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268 Journal of Marine Science and Technology, Vol. 18, No. 2, pp. 268-276 (2010)

DYNAMIC CHARACTERISTICS FOR

MAGNETO-HYDRODYNAMIC WIDE SLIDER

BEARINGS WITH AN EXPONENTIAL FILM

PROFILE

Jaw-Ren Lin* and Rong-Fang Lu*

Key words: magneto-hydrodynamic characteristics,exponential film,

slider bearings, stiffness coefficients, damping coeffi-

cients.

ABSTRACT

On the basis of the magneto-hydrodynamic (MHD) thin-

film lubrication theory, the dynamic characteristics of wide

exponential-shaped slider bearings with an electrically con-

ducting fluid in the presence of a transverse magnetic field are

theoretically investigated. Taking into account the transient

squeezing motion, the MHD dynamic Reynolds-type equation

is derived from the continuity equation and the MHD motion

equations. A closed-form solution for the steady film pressure

and load-carrying capacity and the dynamic stiffness and

damping coefficients are obtained. From the results obtained,

the presence of externally applied magnetic fields signifies an

enhancement in the film pressure. On the whole, the applied

magnetic-field effects characterized by the Hartmann number

provide a significant increase in values of the load-carrying

capacity, the stiffness coefficient and the damping coeffi-

cient as compared to the non-conducting-lubricant (NCL)

case. These improvements of bearing dynamics are more pro-

nounced with increasing Hartmann numbers and decreasing

minimum film thicknesses. To illustrate the use of the presentstudy, a design example is guided. For engineering application,

numerical results are further provided in the Tables.

I. INTRODUCTION

Slider bearings are often designed to support the axial-

component thrust of a rotating shaft. Traditionally, studies of

slider bearings with various film shapes focus upon the

mechanism lubricated with a non-conducting viscous fluid,

such as the studies by Pinkus and Sternlicht [13],Fuller [1],

Williams [21], Hamrock [2], Taylor and Dowson [20], Lin and

Hung [8] and Lin et al. [9]. Further analyses have been pre-

sented considering the temperature variation of fluid film by

Rodkiewicz and Anar [14] and considering the thermal effects

by Talmage and Carpino [19]. To prevent undesirable viscos-

ity change with temperature, the use of an electrically con-

ducting liquid-metal fluid in the presence of externally applied

magnetic fields has been emphasized. These kinds of lubri-

cants possess the high thermal-conductivity and high electri-

cal-conductivity features. Consequently, the heat source gen-

erating from the bearings can be readily conducted away. More-

over, the flow of an electrically conducting liquid-metal fluid

across an externally applied magnetic field will induce an

electrical-field intensity and result in a current density, a Lor-

entz force is then produced acting upon the fluid film. As a

result, the bearing characteristics are therefore improved.

Many authors have investigated the magneto-hydrodynamic

(MHD) performance of bearings lubricated with an electri-

cally conducting fluid in the presence of externally magnetic

fields. A representatively experimental and theoretical re-

search is observed in the MHD hydrostatic bearings by Maki

and Kuzma [11]. Further theoretical studies are found in the

MHD squeeze-film bearings by Shukla [16], Lin et al. [10]

and Hsu et al. [2]; the MHD journal bearings by Kuzma [6],Kamiyama [5] and Malik and Singh [12]; and the MHD slider

bearings by Snyder [18], Hughes [4], Rodkiewicz and Anwar

[15] and Lin [8]. However, the studies of MHD slider bearings

[18, 4, 15, 7] are limited to the steady characteristics in which

the transient squeezing effects are neglected. Recently, Lin

and Hung [9] have analyzed the dynamic characteristics for a

wide exponential film-shaped slider bearing under the

non-conducting-lubricant (NCL) case. To provide more in-

formation for bearing designing and selection, we are moti-

vated to investigate the MHD dynamic characteristics of ex-

ponential film-shaped slider bearings.

On the basis of the MHD thin-film lubrication theory, the

dynamic characteristics of a wide exponential-shaped slider bear-

ing with an electrically conducting fluid in the presence of a

transverse magnetic field are theoretically investigated. Tak-

Paper submitted 10/03/08; revised 06/06/09; accepted 06/08/09. Author for

correspondence: Jaw-Ren Li (e-mail: [email protected]; [email protected]).

*Department of Mechanical Engineering, Nanya Institute of Technology, P. O.

