couette flow of ferrofluid under spatially uniform sinusoidally...

Indian Journal of Pure & Applied Physics Vol. 40, March 2001 , pp. 185- 195

Couette flow of ferrofluid under spatially uniform sinusoidally time-varying magnetic fields

S Jothimani & S P Anjali Devi

Department of Mathematics, Bharathiar University, Coimbatore 641 046

Received 15 May 200 I ; accepted 27 December 200 I

The authors analysed the behaviour of Couette fl ow of a ferrofluid under spati ally uniform sinusoidall y time varying

magnetic fi elds. The imposed magnetic field Hz and magnetic flux density B, are spatially uniform and are imposed on

the system by external sources. The governi ng linear and angul ar momentum conservation equations are solved for fl ow and spin velocity distributions fo r zero and non-zero spin viscosities as a function of magneti c field st rength , phase, frequency, direction coordinates along and transverse to the plates, as a function of pressure gradient along the plates, vortex viscosity, dynamic viscosi ty and fe rrofluid magnetic susceptibility. Solutions for certain limiting cases are also given.

1 Introduction

Technologically, the topic of asymmetric stress in magnetic fluids is intriguing because of its possible applications in the control of heat or mass transfer processes in convective flows, and in the modification of drag in boundary layers. Certain polymers and liquid crystals are expected to display these effects to a small degree, but the magnitude of the effect is large in ferrofluid s. In addition, the subject holds inherent sc ientific interest.

Occurrence of asymmetric stress in magnetic fluids can be due to the influence of unsteady magnetic fields such as alternat ing, travelling or rotating magnetic fields. Okubo et al. 1 studied the stability of the surface of a magnetic fluid layer under the influences of vertical alternating magnetic fields experimentall y and analyticall y. Propagation of waves on the free surface of magnetic fluid under trave ling magnetic field is experimentall y investigated by Kikura et af2. It is found that the surface velocity of the magnetic fluid depends on the intensity and frequency of travelling magnetic fie lds and depth of the channel contained with the magnetic fluid .

The motion of ferroflu id in a trave lling wave magnetic field has been paradox ical as many investigators fi nd a critical magnetic field strength below which the fluid moves opposite to the direction of the trave ll ing wave while above, the ferrofluid moves in the same direction3

·". The

concentration of the suspended magnetic particles, frequency and the fluid viscosity influence the value of the critical magnetic field . Under ac magnetic fields, fluid viscosity acting on the magnetic particles suspended in a ferrofluid causes the magnetization M to lag behind a trave lling H .

A body torque density

T=!l<J(MxH )

arises when M is not collinear with a spatially varying magnetic field H . Under thi s condition , Rosensweig5 has developed fluid mechanical analysis to ex tend traditional viscous fluid flows to account for the non-symmetric stress tensor that results when M and H are not collinear and to then si multaneously sati sfy linear and angu lar momentum conservation equat ions for the ferrofluid.

Ferrofluid motion under travelling wave magnetic fields with sinusoidal time and space dependence has been recently analysed in the study' wherein, the magnetic fi elds were non-uniform and fo rce and torque densities were non-zero .

Recently Zahn and Greerr, have ana lysed ferro hydrodynamic pumping in a spat ially uni form sinusoidall y time-varying magnetic fie lds. But so far, no invest igation has been made on the study of Couette flow of ferrofluid under spatially uniform sinusoidally time-varying magnetic fields when the

186 INDIAN J PURE & APPL PHYS, VOL 40, MARCH 2002

torque density is non-zero but the force along the walls is zero which forms the present work.

2 Formulation of the Problem

Consider the flow of an incompressible, viscous, ferrofluid confined between two parallel rigid plates, the upper plate moving with a constant speed and the lower plate being at rest. The schematic of the problem is shown with the help of Fig. I.

