Ferrohydrodynamic pumping in spatially uniform ... - Semantic Scholar
Journal of Magnetismand Magnetic Materials 149 (1995) 165-173
~
Jeurnalof magnellc inalerlals
ELSEVIER
Ferrohydrodynamic pumping in spatially uniform sinusoidally time-varying magnetic fields Markus Zahn *, Donald R. Greer Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, Laboratoryfor Electromagnetic and Electronic Systems, Cambridge, MA 02139, USA Abstract
Past work has analyzed ferrofluid pumping in a planar duct driven by spatially non-uniform traveling wave magnetic fields. Here we examine a much simpler case where the applied magnetic fields along and transverse to the duct axis are spatially uniform and varying sinusoidally with time. In the uniform magnetic field the magnetization characteristic depends on particle spin but does not depend on fluid velocity. The magnetization force density along the duct axis is zero while the magnetic torque density is non-zero as M and H are not collinear due to a magnetic relaxation time constant as well as due to spatially varying particle spin. The governing linear and angular momentum conservation equations are numerically integrated to solve for flow and spin velocity distributions for zero and non-zero spin viscosities as a function of magnetic field strength, phase, frequency, and direction along and transverse to the duct axis, and as a function of pressure gradient along the duct, vortex viscosity, dynamic viscosity, and ferrofluid magnetic susceptibility. Analytical solutions for simple limiting cases are also given including an effective viscosity that depends on magnetic field strength which can be made zero or negative.
1. Introduction
The motion of a ferrofluid in a travelling 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 travelling wave (backward pumping) while, above, the ferrofluid moves in the same direction (forward pumping) [1,2]. The value of critical magnetic field depends on the frequency, the concentration of the suspended magnetic particles, and the fluid viscosity. Under ac magnetic fields, the 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 H, there is a body torque density T = / x 0 ( M X H ) acting on the ferrofluid even in a uniform magnetic field. Fluid mechanical analysis has been developed to extend traditional viscous fluid flows to account for the non-symmetric stress tensor that results when M and H are not collinear and to then simultaneously satisfy linear and angular momentum conservation for the ferrofluid [3]. Recent analyses on ferrofluid motion in travelling wave magnetic fields with sinusoidal time and space dependence have shown both forward and backward pumping when
* Corresponding author. Email: [email protected]; fax: + 1-617258-6774.
fluid convection and spin effects are included in the study [1]. In that analysis, the magnetic fields were non-uniform and force and torque densities were non-zero. In this paper, we have investigated ferrofluid motion in spatially uniform sinusoidally time-varying magnetic fields where the torque density is non-zero but the force density along the duct axis is zero. By imposing uniform magnetic fields, we have studied how fluid flow is affected by different magnetic field variations including an axial component only, a transverse component only, and both axial and transverse components that are not in phase resulting in a rotating uniform magnetic field. This work extends recent similar analysis which examined the change in effective ferrofluid viscosity under alternating magnetic field, but that work assumed the shear coefficient of spin-viscosity to be zero, only considered a linearly polarized magnetic field and only examined the magnetization characteristic in the small magnetic susceptibility and small spin velocity limits [4].
2. Governing equations
2.1. Magnetization constitutive law The magnetization relaxation equation with a ferrofluid undergoing simultaneous magnetization and reorientation
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(95)00363-0
M. Zahn, D.R. Greer /Journal of Magnetism and Magnetic Materials 149 (1995) 165-173
166
due to fluid convection at velocity v and particle spin at angular velocity to is [1,3,4] OM --+(v. Ot
1
V)M-to×M+-[M-XoH]=O, ~"
where
z = z o ( n + M).
(1)
We would like to solve for M~ and ,~t in terms of the known imposed field amplitudes H~ and /~x. Substituting (5) and (6) into (1) relates the magnetization components to the magnetic field H as
where 1" is a relaxation time constant and X0 is the effective magnetic susceptibility which in general can be magnetic-field-dependent but in this paper will be taken to be constant. For the planar ferrofluid layer shown in Fig. 1, the flow velocity can only be z-directed and the spin velocity can only be y-directed, both quantities only varying with the x coordinate
v = v~(x)i~,
to = toy(X)iy.