Box 324-22-59, Jhongli, Taiwan, R.O.C.

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J.-R. Lin and R.-F. Lu: Dynamic Characteristics for MHD Wide Slider Bearings with an Exponential Film Profile 269

ing account of the transient squeezing motion, the MHD dy-

namic Reynolds-type equation is derived from the continuity

equation and the MHD momentum equations. A closed-formsolution for the pressure is derived and applied to evaluate the

MHD bearing characteristics. Comparing with the NCL case,

the MHD characteristics (including the load capacity, the

stiffness coefficient and the damping coefficient) are presented

for different values of the Hartmann number, the profile pa-

rameter and the minimum film thickness.

II. ANALYSIS

Figure 1 shows the physical geometry of a wide slider

bearing of lengthL. The bearing has a sliding velocity Uand a

squeezing velocityh/tin thexdirection and thezdirection

respectively, in which the exponential-film thickness is de-

scribed by:

( , ) ( ) exp( ln )mx

h x t h t r L

= (1)

In the equation, hm(t) denotes the minimum film thickness

at the outlet and r = h1(t)/hm(t) = [d + hm(t)]/hm(t) is the

inlet-outlet film ratio, where d is the shoulder height repre-

senting the difference between inlet height and outlet height.

The lubricant is taken to be an incompressible isothermal

electrically-conducting fluid with electrical conductivity .An externally uniform transverse magnetic fieldB0 is appliedin thezdirection. It is assumed that the thin-film lubrication

theory by Pinkus and Sternlicht [13] and Hamrock [2] is ap-

plicable, but the Lorentz force is considered, the induced

magnetic field is negligible as compared to the applied mag-

netic field. Under these assumptions, the continuity equation

and the MHD momentum equations in thex- andz-directions

are given by

0u w

x z

+ = (2)

2

0 02( )y

p uB E uB

x z

= + (3)

0p

z

= (4)

The no-slip boundary conditions at the bearing surfaces are:

u = U, w = 0 at z = 0 (5)

u = 0, w = h/t at z = h (6)

Solving (3) together with the boundary conditions, the x-

component velocity is obtained.

L

U

x

z

h hm

h1

B0

h___

t

Fig. 1. Physical geometry of a wide slider bearing with an exponent

tial-film profile lubricated with an electrically conducting fluid.

0 0 0

cosh coth sinhm m m

M M M u U z h z

h h h

=

2

002

my

h pB E

xM

0 0 0

cosh tanh 0.5 sinh 1m m m

M M M z h z

h h h

(7)

where hm0 denotes the steady minimum film thickness at the

exit andMrepresents the Hartmann number defined by

( )1/ 2

0 0/mM B h = (8)

In the present study, the bearing surfaces are assumed to be

perfect insulators and there is no external circuit to the fluid,

the electric field is then approximated by requiring the netcurrent flow to be zero.

( )00

0h

yz

B u E dz=

+ = (9)

Combining the two equations (7) and (9), the expression of

u is re-written as:

0 0

0

sinh( / ) sinh[ ( ) / ]1

2 sinh( / )

m m

m

Mz h M h z hUu

Mh h

=

2

0 0 0 0

0

sinh( / ) sinh( / ) sinh[ ( ) / ]2 cosh( / ) 1

m m m m

m

h h Mh h Mz h M h z hpM x Mh h

(10)

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270 Journal of Marine Science and Technology, Vol. 18, No. 2 (2010)

Integrate the continuity equation (2) with respect toz across

the film thickness.

0 0

h h

z z

u wdz dz

x z= =

=

(11)

Performing the integration together with the boundary con-

ditions (5) and (6), the MHD dynamic Reynolds-type equation

is derived.

( , ) 6 12p h h

A h M U x x x t

= +

(12)

where

2

0 20 0

( , ) 6 coth(0.5 ) 2mm m

h Mh MhA h M h

h hM

=

(13)

Expressed in a non-dimensional form, the non-dimensional

MHD dynamic Reynolds-type equation is written as:

* * ** *

* * * *( , ) 6 12

p h hA h M

x x x t

= +

(14)

where

** * * *

2

6( , ) coth(0.5 ) 2

hA h M Mh Mh

M = (15)

( )* * * * * *( , ) ( ) exp lnmh x t h t x r = (16)

Since the linear dynamic characteristics are evaluated for

the bearing operating under small disturbances about its steady

state, we assume that

* * *

( ) 1 ( ),mh t t= + 1

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where FA and FC are evaluated from the following double

integrals.

