Behaviour of a hydromagnetic squeeze film between porous plates

Behaviour of a hydromagnetic squeeze film between porous plates

Wear, 56 (1979) 327 - 339 0 Elsevier Sequoia !%A., Lausanne 327 - Printed in the Netherlands BEHAVIOUR OF A HYDROMAGNETIC POROUS PLATES SQUEEZE F...

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Wear, 56 (1979) 327 - 339 0 Elsevier Sequoia !%A., Lausanne

327 - Printed

in the Netherlands

BEHAVIOUR OF A HYDROMAGNETIC POROUS PLATES

SQUEEZE

FILM BETWEEN

K. C. PATEL and J. L. GUPTA Department of Mathematics, Birla Vishvakarma Mahavidyalaya (Engineering Vallabh Vidyanagar - 388120, Gujarat (India) (Received

May 12, 1978; in final form December

College),

23, 1978)

Summary The effect of a transverse magnetic field on the behaviour of a squeeze film between porous plates of different geometries was studied. The governing equation for the pressure distribution in the film region can be uncoupled and expressed as a Poisson equation by using an approximation which otherwise does not have any significant effect on the bearing performance characteristics. This greatly simplifies the analysis. Expressions for bearing characteristics are presented in analytical form and the effect of the applied magnetic field is shown graphically.

1. Introduction When two porous surfaces approach each other only part of the fluid between them is squeezed out; the remainder flows out through the porous media and thus the bearing capacity decreases and the film thickness attained in a specific time also decreases. Wu [ 1,2] and Prakash and Vij [ 31 analysed and discussed the behaviour of the squeeze film when one surface was porous and backed by an impermeable solid wall. When fluids such as liquid metals are used as lubricants in high temperature bearings the load-supporting capacity of the bearing is decreased because of their highly conducting properties and low viscosity [4] . It has been established [4, 51 that the effect of an applied magnetic field on porous and non-porous metal bearings is to improve their performance considerably. Reynolds equation for porous bearings is coupled [2,5] but it can be uncoupled by using an approximation due to Morgan and Cameron [6] for thin porous facings which greatly simplifies the analysis 131. The effect of a uniform transverse magnetic field on the behaviour of a squeeze film between porous plates of different geometries has been investigated using the Morgan-Cameron approximation.

328

2. Analysis Consider a fluid film of thickness h between two parallel plates. The lower plate with the porous facing is assumed to be fixed while the upper plate moves along its normal towards the lower plate (Fig. 1). The fluid is electrically conducting and the plates are non-conducting. A uniform magnetic field 3, is applied perpendicular to the two plates. The flow in the porous medium obeys the modified form of Darcy’s law while in the film region the equations of hydroma~etic lubrication theory hold. Following the usual assumptions of hydromagnetic lubrication, the basic equations governing the hydroma~etic flow of the lubricant in the fluid are

SOLlO FACIHG PORGL!5

PEG\OU

SOLI D

FACING

ClRCULAR

RECTANGULAR

%OL\DFACIEIG POROUS

PEGIOH

SOLID

FACIHG

W

ANNULAR

Fig. 1. Geometries and coordinates of the system.

a2u

M8

ay*

hi

_--___~u=--

a2u w ----~=-~ ay2

h:

1 fi

aP

1

ap

P

az

ax

(1)

(2)

329

ap -= ay

0

(3)

_+au+%o au

ax

a2

(4)

ay

and in the porous matrix are ii=---

Kap

p E;=---

(5)

ax 1 + kM2/m

K a3

1 (6)

~1 az 1 + KM2/m

w=--K P

au

1

aj

(7)

ay

av

aiij

(8)

ax+az+-=O ay

where M = M,/h, and M, = B, h, (o/P)~‘~ is the Hartmann number. Using no-slip conditions at the plates the solutions of eqns. (1) and (2) are u=

;

II= E

coshM,y-lcoshMy-l-

cash Mh - 1 sinh Mh

sinh My

(10)

sinh Mh

a2p a22

= -

(9)

p

cash Mh - 1

Substituting eqns. (9) and (10) in eqn. (4) and integrating thickness h gives w,, -w,,

