Indian Jou rn al 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, Bharathi ar University, Coimbatore 641 046
Received 15 May 200 I ; accepted 27 December 200 I
The aut hors analysed the behaviour of Couette fl ow of a ferrofluid under spati all y uniform sinu soidall 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 equati ons 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,
freq uency, direction coordin ates along and transverse to the plates, as a function of pressure gradient along the plates,
vortex viscosity, dynamic viscosi ty and fe rrofluid magneti c susceptibility. Solutions for certain limiting cases are also
given.
1 Introduction
Technologically, the topic of asymmetric stress
in magnetic fluid s is intriguing because of its
poss ible applications in the control of heat or mass
transfer processes in convective flow s, 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
subj ect holds inherent sc ientific interest.
Occurrence of asymmetric stress in magneti c
fluids can be due to th e influence of un steady
magneti c fields such as alternat ing, travelling or
rotatin g magneti c fields. Okubo et al. 1 studied the
stability of the surface of a magneti c fluid layer
under the influences of verti cal altern ating magneti c
fields ex perimentall y and analyticall y. Propagation
of waves on the free surface of magnetic fluid under
trave ling
magneti c
field
is
ex perimentall y
investi gated by Kikura et af2. It is found that the
surface velocity of the magneti c fluid depends on
the intensity and frequency of travelling magneti c
fie lds and depth of the channel contai ned with the
magnetic fluid .
The motion of ferroflu id in a trave lling wave
magnetic field has been paradox ica l as many
investigators fi nd a criti cal 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 magneti c
particles suspended in a ferrofluid causes the
magnetization M to lag behind a trave lling H .
A body torque density
T=!l<J(Mx H )
arises when M is not collinear with a spatially
varying magnetic field H . Under thi s condition ,
Rosensweig 5 has developed fluid mec hani cal
analysis to ex tend traditional viscous fluid flow s to
account for the non-sy mmetric stress tensor th at
results when M and H are not co llinear and to then
linear
and
angu lar
si multaneously
sati sfy
momentum conservation
equat ions for
th e
ferrofluid.
Ferrofluid
motion under travelling wave
magneti c fields with sinu so id al time and space
de pendence has been recently analysed in the study'
wherein, the magneti c fi elds were non-uniform and
fo rce and torque densities were non-zero .
Recently Zahn and Greerr, have ana lysed
ferro hydrod ynamic pumping in a spatially uni form
sinuso idall 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
sinu soi dally 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
I
i
Ferrofluid
d
B=llo(H+M)
. . . (4)
3 Equations 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.
. 0Wy(x)
and
Hz = R [Hz e ion t ]
H
Since the fluid is incompressible :
V.v=O, V.w=O
and the coupled linear and angular momentum
conservation equations for force den sity fand torque
density T for a fluid in a grav ity -gl are 1 :
p[av +(v.V)v]]=
at
-Vp+ f+2t;Vxw+U; +ry)w 2 v-pgi
... (6)
/[ aa~ + (v.w )w]] = T + 2( (V X
----U'z(X)
... (4)
\1-
2
2w) + ry'V w ... (7)
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:
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 vi scos ity 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 a verage magnetic
force and torque densities.
The magnetization relaxation equation 1·5 with a
ferrofluid undergoing simultaneous magnetization
and reorientation due to fluid con vection velocity v
and particle spin at angular ve locity w is :
oM + (v.w)M- w x M + -[MI
XoH =0
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
... (2)
A
A
A
A
l 'u
H=R[(H,. ( x )i+H/)e' ]
3.1 To obtain Mx and M z
... (3)
.. .(8)
JOTHIMANI & ANJALI DEVI: COUETTE FLOW OF FERROFLUID
~
~
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
~
... (9)
iD.M , -w,M z + - - - Hx
·
r
r
~
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
~
... (II)
H, =x -M x
flo
~
187
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:
< fx>
= - :x( ±,UoI Mx 1 } < f z > =
2
0
... ( 16)
Similarly, the torque density 1s g1ven by the
equation:
T=fl 0(M
X
H)= flo( -M xH z + M zHJ]
... ( 17)
The torque is only y-directed:
T, =M zBx -floM.JH z +M z)
.. .( 18)
The time average component of torque density is
then :
~
Solving for M x and M z by using Eq. (II) in
... ( 19)
Eqs (9) and (I 0):
with superscript asterisks indicating the complex
conjugate of a complex amplitude field quantity .