( )

( )

*

* * *

* *1 1

* * * * * *

* *0 0 0( ) ( , )

,

x e

A m A mx x x

h xF h f x h dx dx dx

A h M

= = == =

(26a)

( )

*

* * *

1 1* * * * * *

* *0 0 0

1( ) ( , )

,

x

C m C mx x x

F h f x h dx dx dxA h M

= = == =

(26b)

III. MHD STEADY AND DYNAMIC

CHARACTERISTICSBy letting the minimum film height be constant and the

squeezing velocity be zero, both of the steady film pressure

and the steady load-carrying capacity are evaluated from (20)

and (25), respectively.

( ) ( )* * * * * *0 1 00 006 ( , ) ( , )m A m C mp h f x h c f x h = + (27)

( ) ( ) ( ) ( ){ }* * * * *0 1 00 00 01 6 ( ) ( )m A m A mW F h F h c F h = = + (28)

where the subscript 0 denotes the steady state. The linear dy-

namic stiffness coefficient can be obtained by evaluating the

partial derivative of film force with respect to the minimum

film thickness, and then taking the value under steady state.

( ) ( )*

* *

* *0 00 0

6 6 Ac A mm m

FFS F h

h h

= = +

( ) ( )1 1* *000 0

CC

m m

FcF c

h h

+

(29)

Where

*

* *

* *1

* *

* *2 * *0 00 0

( , )

xeA

x xm m

hF Adx dx

h A h M h

= =

=

(30a)

*

* *

*1

* *

* *2 * *0 00 0

1

( , )

xC

x xm m

F Adx dx

h A h M h

= =

=

(30b)

1

*

0m

c

h

=

* 111 1 12 * *

1 0 0

6[( ) ]

CMAMCM AM m AM

CM m m

fff f h f

f h h

+

(30c)

*

* * *1

*1

* *2 * *0

( )

( , )

eAM

x

m m

h xf Adx

h A M h h

=

=

(30d)

*

*1

*1

* *2 * *0

1

( , )

CM

xm m

f Adx

h A M h h

=

=

(30e)

*** * 2 *2 2 *

* 2

34 coth(0.5 ) csc (0.5 ) 4e

m

hAMh Mh M h h Mh

h M

=

(30f)

Similarly, the linear dynamic damping coefficient can be

obtained by evaluating the partial derivative of film force withrespect to the squeezing velocity, and then taking the value

under steady state.

( )( )

( )

( )( )

*1* 0

* 0 010 0

12

ln 1

AM

c B C

CM

fFD F F

fV

= =

+

(31)

The steady performance and the dynamic characteristics

involving the integrals can be numerically evaluated by the

method of Gaussian Quadrature.

IV. RESULTS AND DISCUSSION

By the definition in (8), the Hartmann number M charac-

terizes the influence of an externally applied magnetic field on

the MHD characteristics of bearings in the presence of an

electrically conducting fluid. When the value ofMis equal to

zero, the MHD dynamic Reynolds-type equation reduces to

the non-conducting-lubricant (NCL) case by Lin and Hung [8].

With the aid of (18), the profile parameter denotes the wedge

effect of an exponential-shaped slider bearing. In the present

analysis, MHD bearing characteristics are presented for the

profile parameter, the steady minimum film height and the

Hartmann number with the following values:

0.2 ~ 2.8, = *0 0.4 ~ 1,mh = 0 ~10.M =

MHD Film Pressure. Figure 2 presents the non-dimensional

steady film pressure as a function of non-dimensional coor-

dinate for different values of the Hartmann number Munder

fixed values of the profile parameter and the steady minimum

film height. Comparing with the NCL case, the effects of the

applied magnetic field (M= 2) are observed to increase the

MHD steady film pressure. Further increments of the film

pressure are obtained with increasing values of the Hartmannnumber (M= 4, 6, 8, 10).