in~lz +
(2)
~
B~=constant,
toyl~[ x .-1-
nx,
(8)
X°I~,
(9)
T
Mz T
T
-
T
where the second term in (1) has zero contribution because v is only z-directed and M can only vary with x. By taking the x-component of (7) and solving for Hx, we obtain
We wish to apply Eq. (1) to the planar ferrofluid layer confined between rigid walls shown in Fig. 1. The imposed magnetic field H z 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 z coordinates, field components can only vary with the x coordinate. Gauss's law for the magnetic flux density and Ampere's law for the magnetic field intensity with zero current density require the imposed fields to be uniform throughout the ferrofluid dB~ =* dx = 0
Xo ^
i 12h~lx - toylQ= z +
2.2. Magnetic fields
V.B=O
(7)
&=.o(&
&=
(lO)
tZo
We then solve for hex and h,lz by using (10) in (8):
Xo[/~(toyZ) + (i/2z+ 1)/~x//% ] Mx= [(toyr)2+ Oar+
1)(inr+ 1 +Xo)]'
(11)
X o [ ( i / 2 z + 1 + Xo)I'tz-BxtoyZ/tZo]
(3)
~tz= [(to:.)2 + ( i n r + 1)(in~-+ 1 +x0)] "
(12)
dH~
V×H=O
~
dx =0
~
H~=constant.
(4)
Eqs. (11) and (12) represent the magnetization of the ferrofluid as a function of the imposed fields H~ and / ~ and as a function of the not yet known spin velocity toy 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 toy. The resulting toy then changes the magnetization. There is a strong magneto-mechanical coupling so that the magnetization and mechanical equations need to be self-consistently satisfied.
There are no y components of the magnetic fields. Therefore, the total magnetic field H and magnetic flux density B inside the ferrofluid layer are of the form
B = ~R([ Bxi x + l~z(X)iz]
ei°t),
(5)
H=9~{[I~x(X)ix+Itzi~]
eint},
(6)
n= = ~[/~=e ~"t]
d ,,'dx)
Ferrofluid
I1= ~.,.-
',o,,(x)
,uz --
~[~:'°'l
1,
y
Z Fig. 1. A planar ferrofluid layer between rigid walls is magnetically stressed by a uniform z-directed magnetic field HZ and uniform x-directed magnetic flux density Bz, both of which vary sinusoidally in time at frequency /2.
M. Zahn, D.R. Greer /Journal of Magnetism and Magnetic Materials 149 (1995) 165-173 2.3. Fluid mechanics
Solving for the x and z components with field components only constant or varying with x,
For incompressible fluids
V.v=O,
dn~
V.w=0
=-~°M~'
,I
]
t~ [ + (v. v).,
t7o d(l .,}
dx
dx
2/x°M~7 '
(17)
dH z L = u0M~ d x = 0.
av
2~V×to+(~+rl)g2v-pgi~,
d [ B x _ Mx~)
=,,oMxT--x dM~
(13)
and the coupled linear and angular momentum conservation equations for force density f and torque density T for a fluid in a gravity field - g i x are [1,3,4]
=-Vp+f+
167
(18)
The time average components of the magnetic force density are then
(14)
(f~)=-~xx
= r + 2 ~ ( v x v - 2o~) + n ' v ~ o ,
4 t%l~xlz
'
(19)
( L ) = 0.
(15) where p is the mass density, p is the pressure, ff is the vortex viscosity, 7/ is the dynamic viscosity, I is the moment of inertia density, and r/' is the shear coefficient of spin-viscosity. We wish to apply these equations to the planar ferrofluid layer confined between rigid walls in Fig. 1. 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 only to the time average magnetic force and torque densities.
(20)
2.4.2. Magnetic torque density Similarly, the torque density is given by the equation
T=t.to(M×H)=I.to(-MxH~+M~H~)i~..
(21)
The torque is only y-directed,
• ~ -- I-%M,, H~ + I-%M~ ---mx T~. /to = M~B x - txoMx(H z + M~).
(22)
The time average component of torque density is then
2.4. Magnetic force and torque densities
1
(Ty)= ~,~[l~zn*
- I,lol~lx (l~z +Mz)]
(23)
2.4.1. Magnetic force density with superscript asterisks ( * ) indicating the complex conjugate of a complex amplitude field quantity. In Fig. 2, the dimensionless torque density ( T y ) = ( T y ) / ( p . o H 2) is plotted as a function of the dimensionless spin velocity
For 0 < x < d, the magnetic force density is given by the equation
f = txo(M. IT)H.
0.3
(16)
i
!
