Download Zahn, M. and P.N. Wainman, Effects of Fluid Convection and Particle Spin on Ferrohydrodynamic Pumping in Traveling Wave Magnetic Fields, Journal of Magnetism and Magnetic Materials 122, 323-328, 1993

Journal of Magnetism and Magnetic Materials 122 (1993) 323-328
North-Holland
Effects of fluid convection and particle spin on ferrohydrodynamic
pumping in traveling wave magnetic fields
M a r k u s Z a h n a n d P e t e r N. W a i n m a n
Massachusetts Institute of Technolo,~', Cambridge, MA 02139, USA
Ferrofluid pumping in a traveling wave magnetic field without a free surface is analyzed by finding the time average
magnetic force and torque and numerically solving the coupled linear and angular m o m e n t u m conservation equations. It is
necessary that the fluid convection and particle spin terms be significant in the magnetization constitutive law and that spin
viscosity be small in order to predict the experimentally observed reverse pumping at low magnetic field strengths and
forward pumping at high magnetic field strengths.
1. Background
The motion of ferrofluid in a traveling wave
magnetic field has been paradoxical as many investigators find a critical magnetic field strength
below which the fluid moves opposite to the
direction of the traveling wave (backward pumping) while above, the ferrofluid moves in the same
direction (forward pumping) [1]. The value of
critical magnetic field depends on the concentration of the suspended magnetic particles and the
fluid viscosity. Under ac magnetic fields, fluid
viscosity acting on the magnetic particles suspended in the ferrofluid causes the magnetization
M to lag behind a traveling H. With M not
collinear with a spatially varying H, there is a
body force density f = / z 0 ( M . IT)H, and a body
torque density T =/~0(M x H ) acting on the ferrofluid. Fluid mechanical analysis has been developed to extend traditional viscous fluid flows to
account for the nonsymmetric stress tensor that
results when M and H are not collinear and to
then simultaneously satisfy linear and angular
m o m e n t u m conservation for the ferrofluid [2].
The most recent and complete experiments
placed a ferrofluid filled circular beaker within a
two-pole, three-phase cylindrical stator which imposes a uniform rotary magnetic field resulting in
a fluid velocity distribution. In the uniform magnetic field there is no body force but there is a
body torque that drives the flow. However, the
torque driven theory predicted a velocity distribution that was opposite to that measured [3]. The
paradox was resolved by self-consistently including magnetic tangential surface stresses on the
free surface whose shape depended on the fluid
level, surface tension, hydrodynamic motions, and
magnetic field.
2. Experiments
To indicate that magnetic tangential surface
stress as the driving mechanism is not the whole
story, experiments were performed confining ferrofluid within tightly spiraled tubing with no free
surface within a four-pole, three-phase stator, as
shown in fig. 1 [4]. In the absence of a magnetic
field, the ferrofluid level on the meter stick equals
the ferrofluid height in the reservoir (h = 0). The
meter stick is inclined at a slight angle 0 to
improve m e a s u r e m e n t accuracy so that a small
vertical displacement h has a much larger horizontal displacement L along the meter stick as
h = L sin 0.
Correspondence to: Dr. M. Zahn, Department of Electrical
Engineering and Computer Science, Laboratory for Electromagnetic and Electronic Systems, Massachussetts Institute of
Technology, Cambridge, MA 02139, USA.
(1)
With an external trigger to an oscilloscope serving as a time reference, an increasing delay of a
small sensing coil voltage as the coil was moved
0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
324
M. Zahn, P.N. Wainman / Ferrohydrodynamic pumping in trat,eling fields
Reservoir
~ ~Meterstick
.......................................................
.......
~Meterstick
............................................
....
t Valve
Stator
Spiral
Tublna
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
i
!
[
L
Fig. I. The s t a t o r of a t h r e e - p h a s e m a c h i n e g e n e r a t e s a r o t a t i n g m a g n e t i c field and s u r r o u n d s a spiralled tube of ferrofluid. The
tubing is c o n n e c t e d to a sight glass along a m e t e r stick at a slight angle 0 from the horizontal. T h e m a g n e t i c force has fluid vertical
d i s p l a c e m e n t h > 0 for forward p u m p i n g and h < 0 for b a c k w a r d p u m p i n g .
within the stator confirmed the direction of magnetic field travel.