Figure 3 shows the non-dimensional steady maximum film

pressure as a function of the non-dimensional minimum film

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x*

0.1

0.3

0.5

0.7

0.9

-1 -0.8 -0.6 -0.4 -0.2 0

0

0.2

0.4

0.6

0.8

1

NCL

M = 2

M = 4

M = 6

M = 8

M = 10

p0*

= 1, hm0* = 1

Fig. 2. MHD steady film pressure *

0p as a function ofx* for differentM

under = 1 and *m0

1.h =

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

0

0.4

0.8

1.2

1.6

2

2.4

NCL

M = 2

M = 6

M = 10

M = 4

M = 8

pM0

*

hm0*

= 1

Fig. 3. MHD steady maximum film pressure *M0p as a function of

*

m0h

for differentMunder = 1.

height for different values of the Hartmann number under

profile parameter = 1. It is observed that decreasing the

minimum film height increases the steady maximum film

pressure. Comparing with the NCL case, the effects of ex-

ternally magnetic fields provide higher values of the steady

maximum film pressure for the bearing with smaller film

heights and larger Hartmann numbers.

Figure 4 depicts the non-dimensional steady maximum film

pressure as a function of the profile parameter for different

0.1

0.3

0.5

0.7

0.9

0

0.2

0.4

0.6

0.8

1

NCL

M = 2

M = 6

M = 10

M = 4

M = 8

pM0

*

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

hm0* = 1

Fig. 4. MHD steady maximum film pressure *M0

p as a function of for

differentMunder *m0

1.h =

M

0.2

0.6

1

1.4

0 1 2 3 4 5 6 7 8 9 10

0

0.4

0.8

1.2

1.6hm0

* = 0.4

hm0* = 0.6

hm0* = 0.8

hm0* = 1.0

W0

*

= 1

Fig. 5. MHD steady load-carrying capacity *0W as a function ofM for

different *m0

h under = 1.

values ofM. Comparing with the NCL case, the effects of the

applied magnetic field (M= 2) for the bearing with an elec-

trically conducting fluid provide a higher maximum film pres-

sure. Moreover, larger increments are obtained for the bearing

with a larger profile parameter ( = 2.8) and a larger Hartmann

number (M= 10).

MHD Load Capacity. Figure 5 presents the non-dimensionalsteady load-carrying capacity as a function of the Hartmann

number for different values of the non-dimensional minimum

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0.1

0.3

0.5

0

0.2

0.4

0.6

NCL

M = 2

M = 4

M = 6

M = 8

M = 10

W0

*

0 0.2 0.4 0 .6 0.8 1 1.2 1.4 1.6 1 .8 2 2.2 2.4 2 .6 2.8 3

hm0* = 1

Fig. 6. MHD steady load-carrying capacity *

0W as a function of for

differentMunder *m0 1.h =

film height under profile parameter = 1. Since the effects of

applied magnetic fields result in a higher film pressure, the

integrated load-carrying capacity is similarly affected. In-

creasing the Hartmann number increases the load capacity. It

is also observed that the increments of load capacity for the

MHD bearing are more accentuated with a smaller squeezing

film height. Figure 6 shows the non-dimensional steady

load-carrying capacity with profile parameter for different val-

ues ofM. Comparing with the NCL case, the effects of the

applied magnetic field (M= 2) are observed to increase the

MHD load-carrying capacity. Further increments of the load-

carrying capacity are obtained with increasing values of the

Hartmann number (M= 4, 6, 8, 10).

MHD Stiffness Characteristics. Figure 7 presents the non-

dimensional stiffness coefficient as a function of the Hartmann

number M for different values of the non-dimensional mini-

mum film height under profile parameter = 1. Bearingstiffness coefficients are observed to increase slightly with

increasing Hartmann numbers. Decreasing the squeezing film

height increases the values of the stiffness coefficients. Totally,

the increments in values of the stiffness coefficients are more

pronounced for MHD bearings with a smaller squeezing-film

height and a larger Hartmann number. Figure 8 shows the

non-dimensional stiffness coefficient as a function of the pro-

file parameter for different values ofM. Under the NCL case,

bearing stiffness is observed to increase with until a critical

value is reached, and thereafter falls as the profile parameter

continues to increase. However, the effects of applied mag-

netic fields (M

= 2, 4, 6, 8, 10) shift the position of profileparameter to obtain the maximum stiffness coefficient at a

larger for MHD bearings. Comparing with the NCL bearing,

higher stiffness coefficients are obtained for the MHD bearing

M

0.5

1

1.5

2.5

3

3.5

4.5

5

5.5

0

2

4

6

hm0* = 1.0

hm0* = 0.8

hm0* = 0.4

hm0* = 0.6

Sc

*

0 1 2 3 4 5 6 7 8 9 10

= 1

Fig. 7. MHD dynamic stiffness coefficient *cS as a function ofM for

different *m0h under = 1.