~
8
X0 [/f zw,r + (iQr +I) x ]
M=
·
flo
2
x
(w,.r) + (iQr + l)(iQr +I+ Xo)
X0 [(i0.r +I+ X0 )Hz -
8
... (12)
It is convenient to express modified pressure as:
w,r]
M=
flo ·
2
z (w,r) + (iQr + l)(iQr +I+ Xo)
x
... (13)
Eqs ( 12) and ( 13) represent the magnetization of
the ferrofluid as a function of the imposed fields
~
3.3 Coupled dimensionless linear and angular
momentum conservation equations
p
'
I
~ 2
= p +-flo
I M x I + pgx
4
... (20)
So that Eqs (6) and (7) in the negligible inertia,
viscous dominated limit become:
~
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
given by the equation:
f=fl 0(M. w) H
IS
... (14)
('r:>
2
c. vz
. ) --+
+T]
,d
/
dw
::~ '
2r~:.----.r up -0
dx2
2
w,-1J - 2
dx
-
2(
(dv
_ z
dx
oz
+ 2w.r
} < T, >= 0
... (2 1)
... (22)
dx
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 magneti c
field strength H0 . Therefore:
INDIAN J PURE & APPL PHYS, VOL 40, MARCH 2002
188
A
A
- 2
- 2
.- - 2
- 2
•
+ 2R[[X0 (wy
- .Q ) + t.O(w, - .Q - 1- X0 )]H zBr]}
H B
Q=Qr , H =-,B = - -
Ho
JloHo
/[(w; -
!'F + I+ X0 ) 2 + (2 + X0 ) 2 fP J
... (29)
Note
A
that
the
phase
re lationship
between
A
H z and Br is very important because, for our case
studies with ry' = O,wr must be zero at the fixed
-
r
'=' -
u-o
-oiJ' - - d- 2-op'
U- 0 ?
, -:~ - -:~·- ,
- u
J10 H 0r oz J1 0 H 0 o z
c
2(
· · .(23)
. . . (24)
d
2
rd-
} < T-
Y
w,
Eq. (29) approximately reduces to :
Hence the dimen sionless flow and spin ve locity
equation s are:
' w,
. - v
17-- · -( ___z +2W
tfX 2
dX
r
boundaries at x=O, and x= d. It is useful to examine
Eq. (29) in the limit of small (V-'. T o first order
Jim
w,. << l
<T"- >""fa +aw,.·
. . . (30)
where
... (3 1)
... (25)
>=0
where
... (26)
-
-
M,=
-
Xo[H zw,. + (i.Q + 1)BJ
_
- ?
.-
.
... (27)
.-
. w; + (t.O + l)(tO +I+ X0 )
Note that a
can be positi ve or negative
depending on the value of .6 compared with I or I
+ Xo· It is inferred that a is positive at high
frequencie s and negative at low frequencies.
4 Solution of the Problem
M = Xo[(iD +_I+ i 0 )Hz - Bxw,]
z
... (28)
w;+ (i.Q + l)(i.Q +I+ Xo)
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 field s. The
primary complexity of the analysi s is that the time
average
torque
density
< T" >
depends
compli cated way on the spin velocity
in
a
w-' which
depends on .X .
Substituting Eqs (27) and (28) into Eq. (26)
gtves:
< f >= Xo {w, [I 8,.1 (w ?,~- 0 +I)
2
2
-'
2
.
.
.
+Iii z 12 (w~ _.Q2 +(I+ Xo)2)]
4.1 Zero spin-viscosity (fj' = 0) solutions
First the simple limiting case of zero spi nviscosity is examined. Solving Eqs (24) and (25),
one can obtai n:
_ ( _)
I o{/
7J dz
- ?