I i

. 0Wy(x) Ferrofluid H d Hz = R [Hz e ion t ]

----U'z(X)

z

Fig. I -Schematic of the problem

For the planar layer of ferrofluid here, the flow velocity can have only the z-component and spin velocity can have only y-component and both vary with respect to the variable x, as:

... (I)

The imposed magnetic field H, and magnetic flux density B. are spatially uniform and are imposed on the system by external sources . Because the imposed fields are uniform with the y- and zcoordinates, field components can only vary with the x-coordinate. Gauss's law for the magnetic flux density and Ampere's law for the magnetic intensity with zero current density require the imposed fields to be uniform throughout the ferrofluid .

There are no y-component of the magnetic fields . The planar ferrofluid layer between two rigid walls is magnetically stressed by a uniform z directed magnetic field H, and uniform x-directed magnetic flux density B., both of which vary sinusoidally in time at frequency . Therefore, the total magnetic field H and magnetic flux density B inside the ferrrofluid layer are of the form:

... (2)

A A A A l 'u H=R[(H,. ( x )i+H/)e' ] ... (3)

and

B=llo(H+M) . . . (4)

3 Equations of the Problem

Since the fluid is incompressible :

V.v=O, V.w=O ... (4)

and the coupled linear and angular momentum conservation equations for force den sity fand torque density T for a fluid in a gravity -gl are1

:

p[av +(v.V)v]]= at -Vp+ f+2t;Vxw+U; +ry)w2v-pgi

... (6)

/[ aa~ + ( v.w )w]] = T + 2( (V X \1- 2w) + ry'V 2w ... (7)

where p is the mass density, p is the pressure, ~ is the vortex viscosity, I is the moment of inertia density, T) is the dynamic viscosity and T)' is the shear coefficient of spin-viscosity.

These equations are applied to the planar ferrofluid layer confined between two parallel walls, lower one being at rest and upper one moving with constant speed. It is assumed that the planar ferrofluid has viscous-dominated flow, so that inertia is negligible and is in steady state, so that the fluid responds only to the time average magnetic force and torque densities.

The magnetization relaxation equation1·5 with a

ferrofluid undergoing simultaneous magnetization and reorientation due to fluid convection velocity v and particle spin at angular ve locity w is :

oM I - + (v.w)M- w x M + -[M- XoH = 0 .. . (8) ar r

where 't is a relaxation time constant, M = (M., 0, M,) is the magnetization vector, Xo is the effective magnetic susceptibility which in general can be magnetic field-dependent but in this work will be taken to be constant and

3.1 To obtain Mx and M z

JOTHIMANI & ANJALI DEVI: COUETTE FLOW OF FERROFLUID 187

~ ~

M, and M z are obtained with the help of

Eqs (2), (3) and (8) . These values are utilized in future in the coupled linear and angular momentum conservation equations.

Substituting Eqs (2) and (3) into Eq. (8) relates the magnetization components to the magnetic field Has:

~ M , _ Xo ~ iD.M , -w,M z +---Hx

· r r ... (9)

~ M X ~ iD.M _ - w,M , + ____£_ = __Q_ H z

- r r ... (I 0)

where the second term in Eq. (8) has zero contribution because v is only z-directed and M can only vary with x. By taking the x-component of Eq. (4) and solving for fl, one can obtain:

~ B ~ H, =-x -Mx ... (II)

flo ~ ~

Solving for Mx and M z by using Eq. (II) in

Eqs (9) and (I 0): ~

X0[/f zw,r + (iQr +I) 8

x ] M= · flo

x (w,.r)2 + (iQr + l)(iQr +I+ Xo) ... (12)

X0[(i0.r +I+ X0 )H z -8

x w,r] M= flo ·

z (w,r)2 + (iQr + l)(iQr +I+ Xo) ... (13)

Eqs ( 12) and ( 13) represent the magnetization of the ferrofluid as a function of the imposed fields ~ ~

H z and B, and as a function of the not yet known

spin velocity w, which can vary with position x. The magnetization gives rise to a torque on the ferrofluid which causes fluid motion and thus a non-zero w,. The resulting w,. then changes the magnetization. There is a strong magneto-mechanical coupling, so that the magnetization and mechanical equations need to be self-consistently satisfied.

