1~. 1. Engng Sci. Vol. 31. No. 4, pp. 593-604, 1993 0020-7225/93 $6.00 + 0.00 Printed in Great Britain. All rights reserved Copyright @ 1993 Pergamon Press Ltd

FERROFLUID LUBRICATION OF EXTERNALLY PRESSURIZED CIRCULAR PLATES AND

CONICAL BEARINGS-f-

DINESH KUMAR Department of Mathematics, Narmada College of Science and Technology, Bharuch, India

PEEYUSH CHANDRA and PRAWAL SINHA Department of Mathematics, Indian Institute of Technology, Kanpur-2OllO16, India

(Communicated by G. A. MAUGIN)

Abstract-Ferrohydrodynamic lubrication, in the presence of a transversely applied magnetic field, of circular plates and conical bearings is analysed. In contrast to earlier analyses this study relaxes the assumption of the magnetization vector being parallel to the magnetic field. Besides, the rotation of magnetic particles is also accounted for. Expressions for various bearing characteristics are obtained using a perturbation scheme in terms of dimensionless Brownian time relaxation parameter. It is observed that the effect of applied magnetic field as well as Brownian relaxation time parameter is negligible on load capacity.

1. INTRODUCTION

In recent years, increasing attention has been focused on the study of ferrofluids. These fluids are stable colloidal suspensions of very fine magnetic particles in a carrier fluid. They exhibit unusual properties under an externally applied magnetic field, that is, they can be confined, positioned or controlled at desired places. This has led to important applications of such fluids in the lubrication of liquid seals, journal bearing, thrust bearing, etc. Several investigators have theoretically analysed ferrofluid lubrication in different bearing configurations, like thrust bearings [l], short bearings [2], seals [3], sliding bearing [4], squeeze film [5], finite journal bearing [6], etc. However no attention seems to have been paid to the analysis of ferrofluid lubrication of externally pressurized bearings.

Externally pressurized oil film bearings have been the topic of numerous investigations over the last three decades. During this period not only has the fundamental understanding of the subject developed to an advanced stage, but also a variety of industrial applications of such bearings has encouraged the development of several practical design procedures.

Externally pressurized bearings, in principle, provide the simplest type of liquid film lubrication. The load carrying film is both created and maintained by external means. The essential property of this lubrication mode is that the load carrying surface is floated irrespective of its motion or the lack of it.

Archibald [7] can be considered to be the pioneer in the investigations on the lubrication of externally pressurized circular plates and conical bearings. Subsequently several researchers have studied these bearings under various situations; e.g. Dowson [8], Dowson and Taylor [9], Krieger et al. [lo], Maki et al. [ll], Kamiyama [12], Chow [13], Agrawal et al. [14], Salem and Khalil [15], Kennedy et al. [16], Sinha et al. [17].

In spite of an abundance of literature on the application of ferrofluids in fluid flows and lubrication theory, there is a dearth of literature as far as the study of ferrofluid lubricated externally pressurized bearings are concerned. It must be emphasized that most of the available studies in this direction use the model suggested by Neuringer and Rosenswig [18], in which the particle rotation is ignored and the magnetization vector is taken to be parallel to the applied magnetic field, thus limiting the range of applicability.

TPaper presented at the International Symposium on Magnetic Fluid Technology and Research, Kurukshetra, India, 21-23 September 1991.

593

594 D. KUMAR et al.

In view of this, in this paper we study the lubrication of externally pressurized circular plates and conical bearings, using a ferrofluid lubricant, under the presence of transversely applied magnetic field. The model suggested by Shliomis [19] is used which accounts for the rotation of magnetic particles. Further, the restriction on the magnetization vector is also relaxed. Various bearing characteristics are obtained through perturbation analysis and elaborated through

graphs.

2. EXTEKNALLY PRESSURIZED CIRCULAR BEARING

2.1 Mathematical formulation

In this section we consider the flow of a ferrofluid in the presence of a transversely applied constant magnetic field between the externally pressurized circular plates (Fig. 1). The flow of lubricant in such a bearing is due to the externally applied pressure. So we consider the flow in radial direction only, and the magnetic field is applied along the Z-direction.

Further, we make the following simplifying assumptions [l, 21: (i) The plates are non-magnetic and non-conducting so that the applied field is not

modified. (ii) Magnetization in the ferrofluid is negligible compared to the applied magnetic field.

(iii) The ferrofluid is saturated so that MO does not depend upon the applied magnetic field. (iv) Thin film lubrication approximations are valid so that the derivatives of velocity,

magnetization, etc. along the film are negligible in comparison to their derivatives across the film.