0.15
I
"
fi-O 0.10
0.2
t'l-O
i
i
A
A
.~.a, v
O0 V
A ,~,,t~ v
0.00 -o,0~
-o.1
I
I -10
0
i
0.2
00,5
0.1
I 0.4
O0
-02
-0.IC
-02
-0.4
-20
1 -I0
1 0
I 10
-20
-I0
0
10
20
-20
t°
20
20 ~y
~tJ
Fig. 2. Dimensionless time average torque density (/~y) = (Ty)/( ~oHo2) versus dimensionless spin velocity &v = to,3- for dimensionless angular frequency ~ = g2~"= 0, 1, 5, 10 with X0 = 1 and with (a) axial magnetic field (H~ = H o, Bx = 0); (b) transverse magnetic field (/4z = O, /~x = ~oHo); and (c) rotating magnetic field (/4z = iHo, /~x =/xoHo)-
M. Zahn, D.R. Greer/Journal of Magnetism and Magnetic Materials 149 (1995) 165-173
168
COy= toy~- with H 0 a nominal magnetic field amplitude. We show three limiting cases with the magnetic field purely axially directed (/7~ =/40, /~ = 0), purely transversely directed (/7~ = 0, B~ =/x0H0), and magnetic field both axially and transversely directed with phase difference ~ / 2 (/7~ = i H 0, /~, = IzoHo ) so that the net magnetic field rotates. Throughout this report, dimensionless variables are indicated with a tilde. Note that for a stationary spin velocity (COy= 0 ) that the slope a r o u n d COy = 0 is negative at low frequencies and positive at high frequencies and that only the rotating magnetic field case in Fig. 2 has non-zero time average torque at COy= 0.
2.4.3. Coupled linear and angular momentum conservation equations It is convenient to define a modified pressure as
1
^
p' =p + -~lx o I M x 12 + pgx
(24)
so that (14) and (15) in the negligible inertia, viscous dominated limit become d 2 vz d toy (ff+'0)-~xZ +2~'~x
Off = O, bz
(25)
where
and
Xo[ (%/7z + (if2+ l)Bx] Atx= [ COre+ (ig~ + 1 ) ( i ~ + 1 + Xo)] ,
Xo[(i~+ I + Xo)/7z-Bxcoy] 3,1,= [ ~2 + (ig) + 1)(ig) + 1 + X0)] .
= yXo [-coy[ i +
+iJ~(CO2- ~ 2 _ 1 _
10 n~-,
/ 7 = Ho'
X
B=
IXoHo
'
Ho '
rr
T
yc = -~,
~ = v:~,
Coy = t o : ,
fy = . o n g ' (27)
2'0
i~/Y a5
~ ' = ~,o/-/o~-'
I
~
-,d2my '0~-~
t-2 oy--
~)Z+l+x °
+(2+x0)2~)2
Note that the phase relationship between /t~ and /~ is very important in the third term of the numerator. Because for our case studies with ~' ~ 0, COy must be zero at the fixed boundaries at x = 0 and x = d, it is useful to examine (33) in the limit of small COy.To first order in COy,(33) approximately reduces to lim ( g ) ~/~0 + oLCOy,
(34)
~y "~" 1
xo [[ ~o
Off
2+
+1 +
[ / 7 : : ]]
[I + Xo + I)2]E+ x2~)2
ixoH~o Oz.
~)
d25z _dcoy 0/5' -d~ -~
+~~
( d~z
o~
O,
)
d2 +2coy + ( / ~ y ) = 0
.
(33)
UoHo~'
We then have dimensionless flow and spin velocity equations
(¢+
/
Xo)I[H~.~, ]]]
where
28r
'0'd 2
= ~oHg~-'
(1 + xo¢]]
xo(co -,v)
2.5. Normalized general equations
~=
,v+ 1)
2[ co -
(26)
It is convenient to express parameters in dimensionless form indicated with tildes, with time normalized to the magnetic relaxation time r, space normalized to duct spacing d, and magnetic field quantities normalized to a nominal magnetic field strength H o
(32)
This set of equations describes the motion of the planar ferrofluid layer confined between rigid walls with imposed spatially uniform, sinusoidally time-varying x and z directed magnetic fields. The primary complexity of the analysis is that the time average torque density (/~y) depends in a complicated way on the spin velocity COy which depends on 2. Substituting (31)-(32) into (30) gives
[ dvz
, dZtoy
'0 --d--~-x2- 2~ ~-~x + 2toy j + (Ty) = 0.
(31)
(28)
(35) Xo [ I/1- I 2 ( ~ 2 - - 1) + I/q* 12[ g)2 - (1 + Xo)2]]
2
[l+xo+ ~212+xg~2 (36)
(29)
Note in particular that as shown in Fig. 2, for purely axial (/~. = 0) or transverse (/Tz = 0) magnetic fields /~o = 0.