3.2 Magnetic force and torque densities

For 0 < x < d, the magnetic force density IS

given by the equation:

f=fl0(M. w) H ... (14)

Solving for the x- and z-components with field components only constant or varying with x, one can get:

.. .( 15)

The time average components of the magnetic force density are then:

< f x > = - :x( ±,Uo I M x 12

} < f z > = 0 ... ( 16)

Similarly, the torque density 1s g1ven by the equation:

T=fl0(M X H)= flo( -M xH z + M zHJ]

The torque is only y-directed:

T, =M zBx-floM.JHz +Mz)

... ( 17)

. . . ( 18)

The time average component of torque density is then :

... ( 19)

with superscript asterisks indicating the complex conjugate of a complex amplitude field quantity .

3.3 Coupled dimensionless linear and angular momentum conservation equations

It is convenient to express modified pressure as:

' I ~ 2 p = p +-flo I M x I + pgx

4 ... (20)

So that Eqs (6) and (7) in the negligible inertia, viscous dominated limit become:

/2 dw ::~ '

(r . ) c. vz 2r .r up -0 ':> +T] --+ ~:.-----dx2 dx oz ... (2 1)

,d 2w, (dv } 1J --

2-- - 2( _ z + 2w.r < T, >= 0

dx dx ... (22)

It is convenient to express parameters in dimensionless form indicated with tildes, with time normalized to the magnetic relaxation time 1:, space normalized to the distance between the walls d and magnetic field quantities normalized to a magnetic field strength H0 . Therefore:


A A

- - H - B Q=Qr,H =-,B =--

Ho JloHo

- 2 - 2 . - - 2 - 2 • + 2R[[X0 (wy - .Q ) + t.O(w, - .Q - 1- X0 )]H zBr]}

/[(w; - !'F + I+ X0 )2 + (2 + X0 )

2 fP J

... (29)

Note that the phase re lationship between A A

Hz and Br is very important because, for our case

studies with ry' = O,wr must be zero at the fixed

boundaries at x=O, and x=d. It is useful to examine · · .(23) Eq. (29) in the limit of small (V-'. To first order w, - 2( oiJ' d op' - u o r- - - ---U --'=' - ? , -:~ - - 2 -:~·- , 0 - u

J10 H0r oz J10 H0 oz c

Hence the dimensionless flow and spin ve locity equations are:

. . . (24)

d2

- rd- } ' w,. - v - -17--· -( ___ z +2W <T >=0 tfX 2 dX r Y

... (25)

where

... (26)

- - -_ Xo[H zw,. + (i.Q + 1)BJ M , = - ? . - . . -. w; + (t.O + l) (tO +I+ X0 )

... (27)

M = Xo[(iD +_I+ i 0 )H z - Bxw,] z w; + (i.Q + l)(i.Q +I+ Xo)

... (28)

Thi s set of equations describes the motion of the planar ferrofluid layer confined rigid plates with imposed spatially uniform, sinusoidally timevarying x- and z-directed magnetic fields. The primary complexity of the analysi s is that the time

average torque density < T" > depends in a

complicated way on the spin velocity w-' which

depends on .X .

Substituting Eqs (27) and (28) into Eq. (26) gtves:

< f >= Xo {w, [I 8,.12 (w?,~- 02 +I) -' 2 . . .

+Iii z 12 (w~ _.Q2 +(I+ Xo)2)]

Eq. (29) approximately reduces to :

Jim <T" >""fa +aw,. w,.<< l - ·

. . . (30)

where

... (3 1)

Note that a can be positive or negative

depending on the value of .6 compared with I or I + Xo· It is inferred that a is positive at high frequencies and negative at low frequencies.

