Indian Jou rnal of Engineering & Materi als Sciences Vol. 10, February 2003, pp. 4 1-49

Generalized Reynolds equation for non-Newtonian ferrofluids

Dinesh Kumar Verma", B V Rathish Kumarb & Prawal Sinha"

"Department of Mathemati cs, Narmada College of Science and Commerce, Zadeshwar, Bharuch 392 0 II, India

"Department of Mathematics, Indian Insti tute of Technology, Kanpur 208 0 16, India

Received 25 Jallllary 2001 .. revised received 30 Septelllber 2002

In th is paper, a genera lized form of Reynolds equation is deri ved for viscoelastic fe rrofluid lubricat ion in the prest:nl"c of a constant magneti c fie ld appli ed transversely to the flow. As a parti cul ar case, one-d imensional form of it is used III

study the characteri sti cs of a squeeze film . It is observed that the effect of viscoelasti c parameter is to reduce the load capac ity of the bearing even in the case of magnetic fluid .

The fl ow behav iour of magnetic fluids has been analytically studied mainl y by using two models -one given by Neuringer and Rosensweig l and the other by Shliomis2

. The fo rmer model takes into account the effects of magneti c body force under the assumption that the magneti zation vector is parallel to the magneti c field vector. The later model removes th is assumption and also takes into account the rotat ion of magnetic particles in the fluid suspension.

In recent years the use of Newtoni an ferromagnetic flui d in bearings has been studied by various researchers)-s. In parti cul ar, Walker and Buckmaster) inves tigated the behaviour of thrust bearing under variable magnetic field . Ti pe i4 deri ved a pressure diffe rential equation equi valent to Reynolds equati on and applied the approx imate form of it to short bearings. The performance of ferromagnetic fluid seals has been studied by Kamiyama et al. 6

. Shukla and Kumars deri ved a generali zed two-dimensional Rey nolds equation fo r a Newtonian ferrofluid in the fo rm of a non-linear integrodi fferenti al equation in presence of transversely applied magnetic field. They employed Shliomis model2 and an iterati ve procedure to determine velocity components. A one-dimensional fo rm of the eq uation was applied to study the characteri sti cs of slider and squeeze film bearings. They pointed out that the load capacity of the bearing increases as the values of ferromagnetic parameters increase. A survey of the analytical studies in ferro lu brication is given by Verma ef a l. 'J.

It may be po inted out here that most commonly used lubricants in bearings under severe operating conditions behave as non-Newtoni an fluids. Kamiyama et al.7 , Kumar and Chandra10 studied the

characteri sti cs of non-Newtoni an fe rrofluid s in seals and bearings respecti vely , using power law fluid model. It has been suggested by Oldroyd 1l le that some fluid suspensions behave as non-Newtonian viscoelastic fluids, showing certain degree of viscous as well as elas tic properties under appli ed stress during motion. He derived general constitut ive relation fo r a vi scoelastic fluid using tensor notat ion. Though the use of viscoelas tic flui ds as lu bri cants has been studi ed in squeeze film and other beari ng l'. lll . the hydrodynami c theory for ferro-viscoe las ti c fl uid has not been developed. However, Raikher and Rusakov l7

.18 have studied the ori entational dy namics

of magnetic particles in viscoelastic base flui d. In thi s paper, therefore a theory fo r thin film lu bri cation using a viscoelas ti c fe rrofluid is presented, and a generali zed two-dimensional Reynolds equation is deri ved using iterati ve procedures. Thi s generalized Reynolds equation can be used to study the characteri stics of bearings and seals, with ferro­viscoelas tic fluid as a lubricant.

Basic Equations The bas ic equations governing the fl ow of an

incompressible non-Newtoni an fe rroflui d are as fo llows,

Eqllation of Illation

DU - - (-)- I - (- -) p-- = - 'lp + 'l.r + /1 /11 M .'l H + - 'l x 5 - /Q Dt 2r

.1

.. . ( I )

42 INDIAN J. ENG. MATER. SCI. , FEBRUARY 2003

where T = (Tij) is the stress tensor in the fluid and can

be defined by a constitutive rel ation for a non­ewtoni an viscoelastic fluid.