Thus, for the steady viscous flow of a ferrofluid the governing equations of motion [19,20] under the above assumptions reduce to the following form:

tBM2au PO~BTS M,=__-- 2az I

HoM,&

M2=Mo--- GM1 au + ~0tBtS - H,M:.

2az I

The equation of continuity is

gpU)=O

-U

f!-f p= P.

p=piqp!- %-----I

(1)

(4)

Fig. 1. Externally pressurixe.d circular plates.

Ferrofluid lubrication of plates and bearings 595

where P is the pressure in the film region, U is the velocity component in the radial direction, M1 and Mz are the magnetization components in the R and 2 directions, respectively. Boundary conditions are

dU ==O at Z=O, U=O at Z=h (5)

and

P=p, at R=R,, P=p, at R=Ri 09

where pe and pin are outlet and inlet pressures and R, and Ri are outlet and inlet radii, respectively.

Introducing the following non-dimensional variables,

P P-P, UrlR,

=- u=(Pin-ps)~~ z=$ R”. R Pin -pi r=-, =;<I e

M M2 ml=--, mz=-

Mll Ml

and non~imension~ parameters

N z= POMOHQR~ (pit, - PeVo

z = ZB(Pin -Pd,

tlR=

A _ ITS

I

the non~imension~ form of equations (l)-(6) is given by

dp #ufNam -Tr+az2 T-z=

o

ml2 au ml =---AN tmlm2

2 dz

au ~‘0 at r=O, u=O at z=l

(7)

(8)

(9)

(10)

(11)

03)

2.2 Analysis

p=O at r=l, p=l at R=l?. 114)

Equations (9) and (11) are coupled non-linear equations. It has been found that the non-~~ension~ parameter z is of the order lo-*, so we solve these equations by the perturbation technique with a perturbation parameter r. The flow variables are taken in the following form,

where f stands for the variable quantities p, u, ml and m2. Using this perturbation scheme and eliminating magnetization vector components of

different orders we obtain following equations, for various orders of z, governing the flow:

05)

=o (16)

5% D. KUMAR et al.

(17)

(18)

Boundary conditions (13) and (14) in perturbed form become,

due au1 -=-= dz dz

O=+$=z at z=O

ug=u1=o=u2=u3 at z=l (19)

Po=Pl =O=p,=p, at r=l

po=1,pI=p2=p3=0 at r=R. (20)

Solving equations (15)-(18) with the corresponding boundary conditions (19), we get following expressions for uo, ul, u2 and u3,

(21)

(22)

(23) u2=[$35N(~+433] (53

u,=[$3- N(d +A&+ N2 G+~+A dpl (’ A 2)2}] (q)+;(%)l(z’_ 1).

(24)

The equation of continuity (12) on integrating over the film thickness gives,

a ’

Z, I rudz=O

which for various orders of r gives

(25)

ruui dz = 0; i = 0, 1, 2, 3.

Using equations (21)-(24) in equation (26) we get equations for pressure of various orders as follows )

$[r(f)]=O

a dp, Ndpo -_-- =

- r dr [{ II o

~,r~~~~(~-~~~+A)~)~]=O

(27)

(28)

(29)

By solving equations (27)-(30) with the boundary conditions (20), the expressions for po, pl, p2 and p3 can be obtained, which finally gives the pressure distribution as follows

1 log; ; 3Nz3

’ -1ogR ----_[ (f-1)+&1)] 160 (log R)3 (31)

Ferrofluid lubrication of plates and bearings 597

Load carrying capacity is defined as [21]

(32)

which on using equation (31) gives the ~mensionless load (w) as follows

W

w = (pi” -pe)R’ =

Elow flux is given by

Q = 2,J’RU dZ 0

which on using equations (21)-(24) yields, the dimensionless flux,

Q* = (Pi” - pe)h3 =

~ux ratio Q =I (Q*/Qz) is given as follows,

Q=Q:+Q:,

where

Q:=[

Q:= 7 [I - A’t(; + A)].

It may be noted here that Q,* is the flux for non-fe~omagneti~ begonia fluid the component of flux independent of A while Q: gives the contribution due to A.

3. CONICAL BEARING

In this section we consider the flow of ferromagnetic fluid throu~ an externally

(W

(35)

(37)

(38)

(39)

case, Q: is

pressurized conical bearing in presence of transversely applied magnetic field [Fig. 2(a)]. The analysis here is similar to that of Shukla [22].

