Effects of fluid convection and particle spin on ferrohydrodynamic pumping in traveling wave magnetic fields

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
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.