Equation of continuity

'Il.u = 0 . .. (2)

Equation of internal angular momentum

-

DS (- -) I (- -) 2 -- = !l lll M X H + - S - I Q + V 'Il S Dt Ts

... (3)

Equations of for electrom.agnetic field

'Il x H =0 (4)

... (5)

Equation of magnetization

DM I (- -) I ( - H 1 &= , Sx M -~ M -Mo IHI . .. (6)

where ry

a- Ps T,. =--, . 15 170

4 3 n a rJ I

T8 =---'-KT

1= 8na5

P,· N

15

I and v = --

2T8

and the vectors ii , Q , M, H , S represent velocity,

vorticity, magneti zati on, magnetic field strength, internal angular momentum respectively . The scalar p represents pressure and M 0 denotes equilibrium

magnetization. The constants !till' T 8' Ts, I, v denote magnetic permeability, Brownian relaxation time of rotati onal diffusion, relaxation time of particle rotation due to frictional resistance of fluid, sum of moment of inertia of particles per unit volume and diffusion coefficient of internal angular momentum respectively . Further, a denotes particle radiu s, N number densi ty of particles, K Boltzmann constant, T temperature of the fluid (assumed to be a constant). Also 170 is the viscosity of the base fluid and P is the

density of suspension. Considering steady state, low Reynolds number

flow and ignoring inertia and angular momentum terms, we have fro m Eq. (3),

s = IQ + J.illl T ,. (M X H ) . . . (7)

which on substituting in Eq. (I) gives

- 'Il p + 'Il.r + !till (M . 'Il )H + & 'Il x (M x H ) = 0 2

Using Eq. (7) in Eq. (6), we get

M =MOI:I+T8QXM+!t1ll~8 Ts (M X H) XM

(8)

. .. (9)

We shall now use these equations for viscoelastic ferrofluid lubrication .

Viscoelastic ferrofluid lubrication Consider the flow of a ferroflui d between two

inclined plane surfaces in presence of a magnetic field applied transversely to the fluid motion .

The physical situation of the system is illustrated in Fig. l a.

~ I I

Z

Ho I I ! I I , , , ! I ,

.Jf Y h (x) /

./ ./

u -.----L

Fig. I a- Ferrofluid slider bearing

z

h

t------... L

Fig. Ib--Ferrofluid squeeze fi lm

-- - -.x

VERMA el al. : GENERALIZED REYNOLDS EQUATION FOR NON-NEWTONIAN FERROFLUIDS 4.3

To simplify the analysis, following assumptions are made, as considered by Walker and Buckmaster) and Shukla and Kumar8

:

(i) The pl ates are non-magnetic and non-conducting so that applied field is not modified.

(ii ) The magnetization in the ferrofluid is negligible compared to the applied magnetic field

IMI«IHI· (iii ) The ferrotluid is saturated so that M 0 does not

depend upon the applied magnetic field H 0 .

(iv) The terms in various equations of the order T~ are retained and other terms of higher order are neglected.

Since, the flow of the fluid is in the x- y plane and the magnetic fie ld is in the z-direction, using the

above assumptions, for U = (u , v, w) and

H = (H" H ", H : ), we have,

/I , v» wand H: » H x' H y '

Also keeping in view the boundary conditions of the magnetic fi e ld components at the plates we can

in fer H:,," Ho and H " H y are negligible as

compared to H 0 .

Thus, Eq. (8) on using (9) is simplified as follows.

- V . P + V . r + {,uIl~T 8 (M . H)} V 2U

.. . ( 10)

- ~,ull/ T 8 V(M . H) X Q = 0 2

and (M .H) = M o Ho -T~ M o HoIQI2 .. , (11 )

Therefore, the Eq . (10) governing the flow of viscoe lastic ferrolubricant under lubricati on assumptions is obtained as follows 8

,

~I Il/ M o H o r ~ 16

... ( 12)

a paT\,: ,ull/ M 0 HoT 8 a2 u --+--+ --a y a z 4 a Z2

,ull/ M o Ho T~ 16

.. , ( 13)

where T xz and T y: can be deri ved from the

constitutive equations of the viscoelastic fluid as follows, Oldroyd 11.12,

where d =~(v .. + v .. ) IJ 2 I .J J.I

( 14)

( 15)

and o Ot

IS the convective derivative and for an

arbitrary tensor Aij'

o Aij aAij 8t = 8t + Aij.1I/ vlI/ - AII/j vi .1I/ - Aill/ V j.1I/

Further, AI' .A2 and ,uo are constants with the

dimensions of time, T/o is the coefficient of vi scos ity.