A representative set of measured vertical displacement h versus rms applied current is shown
in fig. (2). For low currents, the pumping has
h < 0 corresponding to reverse pumping, while
for higher currents h > 0, corresponding to forward pumping. Because there is no free surface,
the pumping must be due to volume forces and
torques.
I
3.1. Magnetization constitutit,e law
l
E
/
/
E
'2
/
/
o
E
~-
-'2
O.
0 ¸
-4
"0
G
0.0
The magnetization relaxation equation with
ferrofluid undergoing simultaneous magnetization and reorientation due to fluid convection at
:
I
m
3. Critique of past models
I
t
i
2.5
5.0
7.5
Current
,
I
E
10.0
12.,5
15.0
17.5
gO.O
(rms amperes)
Fig. 2. R e p r e s e n t a t i v e fluid vertical d i s p l a c e m e n t h versus
s t a t o r c u r r e n t 1 (10 A r m s c o r r e s p o n d s to = 100 G rms).
M. Zahn, P.N. Wainman / Ferrohydrodynarnic pumping in traceling fields
325
velocity v and particle spin at angular velocity ¢o
is
--+(v.lY)M-ooXM+Ot
1[
r
M
d
=0,
(2)
g
where r is a relaxation time constant and Mo/H
= X0 is the effective magnetic susceptibility, which
can be magnetic field dependent.
Prior steady state analysis considered a uniform rotating magnetic field where it was assumed that the magnetization vector rotated with
the same angular velocity as the field, while
lagging the field at a constant phase angle a,
thus yielding a constant torque density Tz =
~oMH sin a. This uniform magnetization assumes that the second and third terms in (2) are
negligible, otherwise the spatial profiles of v and
oJ would make the magnetization a function of
position. Using the parameters of the Moskowitz
and Rosensweig experiment described on p.263
of ref. [5], the uniform spin velocity over most of
the beakers volume is I oJI = 1800 r a d / s making
it comparable to the constant angular speed of
the applied magnetic fields at 100 Hz (628 r a d / s )
and 1000 Hz (6280 rad/s). Thus, gradients in spin
near the wall can have a large effect on the
magnetization characteristic of (2).
3.3. Traveling wave pumping
The velocity and spin fields were also calculated for a planar ferrofluid layer of thickness d
excited by a traveling wave current sheet with
wavenumber k, as shown in fig. 3, again assuming
the convection and spin terms in (2) were negligible [6]. Taking the spin and absolute viscosities to
be equal, ~"= r 1, and with kd = 1, analysis found
in fig. 3 of ref. [6] shows that the peak spin
velocity near the current excitation was
/,* X 0 H t 2 n
^ el(~t., kz)
K = RE{K
}
Y
p_ ~, o¢
Fig. 3. A p l a n a r ferrofluid layer b e t w e e n infinitely m a g n e t i cally p e r m e a b l e walls is m a g n e t i c a l l y stressed by a traveling
wave c u r r e n t sheet.
3.2. Uniform rotating magnetic field
~O') peak
~ z
where X0 is the magnetic susceptibility and Hta n
is the tangential magnetic field which equals the
surface current density. Using as representative
numbers ( = r l =0.0012 kg m -1 s - l , /~0Ht,, =
0.01 T (100 G), and )¢o = 0.3, the peak spin velocity was, ~0pe~k= 400 r a d / s . For alternating fields
of order 60 Hz, such a peak spin field cannot be
neglected in (2). Similarly, the peak linear velocity was calculated to be
~'vz
dgoXoH,2n
= 0.02,
(4)
so that the convection effective frequency vzk =
(t'Jd) = 400 r a d / s also cannot be neglected.
For the case studies investigated, 77' was
nonzero. However, it is interesting to observe
that if rl ' = 0, the magnetization force density
f = go(M. W)H is exactly cancelled by the curl of
the magnetic torque density, V × T= lYX(goM
x H). The velocity field is then independent of
magnetic field strength while if r I' > 0, the pumping is always forward. This implies that reverse
pumping can possibly occur for the more complete analyses including velocity and spin terms in
(2) with rl' very small.