I :rJ
.r:: k _ k
<T,.>ax + 1x+ 2
7J o
·
vz x =--:::-x-- -
... (33)
where k 1 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:
k 1 =U 0
1
op'
1I
-
--:;---:::-+-:;-} <T" >di,k 2 =0
7J dz
7J o
·
So that the ve locity becomes :
JOTHIMANI & ANJALI DEVI: COUETTE FLOW OF FERROFLUID
- V, (X)
-
{)jj' -(= U- oX- + -:=-I -:~X X-
I)
1J oz
. . . (35)
2
2ff
az:
s +ff - >+f
s -
----<T,.
-
I
<T,.
0
-
>ax
.
vz(x =I)= 00 , wy(x = 1) = o
. . . (38)
The Runge-Kutta method cannot directly solve
two-point boundary value problem with boundary
conditions at
= 0 and =I . Rather it requires
x
l
... (36)
It is noted that Eqs (35) and (36) are not known
analytical solutions for
. . . (37)
vzand
x
specification of functions and their derivatives at
= 0 and the system is then numerically integrated
to
= 1 . Thus our procedure was to specify
vl (x = 0) = 0 and (jjy(x = 0) = 0 and to guess the
w =- 0o - -1-[a{/ (2x-l)
-'
vz<x = o) = o, wy(x = o) = o
x
And it is further implied from Eq . (25) that:
189
wr because < f,. >
.
.
.
()jj' .
vanes w1th x and w,. vanes w1th x . When - I s
-
az:
x
derivative values:
D _dv
_z D - dw"
--
,-ax'
ax
At x = 0 and then
2 -
integrate numerically to
x =I
and define:
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
Then, use Newton's method to find best values of
D 1 and D 2 to derive F 1
vzand (jjy
resulting
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-'
x as < f,. >
in Eq.
=D0
and F 2 to zero. The
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)
is the
If < T" > = 0 in Eq. (25) the flow is driven by
only constant given by Eq. (29) and U0 • Then the
the pressure gradient and Eqs (24) and (25) have
analytical solutions of the form:
(36) is a constant, independent of
solution to Eq. (35) is
vl (x) =Vox
and
<T > _o
D
w.(x) = ----h--'
2s
- 2
a-'
=!____E_ff az: 2Ax- {3[Be"-x -
v (x)
z
2
The torque is constant with position if wr (x) is
Ce -ax ]+ D
... (39)
... (40)
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:
where
a=
2s7J
11'<s +7]'),
13
2s7J'
= 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 :
190
8
INDIAN J PURE & APPL PHYS, VOL 40, MARCH 2002
= [A (j3- j3e-" 1
2) + U 1 (e -" -I)]
A2
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.
C = _:_[A_.!_1 (.:.:_j3_-___:j3~e_-'_'+_2__.:_)_+_U_:_1(_I_-_e -_" -=-)]
Az
D = [j3U 1(e" + e-" - 2) + A1(/3'2e"- j3 2e-"- 4 j3)]
Az
where
A2 =2j3(e" +e-" -2)+2(e-" -e" )
(c) Zero pressure gradient
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
<r'
~ =0
a.z
ii =o.o
0.2
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:
-0.2
1 - _ dv, --cs+7J)- +S'w,. =C
2
dX
.
-0.4
.. .(41)
~0
= _
-
~D.
-10
0
for an axia l magnetic fie ld
wJx)=
.
_D
}SIIlh y
where
".(42)
[}Sinhy+(e-y- 1)
e-P' +(1-er)e- P' +U0 (e P' -e- )5') ]
20
-
y(( +7J)smh y
(e r - I)+ 2U0 (xsinh y- sinh }:X)]
10
Wy
Fig. 2 - Effect of .Q on time average torque density
(e-r -l)+(eP' -1)(1- e-Y)-(e-P' -I)
".(43)
I
':---'---:L----'---'--.J....._-~-~
-20
[2x( er -1)
I
i'i =s.o
where Cis a constant. The flow solutions are then:
v,(x)
I
(H~ = /-/0 ,
-
< T,. >
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
sp m
velocities
respectively . Increas ing
U0
increases the flow
JOTHIMANI & ANJALI DEYI: COUETTE FLOW OF FERROFLUID
velocity, whereas it decreases the spin velocity for
the case of non-zero spin velocity .