Using Eqs (14) and ( 15), and lubri cati on assumptions mentioned above, the relevant stress components are g iven by,

au au ( )

2

Txx - 2 AI TXl az = - 2170 A2 az . .. ( 16)

av (av)2 T"y -2A I Tv: -=-217o A2 -.- . a z az

( 17)

,uo ( )aU au TXl + - T xx + T yv - = '17 0 -2 .. az oz . .. ( 18)

T ,,: + ,uo (T xx + T \'\' )aV = '170 dv - 2 .- dz dz

( 19)

where T zz is neglig ible.

It may be observed from Eqs ( 16)-( 19) that these stress components are interre lated. Therefore to ge t

TXl and Tvz ' we substitute Txx and Tyy from Eqs ( 16)

and (17) in Eqs (18) and ( 19) and so lving these

44 INDIAN J. ENG. MATER. SCI., FEBRUARY 2003

equations by the method of iteration while retaining the terms of AI and A2 , we get,

... (20)

(21)

Derivation of generalized Reynolds equation For deriving the Reynolds equation, we first obtain

the equations governing till' 1i()W of ferroviscoelastic fluid between two inclined plates in the presence of transversely applied magnetic field , using Eqs (20) and (2 1) in Eqs ( 12) and (13) as follows ,

... (22)

a avail av -[[ )l[ )2 [ , ? 11 - (ex + (3) az az az + az) == 0

. .. (23)

where

tl M H T 17 == 17 + 1// () U 8 1/ 0 4 '

3 j.-1 11I M o H o T8 a == --------'---'---'--

16 '

f3 == tl{) 1]0 (AI - A2 ),

Solving Eqs (22) and (23) with boundary conditions

II == 0, V == 0

I/==U , v== V

we get,

at Z == he X)} of Z == 0

1/ == - - z - zh + U I - -I (ap) ( 2 ) ( Z )

21] /1 ax It

.. , (24)

... (25 ) and

Eqs (25) and (26) give velocity distributions in x and y directions. It is noted here that these equati ons are integro differential equati ons and can be solved by substituting u and v in the ri ght hand side of these equations by using iterat ive procedure for small ex and f3 .

Integrating the equation of contin uity (2) wit h the boundary conditions,

w == -W" at z == It and

w = ° at Z = 0,

we get

~(h 3 ap)+ ~(1I 3 ap I ax ax ay ay )

.. . (27)

all a " == 61]/1 U - -121] /l W, - 12(a + (3) - f Z F

" d: ax ax 0

where

a " -12(ex+f3)-f z F, d~ ay 0 .,

F, == (~)3 +(~)( av 12 , az az dz)

_ ~ J [ ( ~)3 + ( ~)( av )2] dz II u l az l az l az

. .. (2X)

VERMA el {fl.: GENERALIZED REYNOLDS EQUATION FOR NON-NEWTONIAN FERROFLUIDS 45

F,. :::: ( av).1 + ( av )( au )2 I az az az

- ~ J [( a~ j} + (av)( au )2 ] elz 11 0 ao(. ) az az

0" (29)

Eq. (28) gives the generalized Reynolds equation in the form of integrodifferential equation in two­dimensional form , which can be applied to any bearing situation dealing with viscoelastic ferrofluid under appropriate conditions. Eq. (28) reduces to one derived by Shukla and Kumar8 for the case of a

ewtonian ferrotluid o

One-dimensional form of Reynolds equation The right hand side of Eq. (28) can be simplified in

terms of pressure gradient analytically by using Eqs (25) and (26) in an iterative manner to the desired levels of approximations. We explain the procedure by deriving one-dimensional Reynolds equation as follows .

In one-dimensional form Eq. (28) can be written as fo llows,

+ r ex + ,8 I [i (au Y dz - 3.. J (au Y dZ] \ '10 ) () az) II 0 az)

.. . (30)

Solving Eq. (30) and retaining the terms up to order ex, ,8 and ex ,8 , we have,

1/ :::::-1 (ap) (z 2 - zll) + U (I -3..)

217" ax II

+ -- f C~ dz - -.:: f c; elz (ex + ,8 J [, ~ Ii ]

171/ 0 Iz o

( 6ex,8 1 [, 2 Z Ii 2 ] + ~ f C , F'I dz --I f C, F'l dz

17 11 () I 0

.. 0 (31)

where

F ::::_I_(ap j} (2 Z _ II )} ·'1 817} ax 1/

( )21 1 1 3U ap 1 11 ---- - (2 z -h)---

4n 2 1z ax 3 '/11 ... (32)

3U 2 (ap ) '\ +-- - (2z-Iz) ' 2n h 2 ax 'III

C ::::- - (2z-II)--I (apj U X 217" ax II

... (33)

On substituting Eq o (31) in equation of continuity and integrating, we get one-dimensional Rey nolds equation as follows,

- h - ::::617 U--12/] W a ( 3 ap ) ah ax ax /I ax /I .'