3.1 Slot analysis

The momentum equation governing the flow in a slot [Fig. 2(b)] can be written as follows:

The equation of continuity is

dU_O d;y- * (41:

The equations for the components of the magnetization vector are identical to equations (2: and (3).

598 D. KUMAR et al.

(al (b)

I I

I I h -u I

I I I ---x

(0.0)

I VLP

I Fig. 2(a). Externally pressurized conical bearing; (b) rectangular slot geometry for conical bearing.

The boundary conditions are

U=O at Z=O and h. (42)

Introducing the following non-dimensional variables

P-P,

’ = (Pin - Pe)’

ULrl X z

’ =(Pi, -Pe)h2’ x=-

L’ z=-

h M1

ml=-, WI

m,=$ (43) 0

and non-dimensional parameters

N= POMOHOL h(Pin - Pe)’

t = t&in -PeM Lrl ’

the non-dimensional form of equations (40)-(42) are given by

dp 3% Ndml

-dr+s+--= 2 a2 0

dU -0 z-

u=O at z=O and 1.

*=rlts Z

(45)

(4)

(47)

The dimensionless form of the equations for the components of magnetization vectors are identical to equations (10) and (11).

Adopting the perturbation scheme as outlined earlier in Section 2.2, equation (45) for various orders yields,

dpo #uo o --

& +s=

dpl+ 6 -- -+:g J$ =()

dx az2 [ 1

dps + h -- dx -+;;[$-N_+ dZ2

+[N2A2-i(%)32]=0

(48)

(49)

(50)

(51)

Ferrofluid lubrication of plates and bearings 599

which on using the corresponding perturbed boundary conditions for u, gives

Flux through any cross section of the slot of width b is given as

Q=,g,,,

Equation (56) on perturbation gives

where

i = 0, 1, 2, 3

Q&r) Q: = (pi” _ pe)4;

6 d,

and

Q*=Q;+tQ:+tiQ;+--.

Using equations (52)-(S), equation (57) gives

(52)

(53)

(54)

(55)

(56)

(57)

Q,+ = -; [z] (58)

Q+_;[$;%] (59)

,;=-~[~-sr~_N(!+A)~]] (W

Q;=_~[!-!!&~[!!!$ N(! +A)dr+ N2 %+?+A dpl (’ A 2)!5}]2!!&!33]~ (61)

3.2 Analysis for conical bearing

To obtain flux for conical bearing, the width b in the slot analysis is replaced by

b = 23rX sin (Y in equations (58)-(61). Thus the differential equations governing pO, p,, p2 and p3 are given as follows,

dpo 6Qo* -= -- dx xx sin LY (62)

dpl 6 NQof -= -~ dx x.x sin (Y

-+Q: 4 1 (63)

dp2 -=

dx ---& [ 3Q: - NAQ,*) + Q:]

dt+ -= dx ---& [;(a: - NAQ: + N’A’Q,‘) + Q;] + 40;;5 cu’

(W

(65)

600 D. KUMAR et al.

Boundary conditions for pressure are

p =Pin at X = RJsin cx (66)

p=pe at X = RJsin a. (67)

equation (66) by using non-dimensional variables defined in the Non-dimensionalizing previous section we get

p=l at x=RF, p=O at x=1 (68)

where

Using above boundary conditions, equations (62)-(65) give the expressions for po, pl, pz

and p3. Thus the pressure distribution is given as follows,

log x

‘-1ogR: --- 3Nz3 [_L_jl-,).g-$+(-R-+-$j].

64O(log R;)3 log R; RF2

The flow flux can be subsequently obtained as follows,

n2;;1;;-*2; [l - Nt(l, + A)]. I

(69)

(70)

The flux ratio Q = (Q*/Q~) is given as follows

Q=Q::+Q:, (71)

where

(72)

(73)

(74)

It may be noted here that Qt is the flux for non-ferromagnetic Newtonian fluid case, Q: is the component of flux independent of A while Qi gives the contribution of A.