M. Zahn, D.R. Greer /Journal of Magnetism and Magnetic Materials 149 (1995) 165-173
Note also that a can be positive or negative depending on the value of 1) compared with 1 or 1 + Xo; a is positive at high frequencies and negative at low frequencies.
169
We solve for &y(2) in (37) as
&Y
2~
2 dd:
3. Zero spin-viscosity (h' = O) solutions 1 [~/Y(2£- 1) - ~+. ~/(Tv)
3.1. General solutions
We first examine the simple limiting case of zero spin-viscosity (~/'= 0). We differentiate (29) with respect to 2, and solve for d & y / d 2 d&y
1 d2/~
d.~
2 d.~ 2
1 d +
2(
(37)
d£
Substituting this into (28) and solving for d 2 ~ / d 2 2 , we have d2t3
2 Off
1 d(/~v)
(38)
Integrating this twice, we obtain 1 0/:3'22 - 1 f0i(~y ) d - r + k i - r + k 2 ,
(39)
where k~ and k 2 are constants of integration to be found from the boundary conditions of zero velocity at the rigid boundaries
?:x(£ = 0) = 0,
~;,(2 = 1) = 0.
(40)
Applying these boundary conditions to (39) we have k~
1 Off 1 ~7 O~ + ~- f o (iv) " d£,
k2=O,
Note that (42) and (43) are not known analytical solutions for Vz a n d ( . b y because (Ty) varies with £ because &y varies with 2. However (42) and (43) do help explain key features of ferrofluid flow for ~' = 0. When 0/5'/~2 is large compared with the torque density (Ty), then the flow velocity ~ will be essentially parabolic being maximum at .~ = 0.5 which is the usual Poiseuille flow in a planar duct, while the spin velocity &y will be linear with respect to -,7 being zero at the midpoint £ = 0.5. 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 (42) cancel but there is a constant torque contribution to the spin velocity in (43). If there is no pressure gradient, 8/Y/&? = 0, then the solution f o r ~y in (43) is a constant, independent of 2 as (ify) is only constant given by (33). Then the solution to (42) is i~(2)= 0 and ~ y ( 2 ) = ( T y ) / 2 ~ . The torque is constant with position if &y is constant with position as given by (33). 3.2. Small spin-velocity limit
+-~[xfo(TY)d'~-fo(Ty)d'~]. 1
(43)
(41)
so that the velocity is
1
+ ~ l ( f r ) d£ ].
~
~7
If the torque is linear with position, it contributes parabolically to the flow velocity profile and linearly to the spin-velocity profile in exactly the same way as the pressure gradient so that the net effective viscosity is magnetic field dependent. This occurs in the small spin-velocity
~
(42)
0.41 0.=I oal
\
0.~I gy
.
\~\..
o
\
..~.1 4.11 -o.'
o.1
I~
o~
o.4
~s R
~
o.7
o.B
o.I
'
'
i4
'
'
I
~
I
Fig. 3. Dimensionless flow and spin-velocities of (52)-(55) with (iry) = 0 and (1/~)(0/3'/0~)= 1 for various values of a and fl with a/3= I.
170
M. Zahn, D.R. Greer/Journal of Magnetism and Magnetic Materials 149 (1995) 165-173
limit when (34) is valid. Substituting (34) into (42)-(43) results in the approximate flow and spin-velocity profiles
2 ( 2 - 1) ~ ' Vz(X) =
~kff
a~'
1 ff)y(X) ,~ ( 2 ( -
[
(44) ( ( 2 2 - 1) Off
a ) iT°
T/eft
a_7 '
(45)
where the effective viscosity is (46) To compare with previously published results of the effective viscosity [4], we consider a (~+~)
In the small spin-velocity limit of (34), this equation can be rewritten as
,7-d~- (¢%--~) ~ ay (¢+ ~) ~o
(58)
(57) Note that by integrating (56) over the duct cross-section
M. Zahn, D.R. Greer/Journal of Magnetism and Magnetic Materials 149 (1995) 165-173
with ~z zero at the two walls, C is related to the spinvelocity as (59)
C=ffJ0 wy d$. The flow solutions are then
+ (e v~ - 1)(1 - e -'e) + (e-V~ _ 1)(1 - eV)],
(60) = D [ 1 + (e Z , _ l ) e V , ~ + ( l _ e - / ) e 2 sinh y
t
vii,
]
(61)
where D=
References
2¢2S+2~Yg 1 - a ( s T + ~ ) ' S=I
~'=
(eV+e-~-
2)
3, sinh y ~+ ~ - ~ _
/~'.