1

across the film

Mh - 2 tanh $

Since the upper plate is non-porous wh = 0. The velocity component in the y direction is continuous face between the lower porous plate and the film so that

(11) at the inter-

(12) Hence eqn. (11) becomes CP + -= a2p ax2

az2

dhldt + (k/p) (@lay) 1y=o (1/pM3) (Mh - 2 tanh (Mh/2)}

(13)

330

From eqns. (5) - (8) the fluid pressure in the porous region satisfies the equation

Pji

a2fi

ax2

a2P W

+s+c-= 2

0

(14)

where (15) and $a = kH/h;:

(16)

Since H is very small we can use the Morgan-Cameron eqn. (14) yields -ab ay

=-Y=O

approximation

a%+CP

H

-

c2 ( ax2

and

(17)

az21

where ap/lay = 0 at y = -H has been used. Substitution of eqn. (17) in eqn. (13) gives

dh/dt

6 _+azp=

ax2

az2

(l/pM3){Mh

- 2 tanh (Mh/2)}

Hence the problem reduces to the solution boundary conditions. 2.1. Annular

+ $+,hi/pC2

(18)

of eqn. (18) with the appropriate

discs

The flow is axisymmetric

and hence eqn. (18) reduces to dh/d t

(l/pM3)

{Mh - 2 tanh (Mh/2)}

When eqn. (19) is solved with the boundary

+ &, hi//X2

(19)

conditions

p(a) = p(b) = 0 the expressions for the pressure distribution ity in dimensionless form are

and the load-supporting

capac-

h;p

p =-

p(dh/dt)(a2

- b2)n 1

= 4n [(l/M;){Mh

x

log (a/b)

tanh(Mh/2)}

(r/b)2 - 1

log 0-D)

I

- 2

-

(a/b)2 - 1 I

+ $o/C2]



(21)

331

Wh;

@= -

r.l(dh/dt)(a2

- b2)2n2 1

= 8~[(1/M~){Mh

- 2 tanh (Mh/2)} + tio/C2] ’

(a/l?)2+ 1

x

1 (a/b)2 -

1

1 -

(22)

log (a/b) i

For constant load W the film thickness and the time relation obtained by integrating eqn. (22) which yields

t,/to

AT =

s

Wh;dt

I

/OTa(as - b2)2

1

can be

(23)

where dh

“1 Iho

s

I=-

(l/M;)

1

{Mh - 2 tanh (Mh/2)} + Jlo/C2

(24)

In eqn. (24) the integral on the right-hand side cannot be solved analytically and hence numerical integration is necessary. However, because the results for small values of Me are of little practical interest only large values of Me are considered and eqn. (24) reduces to I=

-M;

M,i

log

MO

- 2 + &M;/C2 -

(25)

2 + JIoM;lC2

where h = h,/h,

(26)

2.2. Circular plates For circular plates the differential

equation

is

dh/d t (~//.LM~) {Mh -- 2 tanh (Mh/2)}+ $oh:/@2 and the boundary p(r) = 0

conditions and

aplar

(27)

when r = 0 are = 0

Solving eqn. (27) using the boundary tribution in dimensionless form:

(28)

conditions

(28) gives the pre&ufe &a-

332

phi

js=-

p(dh/dt)na2 1 - (r/a)2 = 4n [(l/M:) {Mh - 2 tanh (Mh/2)} + tio/c2] and the load capacity

in dimensionless

(29)

form is

Wh; ff7=-._-_ p(dh/dt)n2a4 1 = 8n [(l/M$){Mh The time-height

AT=

- 2 tanh (Mh/2)} + Go/C2 ]

relation

tl’toWhidt s

is

1 (31)

/.lnw -=E?