12( ,8) a [ 115

(ap )} U2

11 (ap 1-- ex + ax 80 I] ~ ax + 41]" ax)

72ex,8 a

It is seen from Eq. (34) th at it is a non- linear integropartial differential equation and the viscoelastic and ferrotluid parameters have additive as well as synergistic effects in thi s equation.

Application to squeeze film

A particular case of squeeze film lubrication is being considered as an application of Eq. (34).

Consider one-dimensional squeezing tlow between two parallel plates (Fig. I b) (11 :::: constant), lhe equation governing the pressure for thi s case can be obtained from Eq. (34) as follows,

~(h .1 dp j dx dX)

d Ii ) dp '

[ - '\ 1 ::::-12110 W, -12(ex+,8) - --(--)

dx 80 1] ·1 dr "

( ,8

J [ 7 ( )5] 72ex d Ii dp

/]" dx 448/] (~ dx .. . (35)

Integrating Eq. (35) and using the condi ti ons p :::: 0

at x :::: 0 and x :::: L , we get the followin g inlegral equation determining p,

46 INDIAN 1. ENG. MATER. seI., FEBRUARY 2003

Solving Eq. (36) by the method of iteration and retaining terms of order up to (X, f3 and af3, we get

pressure in the non-dimensional form as follows,

- 61711 -(I -) 162 ( - f3-) p = -=- x .- x--=- a + II "' 511 7

X [x(l-x){(I-xY +X2}]

_ 23328 aff {I-(1-2xY; } 17511 I I 1711

... (37)

Thus the load capacity of the squeeze fi lm, given by,

L

W = b f prix ()

is obtained in the non-d imensional form as fo llows:

W = ~" -~(a + ff)- 139968 a ff ... (38) h 3 25 h 7 122511 I I 1711

where

_ x - h ph ) WII x = L' h = -, ' p = ; 2' W = I . "

I; 17oW, L u17oW, L

r W L f.1 M H ,2 - f3 W "2 L2 r = B \ N = "' U 0 I, , f3 = ., 4

h/ 170 W, L 170 h;

- _ 17" _ 1 r __ N r 3 17 ,, --- +-,a---.

170 4 16

Results and Discussion Dimensionless parameters

Here, N gives the effect of applied magneti c fi eld.

r is the time relaxation parameter and ff gives the

effect of viscoelastic parameter. It may be noted that the viscosity f/" of the carrier fluid has been modifi ed

in presence of magnetic field . The effect of appli ed magnetic field is to increase the viscos ity of the fluid. Also, it is enhanced due to the ferrofluid parameter

T B' Further, when either N = 0 (i.e. in absence or magnetic field) or T = 0, the Eqs (47) and (48 ) provide the non-dimensional pressure and load capacity for a viscoelas tic fluid. In the case of ff = 0

2~---,--_-, ___ -, ___ .-___ .-__ ~

Ii ~ 0 .1

1.8 N~0 . le+5

r ~ 0.le·3

1.6

14

1.2 ~ , ~

,0.

0.8

0.6

0 .4

0 .2

0.2 04 x 06 0.8

Fig. 2-Effect of 13 on pressure di stributi on

~ O.Oe-O -<>-­~ 0.5e-5 -+­~ 0 .le-4 -8-

~ 0 .2e -4 ~

1.2

VERMA et al.: GENERALIZED REYNOLDS EQUATION FOR NON-NEWTONIAN FERROFLUIDS 47

(i.e. no viscoelastic effect) these equations reduce to those of Shukla and Kumar8.

Taking the representative values, Rosensweigl 9

10-6 . 10-3 k - I - I 10-7 k -2 r tJ :::: s, 170:::: g m s , I-l o:::: g m s

A2, Ho :::: 105 Am-I, M 0:::: 105 A m-I and the bearing

2

h - 0 .1 1.8

1.6

lA

1.2 M

'0 ->< 'D-

0 .8

0 .6

OA

0 .2

0 a 0.2 OA

characteristics hi :::: 10-3 m, L:::: 10- 1 m, W, :::: 10-:; mis,

the parameters N , r and are of the order 1 0 ~, I O -~

respecti vely. The value of the viscoelastic parameter

13 is taken in the range of 10-4 to 10-2 foll owing

Mow l) and Chandra l4 .