We use the solution method of Section 4.1. to numerically solve for the dimensionless flow velocity ?~z and f~y versus position ~. To emphasize magnetic field effects on the flow profile we generally considered small pressure gradients. Fig. 5 shows how the flow profiles approach the zero spin-viscosity limits of Section 3 as ~ ' ~ 0 . Acknowledgements: This work has been supported by NSF Grant No. ECS-9220638 and by the NSF Research Experiences for Undergraduates Program. Stimulating conversations with Dr R.E. Rosensweig of Exxon Research and Engineering Co. are greatly appreciated.
spin-velocity
[2.~(e v - 1)(e -z' - 1)
3'( ~ + ~)sinh 3'
&v(~)
Note that 3' = 0 when ~/en in (46) is zero and that y can become purely imaginary for large enough a , although our approximate solutions are only valid for &y
~
Jeurnalof magnellc inalerlals
ELSEVIER
Ferrohydrodynamic pumping in spatially uniform sinusoidally time-varying magnetic fields Markus Zahn *, Donald R. Greer Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, Laboratoryfor Electromagnetic and Electronic Systems, Cambridge, MA 02139, USA Abstract
Past work has analyzed ferrofluid pumping in a planar duct driven by spatially non-uniform traveling wave magnetic fields. Here we examine a much simpler case where the applied magnetic fields along and transverse to the duct axis are spatially uniform and varying sinusoidally with time. In the uniform magnetic field the magnetization characteristic depends on particle spin but does not depend on fluid velocity. The magnetization force density along the duct axis is zero while the magnetic torque density is non-zero as M and H are not collinear due to a magnetic relaxation time constant as well as due to spatially varying particle spin. The governing linear and angular momentum conservation equations are numerically integrated to solve for flow and spin velocity distributions for zero and non-zero spin viscosities as a function of magnetic field strength, phase, frequency, and direction along and transverse to the duct axis, and as a function of pressure gradient along the duct, vortex viscosity, dynamic viscosity, and ferrofluid magnetic susceptibility. Analytical solutions for simple limiting cases are also given including an effective viscosity that depends on magnetic field strength which can be made zero or negative.
1. Introduction
The motion of a ferrofluid in a travelling 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 travelling wave (backward pumping) while, above, the ferrofluid moves in the same direction (forward pumping) [1,2]. The value of critical magnetic field depends on the frequency, the concentration of the suspended magnetic particles, and the fluid viscosity. Under ac magnetic fields, the 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 H, there is a body torque density T = / x 0 ( M X H ) acting on the ferrofluid even in a uniform magnetic field. Fluid mechanical analysis has been developed to extend traditional viscous fluid flows to account for the non-symmetric stress tensor that results when M and H are not collinear and to then simultaneously satisfy linear and angular momentum conservation for the ferrofluid [3]. Recent analyses on ferrofluid motion in travelling wave magnetic fields with sinusoidal time and space dependence have shown both forward and backward pumping when
* Corresponding author. Email: [email protected]; fax: + 1-617258-6774.
fluid convection and spin effects are included in the study [1]. In that analysis, the magnetic fields were non-uniform and force and torque densities were non-zero. In this paper, we have investigated ferrofluid motion in spatially uniform sinusoidally time-varying magnetic fields where the torque density is non-zero but the force density along the duct axis is zero. By imposing uniform magnetic fields, we have studied how fluid flow is affected by different magnetic field variations including an axial component only, a transverse component only, and both axial and transverse components that are not in phase resulting in a rotating uniform magnetic field. This work extends recent similar analysis which examined the change in effective ferrofluid viscosity under alternating magnetic field, but that work assumed the shear coefficient of spin-viscosity to be zero, only considered a linearly polarized magnetic field and only examined the magnetization characteristic in the small magnetic susceptibility and small spin velocity limits [4].
2. Governing equations
2.1. Magnetization constitutive law The magnetization relaxation equation with a ferrofluid undergoing simultaneous magnetization and reorientation
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(95)00363-0
M. Zahn, D.R. Greer /Journal of Magnetism and Magnetic Materials 149 (1995) 165-173
166
due to fluid convection at velocity v and particle spin at angular velocity to is [1,3,4] OM --+(v. Ot
1
V)M-to×M+-[M-XoH]=O, ~"
where
z = z o ( n + M).