4. Governing equations
4.1. Fluid mechanics
For incompressible fluids,
= 0.02,
(3)
l r . v = 0;
I7. o~ = 0,
(5)
M. Zahn, P.N. Wainman / Ferrohydrodynamic pumping in traveling fields
326
and the coupled linear and angular momentum
conservation equations for force density f and
torque density T for a fluid in a gravity field - gi,,
Substituting (9) into (2) relates the magnetization
to the magnetic field as
([j(.Qr-kt:~r)
are
] ~ x = )(0
p
+
11/7., + ~o,.r/4:}
[j(~Qr - kt'.r) + 1] 2 + (~o,.r) 2
~ +(v.v)v
=A
= - Vp + f + 2~V x oJ + (~ + ~7)V2v - p g i , . ,
(6)
It-~t +(v'V)o~
{
]
= T + M ( l T x v - 2~o) +r/ plT-oj,
,o =,o,,(x)i,..
(7)
(8)
(11)
[j(.Qr-kt':;)+
l]/~:)
[j(~Qr- kv:r) + 112+ (w,.r) 2
wyr/4.,+
= -Ajk,~-B--
where p is the mass density, p is the pressure, ~"
is the vortex viscosity, r/ is the dynamic viscosity,
1 is the moment of inertia density, and r/' is the
bulk coefficient of spin viscosity.
We wish to apply these equations to the planar
ferrofluid layer confined between infinitely permeable walls as shown in fig. 3. At the x = 0
interface a traveling wave current sheet is imposed. We assume that the planar ferrofluid has
viscous dominated flow so that inertia is negligible and is in the steady state so that the fluid
responds to the time average magnetic force and
torque densities. Then the flow and spin velocities are of the form
v=,,:(x)i:;
d2
- Bjk2,
dx
d2
dx
(12)
However, because t,. and wy in eqs. (11)-(12)
depend on x, )~ does not obey Laplace's equation
d[d
[
dx
~-x (1 + A ) +Bjk~(
-jk
1
1
-jk)~(1 + A ) + B ~ - ~
=0.
13)
4.3. Magnetic fbrce and torque densities
For 0 < x < d, the magnetic force density ~s
f=l-%(M" V)H.
(14)
The time average components are then
d/q*
]
( f ~ ) = ½/x°Re M~ d ~ +Jkll21:121"* '
(15)
4.2. Magnetic fields
¢16)
The traveling wave current sheet gives rise to a
traveling wave magnetic field of the form
H = R e ( [ 1 2 1 x ( x ) i , + l q ~ ( x ) i : ] ei[m-kzl},
(9)
which in the current free ferrofluid is curl-free,
allowing definition of a scalar potential X,
VXH=O~H=~"
X.
(10)
where the quantities with asterisks represent
complex conjugates. Similarly, the torque density
is
T =/z0M × H
(17)
M. Zahn, P.N. Wainman / Ferrohydrodynamic pumping in traceling fields
It is convenient to define a pseudo-energy function and a modified pressure as
(.Ix) =
d~
dx
- --;
p'
=p+~+pgx
(18)
and then numerically integrate to x = d and define
F 1 =)~(x = d);
Fz=u=(x=d);
F 3 =wy(x = d).
so that (6)-(7) in the viscous dominated limit
becomes
d2Uz
dwy
(~'+r/)-d-~-x2 + 2~" d~-
, d2toy
r/ d x 2
0p'
0z + ( f ~ ) = 0 ,
[ dv~
2((~x
(19)
)
+2o~y
+(Ty)=0.
(20)
5. rI' 4 : 0 s o l u t i o n s
327
(23)
We would then use Newton's method to find
best values of D 1, D 2, and D 3 to drive F l, F2,
and F~ to zero. As a check, (13) was replaced by
Laplace's equation for ~ and ( f ~ ) and (Z~,) were
replaced by the expressions from the earlier analysis that neglected velocity and spin in (2) [6]. The
numerically obtained solutions were then identical to the analytical solutions from that approximate analysis.
5.1. Solution method
5.2. Case studies
Equations (13), (19) and (20) represent a
sixth-order system where )~ is complex that can
be numerically integrated by the R u n g e - K u t t a
method. At the rigid boundaries the velocity and
spin are zero, while the tangential component of
H is zero at x = d and equal to the surface
current at x = 0 so that the boundary conditions
are
It is convenient to express parameters in nondimensional form as
r
s~ = ~ r ;
~'=
;
d2~"
L,z(x = 0) = 0;
~oy(x = 0) = 0;
I 4 z ( x = O) = - j k ~ ( x
vz(x=d)
=0;
= O) = - I ( y ,
%(x=d)
[c = k d ;
=0;
= d) = O.