0.6

N a 0 .le+5,r 8 0 . le-4,~ a 0.2e-4 ...­N . 0.le+5,r.0.le-4 ,eaOAe·5 -t­N 8 0 .58+5 ,[. 0.3e·4 ,p. 0 .2e-4 -e­N a 0 .5e+5, La 0.3e-4,)I • OAe-5 -><­N a 0 .le+5, Ca 0.3e-4}. 0 .2e-4 -"'­N = 0.5e+5, r- 0.1 e-4,j a OAe·5 ---

O.B 1.2

Fig. 3-Effect of magnetic nuid parameter on pressure di stribution

M

'0

>< ,D-

l A r-----,-----~----,-------,----;-----y--,

N = 0 .lEt5

1.2 T=0.l E-3

;S = 0 .lE·4, h =0.1 -+­;0 g 0.1 E·4, Ii = 0.2 -t­

P = O.lEA , h = 0.3 -e­ii = 0.1 E-4 , Ii = 0.8 -><-

0 .8

0 .6

0.4

1.2

Fig. 4-Effect of film thickness on pressure di stributi on

48 INDIAN J. ENG. MATER. SCI. , FEBRUARY 2003

.., '0

13:

2.2

2

1.8

1.6

1.4

1.2

0.8

h m 0.1 N = 0.le+5, ij = O.Oe·O __ N ~ 0.le+5, i3 " ~ 0.5e-5 -+­B ~ 0. le+5, II = 0.le-4 -B­N = 0.5e+5, jj m O.Oe-O .....­fl = 0.5e+5, /J a 0.5e-5 --­N = 0.5e+5, Jj = 0.le-4 ___

0.6 :---=,":;:--~=---::--'-::-::----' __ -L __ -L-__ ...l.-_--'~ o 2e-05 4 e-05 6e-05 8e·05 0.0001 0.00012 0.000 14 0.00016

Fig. 5-Plol of load (W) versus T

Pressure distribution and load capacity

The effect of various parameters on pressure is shown in Figs 2-4, using the above representative

va lues of N , T and 13. It is noted from Fig, 2, that

for given N, T and h the effect of increase in

viscoelastic parameter {3 is to reduce the pressure.

In Fig. 3, the combined effect of magnetic fluid and viscoelastic parameters on pressure is shown for given film thickness . It is noted that in the case of viscoelastic magnetic fluid, the effect of magnetic fluid parameters on pressure is similar to the case of Newtonian magnetic fluid . The pressure increases as

the applied magnetic field strength increase (i.e. as IV increases), it also increases with time relaxation parameter. However as seen earlier in Fig. 2, pressure reduces as the viscoelastic parameter increases .

Fig. 4 shows the effect of viscoelastic parameter on pressure at different film thicknesses. It is observed that the effect of viscoelastic parameter is more pronounced when the film thickness is smaller. For

large values of h the pressure generated is smal l and

the effect of 13 is negligible.

Load capacity W versus T has been plotted in

Fig. 5 for II (= .1) and various values of IV and {3 . It

is noted that for 13 = 0 and N, W increase linearly

with the increase in T.

For small va lues of N, the increase in W with

respect to T is very small for any value or JJ. however as IV increases the increase in W with T is very appreciable.

Conclusions In this paper, a generali zed Reynolds equati on in

the form of a non- linear integropartial differential equation has been derived by comb ining the ferromagnetic fluid 2 and the constitutive equation or a viscoelastic fluid l

l.12

. This equation is non-linear and both additive and synergistic effects of viscoelas ti c and magnetic fluid parameters arise, though in the original basic equations these effects are on ly addit ive. The procedure for solving thi s eq uati on using iterative method has been explained, in the case of a one-d imensional Reynolds equation. Thi s equation has been used to study the characteri st ics of a squeeze film bearing, which shows that the efl'ect or viscoelastic parameter is to reduce the load capacity of the bearing even in the case of magnetic fluid.

Acknowledgement We thank Prof. J B Shukla and Prof. Peeyush

Chandra, Department of Mathematics. liT Kanpur. for their suggestions and fruitful discLlss ion during the preparation of thi s paper. One of the authors Di ncsh Kumar Verma, is thankful to INSA for awarding

V ERMA el al. : GENERALIZED REYNOLDS EQUATION FOR NON-NEWTONIAN FERROFLUIDS 49

visiting fellowship for the year \998-99 during which thi s study was made.

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