(1)
We would like to solve for M~ and ,~t in terms of the known imposed field amplitudes H~ and /~x. Substituting (5) and (6) into (1) relates the magnetization components to the magnetic field H as
where 1" is a relaxation time constant and X0 is the effective magnetic susceptibility which in general can be magnetic-field-dependent but in this paper will be taken to be constant. For the planar ferrofluid layer shown in Fig. 1, the flow velocity can only be z-directed and the spin velocity can only be y-directed, both quantities only varying with the x coordinate
v = v~(x)i~,
to = toy(X)iy.
in~lz +
(2)
~
B~=constant,
toyl~[ x .-1-
nx,
(8)
X°I~,
(9)
T
Mz T
T
-
T
where the second term in (1) has zero contribution because v is only z-directed and M can only vary with x. By taking the x-component of (7) and solving for Hx, we obtain
We wish to apply Eq. (1) to the planar ferrofluid layer confined between rigid walls shown in Fig. 1. The imposed magnetic field H z 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 z coordinates, field components can only vary with the x coordinate. Gauss's law for the magnetic flux density and Ampere's law for the magnetic field intensity with zero current density require the imposed fields to be uniform throughout the ferrofluid dB~ =* dx = 0
Xo ^
i 12h~lx - toylQ= z +
2.2. Magnetic fields
V.B=O
(7)
&=.o(&
&=
(lO)
tZo
We then solve for hex and h,lz by using (10) in (8):
Xo[/~(toyZ) + (i/2z+ 1)/~x//% ] Mx= [(toyr)2+ Oar+
1)(inr+ 1 +Xo)]'
(11)
X o [ ( i / 2 z + 1 + Xo)I'tz-BxtoyZ/tZo]
(3)
~tz= [(to:.)2 + ( i n r + 1)(in~-+ 1 +x0)] "
(12)
dH~
V×H=O
~
dx =0
~
H~=constant.
(4)
Eqs. (11) and (12) represent the magnetization of the ferrofluid as a function of the imposed fields H~ and / ~ and as a function of the not yet known spin velocity toy 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 toy. The resulting toy then changes the magnetization. There is a strong magneto-mechanical coupling so that the magnetization and mechanical equations need to be self-consistently satisfied.
There are no y components of the magnetic fields. Therefore, the total magnetic field H and magnetic flux density B inside the ferrofluid layer are of the form
B = ~R([ Bxi x + l~z(X)iz]
ei°t),
(5)
H=9~{[I~x(X)ix+Itzi~]
eint},
(6)
n= = ~[/~=e ~"t]
d ,,'dx)
Ferrofluid
I1= ~.,.-
',o,,(x)
,uz --
~[~:'°'l
1,
y
Z Fig. 1. A planar ferrofluid layer between rigid walls is magnetically stressed by a uniform z-directed magnetic field HZ and uniform x-directed magnetic flux density Bz, both of which vary sinusoidally in time at frequency /2.
M. Zahn, D.R. Greer /Journal of Magnetism and Magnetic Materials 149 (1995) 165-173 2.3. Fluid mechanics
Solving for the x and z components with field components only constant or varying with x,
For incompressible fluids
V.v=O,
dn~
V.w=0
=-~°M~'
,I
]
t~ [ + (v. v).,
t7o d(l .,}
dx
dx
2/x°M~7 '
(17)
dH z L = u0M~ d x = 0.
av
2~V×to+(~+rl)g2v-pgi~,
d [ B x _ Mx~)
=,,oMxT--x dM~
(13)
and the coupled linear and angular momentum conservation equations for force density f and torque density T for a fluid in a gravity field - g i x are [1,3,4]
=-Vp+f+
167
(18)
The time average components of the magnetic force density are then
(14)
(f~)=-~xx
= r + 2 ~ ( v x v - 2o~) + n ' v ~ o ,
4 t%l~xlz
'
(19)
( L ) = 0.
(15) where p is the mass density, p is the pressure, ff is the vortex viscosity, 7/ is the dynamic viscosity, I is the moment of inertia density, and r/' is the shear coefficient of spin-viscosity. We wish to apply these equations to the planar ferrofluid layer confined between rigid walls in Fig. 1. 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 only to the time average magnetic force and torque densities.
(20)
2.4.2. Magnetic torque density Similarly, the torque density is given by the equation
T=t.to(M×H)=I.to(-MxH~+M~H~)i~..