0.1

0.3

0.5

0.7

0

0.2

0.4

0.6

NCL

M = 2

M = 4

M = 6

M = 8

M = 10

Sc*

hm0* = 1

0 0.2 0.4 0 .6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Fig. 8. MHD dynamic stiffness coefficient *cS as a function of for

differentMunder*

m01.h =

with the application of magnetic fields. Generally speaking,

the MHD bearing provides a significant improvement in dy-

namic stiffness characteristics, especially for a larger value of

the Hartmann number and the profile parameter.

MHD Damping Characteristics. Figure 9 presents the

non-dimensional damping coefficient as a function of the Hart-mann number Mfor different values of the non-dimensional

minimum film height under profile parameter = 1. Damping

coefficients are observed to increase with increasing values of

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1

2

3

5

7

6

9

10

11

0

4

8

12

hm0* = 1.0

hm0* = 0.8

hm0* = 0.4

hm0* = 0.6

Dc

*

M

0 1 2 3 4 5 6 7 8 9 10

= 1

Fig. 9. MHD dynamic damping coefficient *cD as a function ofM for

different *m0

h under = 1.

the Hartmann numbers. Decreasing the squeezing film height

increases the values of the damping coefficients. It is observed

that the increments in values of the damping coefficients are

more pronounced for MHD bearings operating with smaller

squeezing-film heights and larger Hartmann numbers.

Figure 10 shows the non-dimensional damping coefficient

as a function of the profile parameter for different values of

M. Dynamic damping coefficients are observed to decrease

with increasing values of the profile parameter. Comparing

with the NCL case, the MHD bearing signifies an increase in

values of the damping coefficients by the externally applied

magnetic fields. On the whole, the effects of applied magnetic

fields on the dynamic damping characteristics are more pro-

nounced for the MHD bearing operating at a larger Hartmann

number and a smaller profile parameter.

Design Example and Comparison. To illustrate the use of

the present study for engineering application, a design exam-ple of the exponential-shaped slider bearing lubricated with an

electrically conducting fluid is guided in Table 1. From the

physical quantities given, the profile parameter and the Hart-

mann number are obtained:

= 0.5, 1.0, 1.5, 2.0, 2.5;

M= 0, 1.97, 3.94, 5.91, 7.88, 9.85

With the aid of (27), (28) and (30), the steady load-carrying

capacity and the dynamic stiffness and coefficients of the

MHD bearing are presented in Table 2.

It is useful to compare the present results with previouscontributions. If we neglect the effects of externally applied

magnetic fields, the dynamic characteristics of an exponen-

tial-shaped slider bearing under the NCL case by Lin and

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

NCLM = 2

M = 4

M = 6

M = 8

M = 10

Dc

*

hm0* = 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1 .8 2 2.2 2.4 2 .6 2.8 3

Fig. 10. MHD dynamic damping coefficient *cD with for different M

under *0

1.mh =

Table 1. Example of the exponential-shaped slider bear-

ings lubricated with an electrically conducting

fluid.

Exponential-shaped slider-bearing systemSteady minimum filmthickness

hm0 1.00 104m

Difference between theinlet and outlet filmthickness

d (0.5, 1.0, 1.5, 2.0, 2.5) 104m

Lubricant viscosity 1.55 103Pa s

Electrical conductivity 1.07 106mho/m

Magnetic field B0 0, 0.75, 1.5, 2.25, 3, 3.75 Wb/m2

Hung [8] are recovered. In Table 2, the NCL steady load-

carrying capacity and the dynamic stiffness and coefficients byLin and Hung [8] are also included for comparison. Slight

differences between the zero-M and NCL cases, depending

upon , are observed. This small division can be realized since

the printed values from the computer calculation by Lin and

Hung [8] (the NCL case) have been accurate to two decimal

points and the printed results of the present study (the zero-M

case) have been accurate to four decimal points.

V. CONCLUSIONS

Based upon the MHD thin-film lubrication theory, the dy-

namic characteristics of a wide exponential-shaped slider bear-

ing with an electrically conducting fluid in the presence of a

transverse magnetic field are theoretically analyzed. From the

results and discussion, conclusions can be drawn as follows.

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Table 2. MHD bearing characteristics *0 ,W * ,cS

*cD under

*0 1,mh = and comparison with the NCL case by Lin and

Hung [8].