4 Solution of the Problem

4.1 Zero spin-viscosity (fj' = 0) solutions

First the simple limiting case of zero spinviscosity is examined. Solving Eqs (24) and (25), one can obtain:

_ ( _) I o{/ - ? I :rJ - .r:: k _ k vz x =--:::-x-- - <T,.>ax + 1x+ 2

7J dz 7J o · ... (33)

where k1 and k2 are constants of integrati on to be found from the boundary conditions at the rigid boundaries:

. .. (34)

Applying these boundary conditions to the above equation, one can get:

- 1 op' 1 I -k1 =U0 --:;---:::-+-:;-} <T" >di,k2 =0

7J dz 7J o ·

So that the ve locity becomes:


- - - - I {)jj' -(- I) V, (X) = U oX + -:=- -:~- X X -- 1J oz

. . . (35)

And it is further implied from Eq . (25) that:

w =-0o --1-[a{/ (2x-l)

-' 2 2ff az:

- I l s +ff - -----<T,. >+f <T,. >ax s - 0 -

... (36)

It is noted that Eqs (35) and (36) are not known

analytical solutions for vz and wr because < f ,. >

. . - - . . - ()jj' . vanes w1th x and w,. vanes w1th x . When -Is - az: large compared with the torque density < T,. > then

the flow velocity vz will be essentially parabolic,

while the spin velocity (jj-' will be linear with

respect to x . If the time average torque density is constant

with position, then the torque has no contribution to the flow velocity as the two torque terms in Eq . (35) cancel, but there is a constant torque contribution to the spin velocity in Eq. (36). If there is no pressure

gradient, ::' =0, then the solution for w-' in Eq.

(36) is a constant, independent of x as < f ,. > is the

only constant given by Eq. (29) and U0 • Then the

solution to Eq. (35) is vl (x) =Vox and

<T > D w .(x) = ----h-- _o -' 2s 2

The torque is constant with position if wr (x) is

constant with position as given by Eq . (29).

4.2 Non-zero spin-viscosity solutions

(a) General solution method

A real physical system will have 17' -:t- 0, so that

Eq . (24) and (25) are a fourth-order system that can be integrated numerically by the Runge-Kutta method . The boundary conditions are:

vz<x = o) = o, wy(x = o) = o

vz (x =I)= 00 , wy(x = 1) = o

. . . (37)

. . . (38)

The Runge-Kutta method cannot directly solve two-point boundary value problem with boundary conditions at x = 0 and x =I . Rather it requires

specification of functions and their derivatives at x = 0 and the system is then numerically integrated to x = 1 . Thus our procedure was to specify

vl (x = 0) = 0 and (jjy (x = 0) = 0 and to guess the

derivative values:

dv dw" D __ z D ----,-ax' 2 - ax At x = 0 and then integrate numerically to x =I and define:

Then, use Newton's method to find best values of

D 1 and D2 to derive F 1 = D 0 and F2 to zero. The

resulting vz and (jjy would be the solutions

describing the ferrofluid motion. As a check to the numerical method, several cases are examined with

< T,. > constant is found in excellent agreement

between analytical closed-form solutions of Eqs (25) and (26) to the numerical results.

(b) Zero magnetic torque density ( < f ,. > =0)

If < T" > = 0 in Eq. (25) the flow is driven by

the pressure gradient and Eqs (24) and (25) have analytical solutions of the form:

- 2 a-' v (x) = !____E_- 2Ax- {3[Be"-x - Ce -ax]+ D (39)

z ff az: ...