(21)
The torque is only y-directed,
• ~ -- I-%M,, H~ + I-%M~ ---mx T~. /to = M~B x - txoMx(H z + M~).
(22)
The time average component of torque density is then
2.4. Magnetic force and torque densities
1
(Ty)= ~,~[l~zn*
- I,lol~lx (l~z +Mz)]
(23)
2.4.1. Magnetic force density with superscript asterisks ( * ) indicating the complex conjugate of a complex amplitude field quantity. In Fig. 2, the dimensionless torque density ( T y ) = ( T y ) / ( p . o H 2) is plotted as a function of the dimensionless spin velocity
For 0 < x < d, the magnetic force density is given by the equation
f = txo(M. IT)H.
0.3
(16)
i
!
0.15
I
"
fi-O 0.10
0.2
t'l-O
i
i
A
A
.~.a, v
O0 V
A ,~,,t~ v
0.00 -o,0~
-o.1
I
I -10
0
i
0.2
00,5
0.1
I 0.4
O0
-02
-0.IC
-02
-0.4
-20
1 -I0
1 0
I 10
-20
-I0
0
10
20
-20
t°
20
20 ~y
~tJ
Fig. 2. Dimensionless time average torque density (/~y) = (Ty)/( ~oHo2) versus dimensionless spin velocity &v = to,3- for dimensionless angular frequency ~ = g2~"= 0, 1, 5, 10 with X0 = 1 and with (a) axial magnetic field (H~ = H o, Bx = 0); (b) transverse magnetic field (/4z = O, /~x = ~oHo); and (c) rotating magnetic field (/4z = iHo, /~x =/xoHo)-
M. Zahn, D.R. Greer/Journal of Magnetism and Magnetic Materials 149 (1995) 165-173
168
COy= toy~- with H 0 a nominal magnetic field amplitude. We show three limiting cases with the magnetic field purely axially directed (/7~ =/40, /~ = 0), purely transversely directed (/7~ = 0, B~ =/x0H0), and magnetic field both axially and transversely directed with phase difference ~ / 2 (/7~ = i H 0, /~, = IzoHo ) so that the net magnetic field rotates. Throughout this report, dimensionless variables are indicated with a tilde. Note that for a stationary spin velocity (COy= 0 ) that the slope a r o u n d COy = 0 is negative at low frequencies and positive at high frequencies and that only the rotating magnetic field case in Fig. 2 has non-zero time average torque at COy= 0.
2.4.3. Coupled linear and angular momentum conservation equations It is convenient to define a modified pressure as
1
^
p' =p + -~lx o I M x 12 + pgx
(24)
so that (14) and (15) in the negligible inertia, viscous dominated limit become d 2 vz d toy (ff+'0)-~xZ +2~'~x
Off = O, bz
(25)
where
and
Xo[ (%/7z + (if2+ l)Bx] Atx= [ COre+ (ig~ + 1 ) ( i ~ + 1 + Xo)] ,
Xo[(i~+ I + Xo)/7z-Bxcoy] 3,1,= [ ~2 + (ig) + 1)(ig) + 1 + X0)] .
= yXo [-coy[ i +
+iJ~(CO2- ~ 2 _ 1 _
10 n~-,
/ 7 = Ho'
X
B=
IXoHo
'
Ho '
rr
T
yc = -~,
~ = v:~,
Coy = t o : ,
fy = . o n g ' (27)
2'0
i~/Y a5
~ ' = ~,o/-/o~-'
I
~
-,d2my '0~-~
t-2 oy--
~)Z+l+x °
+(2+x0)2~)2
Note that the phase relationship between /t~ and /~ is very important in the third term of the numerator. Because for our case studies with ~' ~ 0, COy must be zero at the fixed boundaries at x = 0 and x = d, it is useful to examine (33) in the limit of small COy.To first order in COy,(33) approximately reduces to lim ( g ) ~/~0 + oLCOy,
(34)
~y "~" 1
xo [[ ~o
Off
2+
+1 +
[ / 7 : : ]]
[I + Xo + I)2]E+ x2~)2
ixoH~o Oz.
~)
d25z _dcoy 0/5' -d~ -~
+~~
( d~z
o~
O,
)
d2 +2coy + ( / ~ y ) = 0
.
