents the first order

perturbation along ε0 and directions

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1. Solid Housing 2. Flexible Liner 3. Journal 4. Lines of centers

Fig. 1 Schematic diagram of Journal Bearing

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Fig. 2 Variation of critical mass parameter versus F taking Newtonian fluid as lubricant

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Fig. 3 Variation of critical mass parameter as a function of lm for Rigid Bearing taking

micropolar fluid as lubricant for various values of N2 at ε0 = 0.5 and L/D = 1.0

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Fig. 4 Variation of critical mass parameter as a function of Sommerfeld number for various

values of F at L/D =1.0, ν=0.4, lm=40.0, N2=0.5, H/R=0.3

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Fig. 5 Variation of critical mass parameter as a function of Sommerfeld number for various

values of lm at L/D=1.0, ν=0.4, F=0.2, H/R=0.3, N2=0.5

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Fig. 6 Variation of critical mass parameter with respect to lm for various values of F at L/D=1.0,

ν=0.4, H/R=0.3, ε0=0.5, N2=0.5

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Fig. 7 Variation of critical mass parameter versus F for various values of ε0 at L/D=1.0, ν=0.4,

H/R=0.3, lm=40.0, N2=0.5

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Fig. 8 Variation of whirl ratio versus lm for various values of F at L/D=1.0, ν=0.4, H/R=0.3,

ε0=0.5, N2=0.5

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Fig. 9 Variation of whirl ratio with F for various values of ε0 at L/D=1.0, ν=0.4, H/R=0.3,

lm=40.0, N2=0.5

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Table 1 Values of parameters Λ and N of some fluids

Fluid Λ, m. N

α = 0 α = 0.9 α = 0 α = 0.9

P1 3.34095 x 10−9 5.79745 x 10−9 0.50782 0.88921

P2 6.1088 x 10−9 1.022 x 10−8 0.53432 0.89443

water 9.2614 x 10−9 1.3392 x 10−8 0.6483 0.93744

E1 1.2713 x 10−5 1.867 x 10−5 0.63565 0.93352

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