... (40)

where

a= 2s7J 2s7J'

11'<s +7]'), 13 = 11<s +1J)

The coefficients A , B, C and D are found by applying the zero slip boundary conditions at the plates given by Eqs (37) and (38) and they yield :


8 = [A1 (j3- j3e-" - 2) + U1 (e -" -I)]

A2

C = _:_[ A_.!_1 (.:.:_j3_-___:j3~e_- '_' +_2__.:_)_+_U_:_1 (_I _-_e -_" -=-)]

Az

D = [j3U1 (e" + e-" - 2) + A1 (/3'2e"- j3 2e-"- 4j3)]

Az

where

A2 =2j3(e" +e-" -2)+2(e-" -e" )

<r' (c) Zero pressure gradient ~ = 0

a.z At the opposite limit, where the flow is driven

by the magnetic torque, one can take the pressure gradient to be negligibly small. Then Eq. (24) can be integrated once to yield:

1 - _ dv, ---cs- +7J)-- +S'w,. =C 2 dX .

.. . (41)

where Cis a constant. The flow solutions are then:

v,(x) = _ ~D. [2x( er -1) - y(( +7J)smh y

(e-r -l)+(eP' -1)(1- e-Y)-(e-P' -I)

(e r - I)+ 2U0 (xsinh y- sinh }:X)]

wJx)= _D [}Sinhy+(e-y- 1) . }SIIlh y

e-P' +(1-er)e- P' +U0(e P' -e- )5') ]

where

".(42)

".(43)

5 Results and Discussion

In general, the results are identical to that of Zahn and Greer" when the upper plate is at rest for all the cases.

Non-dimensional time average magnetic torque density is given by Eq. (29). In order to have the physical insight about the variation of this, time average magnetic torque densi ty is plotted as a function of the dimensionless spin velocity for different orientations of the applied magnetic field by fixing certain values for the parameters involved.

0-4 ,-----------------, Xo = 1.0

0.2 ii = o.o

I ~0 I

i'i =s.o I -0.2

-0.4 ':---'---:L----'---'--.J....._-~-~ -20 -10 0 10 20

Wy

- -Fig. 2- Effect of .Q on time average torque density < T,. >

for an axial magnetic fie ld (H~ = /-/0 , B,. = 0)

Figs 2-4 illustrate the dimensionless time average torque density as a function of dimensionless spm velocity for various dimensionless angular frequencies for three different cases, namely (a) axial magnetic field , (b) transverse magnetic field, and (c) rotating magnetic field. It is seen that for a stationary spin velocity

(w = 0) the slope around w = 0 is negative at low

frequencies and positive at high frequencies and that only the rotating magnetic field case has non-zero

time average torque at w = 0.

Figs 5 and 6 portray the effect of speed of the upper plate overflow and spm velocities

respectively . Increas ing U0 increases the flow

JOTHIMANI & ANJALI DEYI: COUETTE FLOW OF FERROFLUID 191

velocity, whereas it decreases the spin velocity for the case of non-zero spin velocity .

0.3r-- - - - ----- --- ---..,

0.2

0.1

<ly)

-0.1

20

- -

12 0

8.0

N N ap' = 0.01 Xo = 1. 0 I ( = 1. 0 ' ll = 1. 0' az

N

U0 = 10.0

Uo = 3.0

Uo = 1.0 ~~~~======::;oo = 0.1

Uo = 0.0

o.o

Fig. 5- Flow velocity profiles for different U0 when if ofc 0

2 . 0---------~------, r a~,

. Xo=1.0 , (=1.0, ~=1·0, ~= 0.01

Fi g. 3-Effect of Q on < Tr > for a transverse magnetic 1 .. 0

field (H, = 0, Bx = flo Ho)

Q.6 X0: 1.0

0.4

0.2

"' (Ty) o.o

-0.2

-Q.4

-Wy

- -Fig. 4- Effect of Q on < Tr > for a rotating magnetic field

(H, = iHo, Bx = ~to Ho)

N Uo = o.o ~ Uo = 0.1 · uY y

Uo:. J.o -2.0 Uo = 3.0

-3.0 Uo = 10.0

-4.0

-S.O'L-----,JL_----l...,--_J_.,.-------.L..---,-L---L-0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