Xo = 1. 0
12 0
N
I
(
=1. 0 '
N
ll = 1. 0'
191
ap' = 0.01
az
N
U0 = 10.0
0.3r-- - - - - - - -- - -- ---..,
8.0
0.2
Uo = 3.0
Uo =1.0
~~~~======::;oo = 0.1
Uo =0.0
0.1
<ly)
-0.1
o.o
20
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
field
(H, =0,
1. 0
Bx = flo Ho)
Q.6
Uo =o.o
X0 : 1.0
N
Uo = 0.1 ·
Uo:. J.o
~
uY y
0.4
-2.0
0.2
"'
(Ty)
Uo = 3.0
Uo =10.0
-3.0
o.o
-4.0
-0.2
-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
-Q.4
Fig. 6 - Spin velocity profiles for different
-
Wy
-
Fig. 4 - Effect of Q on
-
< Tr > for a rotating magnetic field
(H, =iHo,
Bx = ~to Ho)
{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
INDIAN J PURE & APPL PHYS , VOL 40, MARCH 2002
192
1-2 .---- - - - - - - - - - - ..-------;
~
a'i)·
Xo=1.0, (: 1.0 , ~:1 . 0, az=0.01
0.16 .------~--"'-----;;d...-;P,·-------,
Xo::: 1. o, ("' 1. 0 , ~ =1. 0, ~:: 0.01
1.0
0.12
0.8
0.08
~ = 0.001
1
\_1·=0.1 ~'= 0.2,
j
o.
0.4
0.2
~·:: 0.02 , ~ :0.05
o.o
...
Fi g. 7 - Effect of
if
A
magnetic field
X
over now veloci ty profil es for an axial
-
(H, = !-/0 , B, = 0) when U 0 = I .0
o.2 .------.-----:;;a"'~------,
Xo=1 .0 , (=1.0, n=l.D,
..!:0.01
1
if
Fig. 9 - Flow velocity profi les for different
A
magneti c field
(If, = if-10 , B,
for a rotat in g
-
= flo f-lo) when
U0
= 1.0
az
o.o
N
r T)' = o.ool
/~'= 0 01
r 1' = 0.02
~· =
r~' =
o.os
o.l
r ~'= 0.2
-0.6
o.o
-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. 8 - Effect of
if
magnetic field
over spin velocity profiles for an axia l
(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
Fig. I 0 - Spin velocity profiles for difTcrcnt
if
A
magnetic field
(If, = if-1 0 , B,
for a rot ati ng
-
= Po !-/0 ) wh en
U0
= 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
JOTHIMANI & ANJALI DEVI : COUETTE FLOW OF FERROFLUID
193
0.6.--- - -- -- - - - - - - - - ,
both increasing and decreas ing trend, whereas spin
ve loc ity increases for increasing r( .
----- Uo = 0.1
- - Uo = o.o
2.0 ~------------:;;;- -----,
X0 = 1. 0, f=1 .0, ~=1.0, ;~·=0 . 01
a= 30
0.2
o.o
-
Wy 0.0
~· = 0.2
""---~' = 0. 1
Wy
-0.2
~· = o.os
- 0.4
~· = 0.02
~·= 0.01
- 6.0
-o. 6o.ol___j[____l:--~:----7:;:--~--7:;--~
0.2
0.4
0.6
o.s 1.0
1.2 1.4
~ 1 = 0.001
X
-8.0 L--.,:L:----::-J..,----:.1.::----::L::----:-~---;-~-;-l
o.o 0.2 0.4 0.6
O.B
1.0
1.2
1.4
Fig. 13 - Spin velocit y profiles for different a
~
when
X
Fig. II -Effect of
if
over spin ve locity profi les for a rotatin g
magnetic field (If: = iH0 ,
B_, = p
0
H0 ) when
U0
< T,. >
=0
16.0 .----- -- - - - ------::;-----,
8p'
- = 1.0
az
= 10.0
12.0
0.2 r - -- -- - ------ Uo = 0.1
- - U0 =
0.1
- - - --
-
- --.