The load capacity is defined as,

W = Z(p, - p,)R: + 2~a sin2cr I

R&in a

(P -P&f dx Rdsinn

(75)

which on using equation (69) gives the dimensionless load as follows

W

w = (pin - ~3: =

IG(R: - 1) 3N3tt3

2(log RF) - 32O(log R:)2 l- 4(loglR:)2(1 - R12)(& - l)] . (76)

Ferrofluid lubrication of plates and bearings 601

4. RESULTS AND DISCUSSION

4.1 Nondimensional parameters

The bearing characteristics are the functions of the non-dimensional parameters N, r and A. These parameters are defined in equation (8) for circular bearings and in equation (44) for

conical bearings. Using the following representative values of magnetic fluid parameters H,, - 105 A/m,

M,, - lo4 A/m, tg - low6 s, tS - 10-l’ s, p. - lob7 kg/s* A*, I - 10-14/m, the parameters, N, r

and A are found to be of orders lo’, lo-* and 1 respectively. It may be pointed out here that there are certain restrictions on the values of N, r and A in order to have non-negative flux. As can be seen from the expressions of flux ratio (36) and (71), there is a possibility of 0 becoming negative if A, N and r are large. Hence for large values of A, t is taken to be small (-lo-*) while for small values of A, t can take values up to 0.9 for N = 5. However, if N = 10, the values of A and r have to be still smaller.

It must be mentioned here that the qualitative behaviour of both geometries is similar hence only conical bearings are discussed.

4.2 Pressure distribution and load capacity

It is seen that the pressure distribution and load for circular and conical bearings up to the third order approximation are independent of the parameter A [refer to equations (31), (33), (69), (76)]. Further, the coefficients of t and r* are zero, and so the effect of N and r comes only with the third order of r. Thus the effect of magnetic field (Ho) and Brownian relaxation time parameter (rn) on pressure and load in both the cases is very small. This effect is shown in Table 1 for pressure and in Figs 3 and 4 for load.

The non-dimensional load capacity (w) vs r has been shown in Figs 3 (for small values of t) and 4 (for large values of t) for various values of N. It is seen from these figures that the load capacity (w) increases with the increase in N and r. The increase with respect to t in both cases is not linear and the variation from Newtonian case (N = 0) is seen only at higher values of r. However it may be noted that the ordinate of these figures is highly enlarged and actually the variation is only of the order lob4 for small values of r. Hence, the effect of applied magnetic field as well as Brownian relaxation time parameter is negligible on load for an externally pressurized bearing.

4.3 Flux ratio

Non-dimensional flux ratio (0) with respect to t has been plotted in Fig. 5 for different N and A = 0. The effect of A on e vs r for N = 5.0 has been shown in Figs 6 and 7 for small and large values of t respectively. It is observed from Fig. 5 that the flux ratio decreases with the increase in N and r in both the cases. This decrease becomes sharper as N increases. From Figs 6 and 7 it is observed that even in the case of A # 0, (2 decreases with r. The qualitative behaviour of 0 for large and small values of A is similar and the effect of A on 0 depends upon the values of r. 0 increases with A up to certain values of t and then shows a decreasing trend with respect to A, e.g. from Fig. 7 it is seen that Q increases with A if A 50.2 and t 10.25 or if A ~0.4 and r I 0.2 while Fig. 6 gives an increase with A if A 5 3 and r I 0.038, etc.

Table 1. Non-dimensional pressure for externally pressurized conical bearing

(~$9 N=O (0.01, 5) (0.05,5) (0.05, 10) (0.1, 5) (0.2,5)

0.1 1.00 1.00 1.00 1.00 1.00 1.00 0.3 0.5228787 0.5228788 0.5228887 0.5228987 0.5229587 0.5235185 0.5 0.3010300 0.3010300 0.3010364 0.3010428 0.3010814 0.3014416 0.7 0.1549019 0.1549021 0.1549053 0.1549054 0.1549568 0.1551215 0.9 0.0457574 0.0457575 0.0457585 0.0457579 0.04576573 0.0458234

D. KUMAR et al.

0.67544

xl 0.67540

. A = 0.0 - N-0.0 I

--- N=5.0 . N= 10.0

i

/ .

0.67536 ( 0.01 0.03 0.05 0.07

T

Fig. 3. Variation of dimensionless load capacity, w vs r, for low values of t.

1.00

0.9:

a

0.9c

0.8:

‘\ ‘Y\ l \

A= 0.0 \ - N=O.O --- N-5.0

. N= 10.0 ‘\

‘\ . -\

\ t

I I I I 0.01 0.03 0.05 0.07

r

0.730

r

A= 0.0 0.710 -

I-r ;:g:; ! . N= 10.0

F/ !

0.690 -

o.670 I__- 0 0.2 0.4

T

Fig. 4. Variation of dimensionless load capacity, w vs r, for high values of r.