~

X

Fig. 6- Spin velocity profiles for different {j 0 when if ofc 0

When r( '# 0, the effect of spin viscosity, over

flow velocity and spin velocity is shown with the help of Figs 7-1 I. It is interesting to note that the profiles for flow velocity and spin velocity are similar in both the cases of tangential magnetic field and normal magnetic field and hence the figures are

192 INDIAN J PURE & APPL PHYS , VOL 40, MARCH 2002

1-2 .--------------..-------; ~ a'i)·

Xo=1.0, (: 1.0 , ~:1 . 0, az=0.01

1.0

0.8

~1= 0.001

0.4

0.2

Fi g. 7- Effect of if over now veloci ty profil es for an axial

A -

magnetic field (H, = !-/0 , B, = 0) when U 0 = I .0

o.2 .------.-----:;;a"'~------,

Xo=1 .0 , (=1.0, n=l.D, ..!:0.01 1 az

o.o

-0.6

Fig. 8- Effect of if over spin velocity profiles for an axia l

magnetic field (If, = !-/0 , Bx = 0) when iJ 0 = 1.0

displayed here on ly for the case of tangential magnetic fie ld and this trend is maintained even for the higher values of the speed of the upper plate. In addition, it is seen that the effect of moving of the

0.16 .------~--"'-----;;d...-;P,·-------,

Xo::: 1. o , ("' 1. 0 , ~ = 1. 0, ~:: 0.01

0.12

0.08

o.

o.o

...

\_1·= 0.1 j ~'= 0.2,

~·:: 0.02 , ~ :0.05

X

Fig. 9- Flow velocity profi les for different if for a rotat ing

A -

magneti c field (If, = if-10 , B, = flo f-lo) when U 0 = 1.0

o.o

N

r T)' = o.ool

/~'= 0 01

r 1' = 0.02

~· = o.os r~' = o.l r~'= 0.2

-4. 0'-;;-----:::':::---::"-:-1 ---:;-l-;::-----::l';;-----:~--:-1=--....,-J o.o 0.2 0.4 0.6 0.8 1.0 1.2 1.4

X

Fig. I 0- Spin velocity profiles for difTcrcnt if for a rotati ng A -

magnetic field (If, = if-10 , B, = Po !-/0 ) when U 0 = 1.0

upper plate is even to change the trend of profiles of both the flow and spin velocities , wh1ch are vividly seen through Figs 7 and 8. Further, it is noted that as spin viscosity increases, flow veloci ty profiles have


both increasing and decreas ing trend, whereas spin ve loc ity increases for increasing r( .

2.0 ~------------:;;;-------,

X0 = 1. 0, f=1.0, ~=1.0, ;~·=0 . 01

o.o

~· = 0.2

Wy

- 6.0

""---~' = 0. 1

~· = o.os ~· = 0.02

~·= 0.01

~1 = 0.001

-8.0 L--.,:L:----::-J..,----:.1.::----::L::----:-~---;-~-;-l o.o 0.2 0.4 0.6 O.B 1.0 1.2 1.4 ~

X

Fig. II -Effect of if over spin velocity profi les for a rotating

magnetic field (If: = iH0 , B_, = p0 H0 ) when U0 = 10.0

0.2 r-------- ----- - ---.

0.1

-0.1

----- Uo = 0.1

-- U0 = o.o

0.2

8p' - = 1.0 az

0.4 0.6 1.2

X

Fig. 12- Flow velocity profiles for different a

when < T,. > = 0

1.4

Figs 9-11 illustrate the effect of spin viscosity over flow ve loc ity and spi n ve loci ty respectively, for a rotating magnetic fie ld . The effect of speed of the upper plate is to suppress the va lues in general over the flow velocity. As spin viscos ity increases, flow ve locity increases as well as decreases.