1

(30)

where a is the radius of the plate and I is given by eqn. (25). 2.3. Elliptical plates The differential equation

a2p

a2p

ax2

az2

-+_=

and the boundary 4hZl)

is dhld t

(1/pM3){Mh - 2 tanh (Mh/2)} + tioh$/pC2 conditions

(32)

are

= 0

where

xf

2;

a”+bz=l

(33)

2.3.1. Pressure i; =-

ph: p(dh/dt)nab

(34)

333

2.3.2. Load capacity jjT=_

Wh; p(dh/dt)n2a2b2 Mh

(35)

2.3.3. Response time

(36)

where 2a and 2b are the major and minor axes of the elliptical given by eqn. (25). 2.4. Triangular plates The governing differential dition p(xl, zl) = 0 where (Xl -4(x1

--J221

equation

+2a)(x1

+

plate and I is

is eqn. (32) with the boundary

4321

+2c)=

and a is the length of each side of the equilateral

0 triangle whose equation

(x - a)(x - 432 + 2a)(x + 432 + Zc) = 0 The point of intersection origin.

con(37) is (33)

of the medians of the triangle is selected as the

2.4.1. Pressure jj =-

ph: p(dh/dt)3d3a2 Mh-2tanh

Mh 2

(39) 2.4.2. Load capacity gi=-

Whg 27p(dh/dt)a4

334

J3

1

M$

=-

-1

Mh - 2 tanh 60 1$j

(40)

i

24.3. Response time AT=

tl’to Wh;dt _=2Qa4

d/3 I

s

(41)

60

1

where I is given by eqn. (25).

2.5. Ret tang&r plates Solving eqn. (32) with the boundary

conditions

=0

(42)

p(x, k b/2) = 0

(43)

p(*a/2,2)

gives the pressure distribution

as

phi

p=-

p(dh/dt)ab =

& [i - (g t

n=O

n cash {(2n + l)(zx/b)}

+_$ p) x

(2n + 1)3 cash ((2n + l)(na/2b)}

Mh-2tanh

The dimensionless

cash {(2n f l)(nz/b))

1 x

-1

Mh

(44)

2

form of the load-supporting

capacity

is

Wh;

jjY=_

p(dh/dt)a2b2 m tanh ((2n + l)(na/2b)) x (2n + 1)’ n “co

n4 a =_--12 b

16

X [n4(;i2

The time-height

&(Mh-2tanh

relation

is

f)

x

+$I]-’

(45)

335

‘l”o Wh;dt

AT=

-

s

pa2b2

1

=

1

n4 a ---n4(a/b)’ [ 12 b

m tanh ((2n + l)(na/2b)} L: (2n + 1)5 71 n=O

16

1 I

(46)

where a and b are the sides of the rectangular plate and I is given by eqn. (25).

3. Results and discussion For all geometries considered the general formula for the load is w=-

12c((dh/dt)A= ha sJ

where A is the characteristic area and J is a parameter characterizing the hydromagnetic effects and is defined as Mh-2tanh

Mh 2

which in the limits as M. + 0 becomes (1 + 12$)-l where $ = $o(ho/h)3. The shape factors for different geometries are given in Table 1. TABLE 1 Bearing geometry

Characteristic area A

Annular

(a2 - b2)n

Shape factor S

Circular

Elliptic

nab

1 a/b 4n (a/b)= + 1

Triangular

3J3a=

J3/6Q

Rectangular

ob

1

lr4 (1 --12 b

n4(a/b)=

--

16 n

5 n=O

tanh b;W;W2b)} I

336

The time-height formula At=

relation

for all the geometries

considered

has the general

PA2 IS Wh;

where I is given by eqn. (25). Thus in the limit as M, + 0 the results of Prakash and Vij [3] are confirmed by the present results. To compare the present analysis using the Morgan-Cameron approximation for small H with that of Sinha and Gupta [ 51 the results for the time taken to attain a specified film thickness for the case of rectangular plates are given in Table 2. The difference between the two sets of results is less than 0.04% which indicates that the approximation used is not likely to cause any significant error for most bearing applications. TABLE 2 Time required to reduce the film thickness from ho to hl for a rectangular plate