1 .oo

0.99

0.91

0.69 I I I I 0.01 0.03 0.05 0.07

T

Fig. 5. Variation of Bux ratio, a vs r, for a fixed Fig. 6. Variation of flux ratio, e vs t, for a fixed value of A. value of N.

Ferrofluid lubrication of plates and bearings 603

ij 0.6 - ‘\ ‘. N- 5.0

- A=O.O \ \

-.- A-O.2 ---- A-O.4

‘\ ‘\

0.2 - 0 0.2 0.4

T

Fig. 7. Variation of flux ratio, 0 vs t, for high values of r.

REFERENCES

(11 J. S. WALKER and J. D. BUCKMASTER, Int. 1. Engng Sci. 17, 1171 (1979). [2] N. TIPEI, J. Lub. Tech&. (ASME) 104, 510 (1982). [3] S. KAMIYAMA, T. OYAMA and J. HTWE, Basic study on the performance of magnetic fluid seals. In Proc.

JSLE hit. Trib. Conf., p. 985, Tokyo (1985). [S] S. MIYAKI, SADAO and TAKAHASHI, ASLE Tram 28,83 (1985). [S] P. D. S. VERMA, ht. J. Engng Sci. 24,395 (1986). [6] F. SORGE, J. Tri6. (ASME) 109,7l(l987). (7) F. R. ARCHIBALD, Tram ASME (F) 78,29 (1956). [8] D. DOWSON, Tram ASME, J. &sic Engng 83,227 (1961). [9] D. DOWSON and C. M. TAYLOR, Trans. ASME, Paper No. 66Lubs-15.

[lo] R. J. KRIEGER, H. J. DAY and W. F. HUGHES, J. Lub. Technol. (ASME) 89,307 (1967). [ll] R. E. MAKI, D. C. KUZMA and R. J. DONNELLY, J. FZuid Mech. 30,83 (1967). [12] S. KAMIYAMA, Trans. ASME (F) 91,589 (1969). [13] C. Y. CHOW, Appl. Sci. Res. #), 40 (1969). [14] V. K. AGARWAL, K. L. GANJU and S. C. JETHI, Wear 19,259 (1972). [U] E. SALEM and F. KHALIL, Wear 56,251 (1979). [16] J. S. KENNEDY, SINHA PRAWAL and C. M. RODKIEWICZ, J. Trib. (ASME) 110,201 (1988). [17] SINHA, PRAWAL and C. M. RODKIEWICZ. 1. Trib. (ASME) lW,339 (1991). [18] J. L. NEURINGER and R. E. ROSENSWIG, Phys. Fluids 7, 1927 (1964). (191 M. I. SHLIOMIS, Soviet Phys. JETP 34,129l (1972). [20] S. KAMIYAMA, K. KOIKE and N. IIZUKA, Bull. JSME Z&128.5 (1979). [21] D. D. FULLER, Theory and Pructice of Lubricufion for Engineem. Wiley, New York (1956). [22] J. B. SHUKLA, Wear 6,371 (1963).

(Received 5 March 1992; accepted 20 May 1992)

Nomenclature overleaf

604 D. KUMAR et al.

NOMENCLATURE

A

b

6 h ho H

HO 1

= dimensionless relaxation time parameter

= $d;:~;::)sjlot

Q: = flux of the ith order, i = 0, 1, 2, 3 Q = tlux ratio (Q*/Q$) R = radial coordinate

= dimensionless width of the slot = dimensionless radial coordinate = film thickness ;r = radii ratio for circular plates = minimum film thickness R: = radii ratio for conical bearings = applied magnetic field R,, Ri = outlet and inlet radii = constant applied magnetic field strength = sum of moments of inertia of particles/unit

U, V = velocity components u, u = dimensionless velocity components

volume L = length of the rectangular slot [Fig. 2(b)]

= magnetization vector (MI, 0, Ma) = equilibrium magnetization . __

w = load capacity W = dimensionless load capacity X, Z = Cartesian coordinates x, r = dimensionless Cartesian coordinates

M,, Ms = components of magnetizatron vector M

m,,m, - - dimensionless components of M N = dimensionless parameter [equation (8)] P = pressure

P = dimensionless pressure

Pm Pin = outlet and inlet pressure respectively

Pi = pressure of the ith order, i = 0, 1, 2, 3 = flux across a cross-section = dimensionless flux

ff = semi-vertical angfe of the conical bearing [fig. 2(a)]

tl = viscosity coefficients PO = permeability of free space ru = Brownian relaxation parameter TS = relaxation time parameter due to rotation t = dimensionless parameter corresponding to

TB