--- NCL M= 0 M= 1.97 M= 3.94 M= 5.91 M= 7.88 M= 9.85

= 0.5 0.13 0.1320 0.1450 0.1784 0.2224 0.2711 0.3220

= 1.0 0.16 0.1622 0.1854 0.2399 0.3073 0.3797 0.4545

= 1.5 0.16 0.1644 0.1957 0.2638 0.3440 0.4288 0.5157

= 2.0 0.16 0.1582 0.1957 0.2725 0.3599 0.4513 0.5446

*0W

= 2.5 0.15 0.1496 0.1920 0.2742 0.3657 0.4606 0.5571

= 0.5 0.26 0.2639 0.2651 0.2768 0.3037 0.3419 0.3863

= 1.0 0.32 0.3244 0.3279 0.3539 0.4027 0.4640 0.5320

= 1.5 0.33 0.3289 0.3353 0.3750 0.4386 0.5134 0.5940

= 2.0 0.32 0.3164 0.3261 0.3769 0.4500 0.5325 0.6201

*cS

= 2.5 0.30 0.2992 0.3122 0.3714 0.4505 0.5375 0.6289

= 0.5 0.65 0.6509 0.7154 0.8799 1.0968 1.3373 1.5884

= 1.0 0.47 0.4681 0.5349 0.6922 0.8866 1.0957 1.3113

= 1.5 0.36 0.3589 0.4271 0.5758 0.7509 0.9360 1.1256

= 2.0 0.29 0.2880 0.3563 0.4960 0.6552 0.8216 0.9914

*cD

= 2.5 0.24 0.2389 0.3066 0.4378 0.5838 0.7353 0.8894

Taking into account the transient squeezing action, the

MHD dynamic Reynolds-type equation has been derived for

the study of a MHD exponential-shaped slider bearing. A

closed-form solution is obtained for the steady load capacity,

the stiffness coefficient and the damping coefficient. Com-

paring with the NCL case, the MHD exponential-shaped bear-ing provides an increase in the steady load and the dynamic

coefficients. On the whole, the effects of externally applied

magnetic fields on the steady load and the dynamic stiffness

coefficient are more pronounced with larger values of the

Hartmann number and the profile parameter and small values

of the minimum film thickness; but, the improvements of

dynamic damping characteristics are emphasized for bearings

designed at smaller profile parameters. To illustrate the use of

the present study, a design example is guided. Further results

are also provided in Tables for engineering application.

APPENDLX I: NOMENCLATURE

A,A* function defined in equation (12),

A*(h*,M) = 3 0( , ) / mA h M h

B, L width and length of the bearing

B0 applied magnetic field

d difference between the inlet and outlet film thickness

Dc,*

cD damping coefficient,* 3 3

0/c c mD D h L B=

Ey induced electric field in the y-direction

F, F* magneto-hydrodynamic film force,* 2 2 * * *

0/ ( , )m mF Fh UL B F h V = =

h film thickness, ( )( , ) ( ) exp / lnmh x t h t x L r =

*h non-dimensional film thickness,* * * * *

0/ ( ) ( )m m eh h h h t h x=

1h inlet film thickness, 1( ) ( )mh t d h t = +

*

eh non-dimensional function,

( )* * *( ) exp ln 1eh x x = +

,mh *

mh

minimum squeezing film thickness,* *

0( ) ( ) / m m mh t h t h=

0mh steady reference minimum film thickness at the exit

*

0mh non-dimensional steady reference minimum film

thickness, * *0 0( )m mh h=

M Hartmann number, ( )1/ 2

0 0mM B h =

p,p* film pressure, * 20 /mp ph UL=

*

0 0,p p steady film pressure,* 2

0 0 0 /mp p h UL=

r inlet-outlet film ratio, 1( ) / ( )mr h t h t = =

[ ( )] / ( ) 1m md h t h t + +

*,c cS S stiffness coefficient,

* 2 2

0/c c mS S h UL B=

t, t* time, * /t Ut L=

u, w velocity components in thex andz directions

U sliding velocity of the lower part

V* non-dimensional squeezing velocity, * * */mV dh dt =

W0,*

0W steady load-carrying capacity,

* 2 2

0 0 0/mW W h UL B=

x,y,z Cartesian coordinates

x* non-dimensional coordinate, * /x x L=

profile parameter, 0/ md h =

lubricant viscosity

electrical conductivity

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Subscript

0 the steady state

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