o.o
s.o
8p'
- = 1.0
az
a= 15 , a= 20, a = 30
-0.1
- 4' ~-l,.o---:lo.2=-----::o~.4:---o;;-1.74 -,of<.s,--,lt;;o ---;1;1::.z>-;"u
X
0.2
0.4
1.2
0.6
1.4
X
Fig.l 4-Flowvclocityproli lesfordifferentawhen U 0 = 10
and
< T,. >
=0
Fig. 12 - Flow velocity profiles for different a
when
< T,. >
=0
Figs 9-11 illustrate the effect of spin viscosity
over flow ve loc ity and spi n ve loci ty respectively ,
for a rotatin g magnetic fie ld . The effect of speed of
the upper plate is to suppress the va lues in general
over the flow veloc ity. As sp in viscos ity increases,
flow ve locity increases as well as decreases.
However, this is just the reverse for the case of
tangential or norma l 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 sp in veloc ity is noted through
Fig. I I . As 11' increases, sp in velocity increases,
whi ch is just the opposite of when the speed of the
INDIAN J PURE & APPL PHYS, VOL 40, MARCH 2002
194
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 ..-- - -- - - -- - --
---------,
ap·
az-
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.
- . 1.0
12.0,.----------------~
8p'
- = o.o
o.o
a'Z
10.0
-2 0
1: 1.5 i
8.0
Wy
1" i
-4.0
~
Wy 6.0
-6.0
4.0
-8.0 L.-__.J.__::-'-:--7-:---=-'::---:-'=--;-';~--;-'
o.o 0.2 0.4 0.6 0.8 1.0 1. 2 1.4
-1 = 1.5
2.0
N
X
Fig. 15 - Spin velocity profi les for different a when U 0
and
< Tr >
= I0
0.2
o.s
o.s
1.0
1.2
X
=0
Fig. 17 -
4-0r--- --
0.4
- - - - -- -- ------,
ap'
az = o.o
Spin velocity profiles fo r different y
Figs 16 and 17 di sclose the sign ificant 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 th e
trend for imagin ary 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 sp in
ve loci ty but does not depend on flo w ve loc ity.
2. The magnetic force density along the plates is
zero, while the magneti c torque den sity is non-ze ro.
X
Fig. 16 -
Flow velocity pro tiles for different y
Dimensionless flow and spi n ve loc iti es for
various values of the non -dimen sional parameter a
and U0 (speed of th e upper pl ate) with zero
magn etic torqu e dens ity are shown throu gh Figs 12-
3. The effect of the upper plate acce lerates the flow
ve locity, while it decelerates the sp in ve loc ity when
the spin viscos ity and pressure gradi en t are non-zero
and magneti c torque den si ty is zero.
4. When the speed of the upper plate is re latively
hi gher, 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 press ure gradi ent.
JOTHIMANI & ANJALI DEVI : COUETTE FLOW OF FERROFLUID
195
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.
References
2
Kikura H, Sawada T, Tanahashi T & Seo L S, J Magn
Ma gn Mater, 85 ( 1990) 167.
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.
3
Zahn M & Wai nman P N, J Magn Magn Mat er, 122 ( 1993)
323.
4
Calugaru G H, Cotae C, Badescu R, Badescu V & Luca E,
Rev Rown Phys, 21 ( 1976) 439.
5
Rosensweig R E, Ferrohydrodynam ics
University Press, Cambrid ge), 1985.
6
Zahn M & Greer D R, J Magn Magn Mater, 149 ( 1995)
165.
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.
Okubo M, Ishibashi Y, Oshima S, Katakura H & Yamane
R, J Magn Magn Mater, 85 ( 1990) 163.
(Cambrid ge