0.6.----------------, ----- Uo = 0.1

-- Uo = o.o a= 30

0.2

-Wy 0.0

-0.2

-0.4

-o. 6 l___j[____l:--~:----7:;:--~--7:;--~ o.o 0.2 0.4 0.6 o.s 1.0 1. 2 1.4

X

Fig. 13-Spin velocity profiles for different a

when < T,. > = 0

16.0 .------- - --------::;-----, 8p' - = 1.0 az

12.0

s.o

a= 15 , a= 20, a = 30

- 4' ~ -l,.o---:lo.2=-----::o~. 4:---o;;-1.74 -,of<.s,--,lt;;-o ---;1;1::. z>-;"u

X

Fig.l4-Flowvclocityproli lesfordifferentawhen U0 = 10

and < T,. > = 0

However, this is just the reverse for the case of tangential or normal magnetic field. Increasing rt' decreases the spin ve locity and thi s is shown through Fig. I 0.

Distinguishing the effect of the speed of the upper plate over spin velocity is noted through Fig. I I . As 11 ' increases, spin velocity increases, which is just the opposite of when the speed of the


upper plate is small. However, this trend is similar for the case of tangential or normal magnetic field , whatever be the speed of the upper plate (small or relatively large).

2.0 ..------------ ---------,

o.o

-2 0

Wy

-4.0

-6.0

ap· -. 1.0 az-

-8.0 L.-__.J.__::-'-:--7-:---=-'::---:-'=--;-';~--;-' o.o 0.2 0.4 0.6 0.8 1.0 1. 2 1.4 N

X

Fig. 15- Spin velocity profi les for different a when U 0 = I 0

and < Tr > = 0

4-0r----- --------------, ap' az = o.o

X

Fig. 16 - Flow velocity pro tiles for different y

Dimensionless flow and spin ve lociti es for various values of the non-dimensional parameter a

and U0 (speed of the upper pl ate) with zero

magnetic torque dens ity are shown through Figs 12-

15. Increasing speed of the upper enhances the flow velocity, whereas it has the opposite trend over the spin velocity so as to decrease it and these are depicted in Figs 12 and 13 . Dominating feature of the effect of the speed of upper plate is obviously displayed through Figs 14 and I 5.

12.0,.----------------~

10.0

8.0

~

Wy 6.0

4.0

2.0

0.2 0.4 o.s o.s X

8p' - = o.o a'Z

1: 1.5 i

1" i

-1 = 1.5

1.0 1.2

Fig. 17 - Spin velocity profiles fo r different y

Figs 16 and 17 di sclose the significant effect due to the speed of the upper wa ll over the flow and spin velocities with zero pressure gradi ent. The effect of moving of the upper pl ate is to even change the nature of ve locity profiles, so as to even reverse the trend for imaginary y.

6 Conclusion

The following conclusions can be drawn:

I. In genera l, in the uni form magnetic field , the magnetization characteristi c depends on fluid spin ve loci ty but does not depend on flow veloc ity.

2. The magnetic force density along the plates is zero, while the magneti c torque den sity is non-zero.

3. The effect of the upper plate acce lerates the flow velocity, while it decelerates the spin ve loc ity when the spin viscos ity and pressure gradien t are non-zero and magnetic torque densi ty is zero.

4. When the speed of the upper plate is re latively higher, its effect is so sign ificant over the flow and spi n velocit ies so as to reverse their nature for the case of zero pressure gradi ent.


5. When the spin viscosity is not uniform, the moving plate effect over the flow velocity is so dominant , as to accelerate it and also to alter its trend whereas its influence over spin velocity is to decelerate it.

6. It is interesting to note that this moving plate effect is so influencing and enhancing for the case of rotating magnetic field in all cases.

7. It is further seen that, under a relatively higher speed of the upper plate, the rotating field behaviour is analogous to that of transverse or axial magnetic field over spin velocity.

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couette flow of ferrofluid under spatially uniform sinusoidally...

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