However, the R u n g e - K u t t a method cannot directly solve systems with boundary conditions at
x = 0 and x = d. Rather, it requires specification
of functions and their derivatives at x = 0, and
the system is then numerically integrated to x = d.
Thus our procedure was to specify vz(x = 0) = 0,
wy(x = 0) = 0, and ~ ( x = 0) = - j K y / k
and to
guess derivative values
Dt = ~d)~
x x=0 ;
D3
= dwy
d x x=0
1~0)(0 [ /~y [ 2,7.
0 -
(21a)
(21b)
lq~(x = d) = - j k ~ ( x
(=
D2 = -~-x
dcz x=O;
(22)
;
(24)
(
To emphasize the magnetic field effects on
pumping, we set the pressure gradient to zero
and solve for the flow rate per unit depth
Q =
c,z dx.
(25)
The key magnetic field p a r a m e t e r is f, so fig. 4
shows representative non-dimensional flow rate
versus log (. With ~ ' > 0 . 0 4 , the flow rate is
positive (forward pumping) for all cases investigated which includes 10 -3 _< ~ < 10 3, 10 3 < ~ <
103, 1 / 2 </~ < 2, all with ~ = 2.
Because we expect reverse pumping with ~ '
close to zero, we tried to run case studies with
small ~', however, because it is the coefficient of
the highest-order derivative it was not possible to
obtain numerical convergence for ~ ' < 0.04.
M. Zahn. P.N. Wainman / Ferrohydrodynamic pumping in trat'eling fields
328
6. ~' = 0 solutions
O,OZ
0,0!
We thus set 7' = 0 in (20), but this lowers the
system to fourth order and the two boundary
conditions on w>. at x = 0 and x = d must bc
dropped. The Runge-Kutta integration can proceed for c. and )~, but solving (20) for ~oy gives a
nonlinear algebraic equation rather than a differential equation so that Newton's method is used
to solve for coy. Figure 5 now shows that the flow
rate is negative (reverse pumping) for high enough
corresponding to low magnetic fields while the
flow rate is positive (forward pumping) for low ~:
corresponding to high magnetic fields, in agreement with experimental observations.
/
/
0.00
I
I
I
I
I
D
1
2
3
4
i
-2
-1
7. Concluding remarks
Fig. 4. Nondimensional flow rate 0 = Q~/(dZtxl~Xc~l I£~ 12)
versus log((-) with ~ ' = 1, ~(} = 1, /~ = 1, ?)p'/~z = 0, X0 = 1,
and ~ = 2.
"Q ( X 1D-3 )
0.1
I
I
I
I
0.0
This work was supported by the National Science Foundation under Grant No. ECS 8913606
and was the subject of P. Wainman's MIT bachelor's thesis.
-0.1
-0,2
References
\
\
-0.3
-0.4
I
2
I
4
,
6
I
8
,
10
12
T
Fig. 5. 0
If the fluid convection and spin terms are
small in the magnetization constitutive law of (2)
compared to the time rate of change of magnetization, there is only forward pumping if spin
viscosity 7' is nonzero. If 7 ' = 0 there is no
pumping at all. This work has shown that in order
to obtain reverse pumping it is necessary that the
fluid convection and spin terms be significant in
the magnetization constitutive law and that ~'
must be small.
versus
f=(/(izoxoil~>.12r)
with ~ ' = 0 ,
•=1,
= 1, ~}p'/az = 0, X{l = 1, and ~ = 2 showing reverse pumping
at high ~-(Iow magnetic field).
[1] G.H. Calugaru, C. Cotae, R. Badescu, V. Badescu and E.
Luca, Rev. Roum. Phys. 21 (1976) 439.
[2] R.E. Rosensweig, Ferrohydrodynamics (Cambridge University Press, Cambridge, 1985), ch. 8.
[3] R.E. Rosensweig, J. Popplewell and R.J. Johnston, J.
Magn. Magn. Mater. 85 (1990) 171.
[4] D. Misra, BS thesis, MIT, 1990.
[5] R. Moskowitz and R.E. Rosensweig, Appl. Phys. I,en. 1
(1967) 310.
[6] M. Zahn, J. Magn. Magn. Mater. 85 (1990) 181.