0.0001 0.0010 0.0100 0.1000 1 .oooo

Sinha and Gupta’s analysis

Present analysis

Percentage difference

17.262696 8.536220 2.208932 1.411020 1.026716

17.268447 8.537181 2.209201 1.141405 1.027112

0.033 0.011 0.012 0.027 0.038

MO = 30r-n hl/hu = 0.5, a/b = 2, m = 0.6, H/b = 0.01

Values of the dimensionless load capacity cii and the time AT taken to attain a specified film thickness for different values of the Hartmann number MO are presented in Tables 3_and 4 respectively. The effect of an applied magnetic field is to increase W and AT substantially. There is a further increase as the intensity of the applied magnetic field is increased. Figure 2 and Table 5 show that the effect of permeability is to decrease wand AT with increasing 1~ for all the geometries considered. The effect of plate shape on the time-height relation is shown in Fig. 3. The maximum transient time is obtamed with triangular plates. The time to squeeze out all the fluid in the present case is not infinite as is the case for non-porous bearings.

Nomenclature a

b h h0

hl H

outer radius of the disc inner radius of the disc film thickness initial film thickness film thickness after time At thickness of the porous facing

337 TABLE 3 Values of lfor

MO 10 20 30 40 50

h= 0.5 and J/ = 0.0001

Geometry Circular

Annular

Elliptic

Triangular

Rectangular

12.7255 36.1818 68.5143 104.7254 141.1153

2.8502 8.1037 15.3452 23.4555 31.6058

10.1804 28.9454 54.8114 83.7804 112.8922

249.2807 708.7656 1342.1267 2051.4680 2764.3110

9.1423 25.9938 49.2222 75.2371 101.3805

TABLE 4 Values of AT when E= 0.5 and $ = 0.0001

MO 10 20 30 40 50

Geometry Circular

Annular

Elliptic

Triangular

Rectangular

3.8217 12.0841 24.0336 38.2109 53.3266

0.8559 2.7065 5.3835 8.5582 11.9436

3.0574 9.6672 19.2292 30.5688 42.6612

74.8620 236.7154 470.8535 748.5141 1044.6158

0.9804 3.1001 6.1665 9.8029 13.6808

TABLE 5 Values of

J/

AT for MO = 20

and h = 0.5

Geometry Circular

Annular

Elliptic

Triangular

Rectangular

0.0001 0.001 0.01 0.1 1.0

12.0841 7.8271 2.1836 0.8056 0.6470

2.7065 1.7531 0.4891 0.1804 0.1449

9.6672 6.2618 1.7470 0.6444 0.5176

236.7154 153.3260 42.7752 15.7797 12.6750

3.1001 2.0080 0.5602 0.2067 0.1660

K

permeability of the porous material porosity pressure in the film region pressure in the porous region dimensionless pressure radial coordinate time time required for the film thickness to decrease to a value hl

m P

d P r fit

338

3!

3c

2.f

c-G 7-C

If

4CI-

5

C

ow

01

‘001

Fig. 2. Dimensionless load w us. $ for different geometries.

/-

ANNULAR

MO.20 y

= o-01

Q=OO\

15

5

20

At

Fig. 3. Dimensionless time AT us. dimensionless film thickness 6 for different geometries.

339

AT u, u, w u, 5, w w0, Wh ip x, Y, a I-( u J/O

dimensionless time velocity components of the fluid in the film region velocity components of the fluid in the porous region values of w at y = 0 and y = h respectively dimensionless load capacity space coordinates viscosity of the fluid conductivity of the fluid KH/hi

References 1 H. Wu, Squeeze film behaviour for porous annular disks, J. Lubr. Technol., 62 (1970) 593 - 596. 2 H. Wu, An analysis of the squeeze film between porous rectangular plates, J. Lubr. Technol., 94 (1972) 64 - 68. 3 J. Prakash and S. K. Vij, Load capacity and time-height relations for squeeze films between porous plates, Wear, 24 (1973) 309 - 322. 4 R. A. Elco and W. F. Hughes, Magnetohydrodynamic pressurization in liquid metal lubrication, Wear, 5 (1962) 198. 5 P. C. Sinha and J. L. Gupta, Hydromagnetic squeeze films between porous annular disks, J. Math. Phys. Ski., 8 (5) (1974) 413 - 422. 6 V. T. Morgan and A. Cameron, Mechanism of lubrication in porous metal bearings, Cont. on Lubrication and Wear, Inst. Mech. Eng., London, 1957, Paper 89, pp. 151 157.