(33)
UoHo~'
We then have dimensionless flow and spin velocity equations
(¢+
/
Xo)I[H~.~, ]]]
where
28r
'0'd 2
= ~oHg~-'
(1 + xo¢]]
xo(co -,v)
2.5. Normalized general equations
~=
,v+ 1)
2[ co -
(26)
It is convenient to express parameters in dimensionless form indicated with tildes, with time normalized to the magnetic relaxation time r, space normalized to duct spacing d, and magnetic field quantities normalized to a nominal magnetic field strength H o
(32)
This set of equations describes the motion of the planar ferrofluid layer confined between rigid walls with imposed spatially uniform, sinusoidally time-varying x and z directed magnetic fields. The primary complexity of the analysis is that the time average torque density (/~y) depends in a complicated way on the spin velocity COy which depends on 2. Substituting (31)-(32) into (30) gives
[ dvz
, dZtoy
'0 --d--~-x2- 2~ ~-~x + 2toy j + (Ty) = 0.
(31)
(28)
(35) Xo [ I/1- I 2 ( ~ 2 - - 1) + I/q* 12[ g)2 - (1 + Xo)2]]
2
[l+xo+ ~212+xg~2 (36)
(29)
Note in particular that as shown in Fig. 2, for purely axial (/~. = 0) or transverse (/Tz = 0) magnetic fields /~o = 0.
M. Zahn, D.R. Greer /Journal of Magnetism and Magnetic Materials 149 (1995) 165-173
Note also that a can be positive or negative depending on the value of 1) compared with 1 or 1 + Xo; a is positive at high frequencies and negative at low frequencies.
169
We solve for &y(2) in (37) as
&Y
2~
2 dd:
3. Zero spin-viscosity (h' = O) solutions 1 [~/Y(2£- 1) - ~+. ~/(Tv)
3.1. General solutions
We first examine the simple limiting case of zero spin-viscosity (~/'= 0). We differentiate (29) with respect to 2, and solve for d & y / d 2 d&y
1 d2/~
d.~
2 d.~ 2
1 d +
2(
(37)
d£
Substituting this into (28) and solving for d 2 ~ / d 2 2 , we have d2t3
2 Off
1 d(/~v)
(38)
Integrating this twice, we obtain 1 0/:3'22 - 1 f0i(~y ) d - r + k i - r + k 2 ,
(39)
where k~ and k 2 are constants of integration to be found from the boundary conditions of zero velocity at the rigid boundaries
?:x(£ = 0) = 0,
~;,(2 = 1) = 0.
(40)
Applying these boundary conditions to (39) we have k~
1 Off 1 ~7 O~ + ~- f o (iv) " d£,
k2=O,
Note that (42) and (43) are not known analytical solutions for Vz a n d ( . b y because (Ty) varies with £ because &y varies with 2. However (42) and (43) do help explain key features of ferrofluid flow for ~' = 0. When 0/5'/~2 is large compared with the torque density (Ty), then the flow velocity ~ will be essentially parabolic being maximum at .~ = 0.5 which is the usual Poiseuille flow in a planar duct, while the spin velocity &y will be linear with respect to -,7 being zero at the midpoint £ = 0.5. 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 (42) cancel but there is a constant torque contribution to the spin velocity in (43). If there is no pressure gradient, 8/Y/&? = 0, then the solution f o r ~y in (43) is a constant, independent of 2 as (ify) is only constant given by (33). Then the solution to (42) is i~(2)= 0 and ~ y ( 2 ) = ( T y ) / 2 ~ . The torque is constant with position if &y is constant with position as given by (33). 3.2. Small spin-velocity limit
+-~[xfo(TY)d'~-fo(Ty)d'~]. 1
(43)
(41)
so that the velocity is
1
+ ~ l ( f r ) d£ ].
~
~7
If the torque is linear with position, it contributes parabolically to the flow velocity profile and linearly to the spin-velocity profile in exactly the same way as the pressure gradient so that the net effective viscosity is magnetic field dependent. This occurs in the small spin-velocity
~
(42)
0.41 0.=I oal
\
0.~I gy
.
\~\..
o
\
..~.1 4.11 -o.'
o.1
I~
o~
o.4
~s R
~
o.7
o.B
o.I
'
'
i4
'
'
I
~
I
Fig. 3. Dimensionless flow and spin-velocities of (52)-(55) with (iry) = 0 and (1/~)(0/3'/0~)= 1 for various values of a and fl with a/3= I.
170
M. Zahn, D.R. Greer/Journal of Magnetism and Magnetic Materials 149 (1995) 165-173
limit when (34) is valid. Substituting (34) into (42)-(43) results in the approximate flow and spin-velocity profiles
2 ( 2 - 1) ~ ' Vz(X) =
~kff
a~'
1 ff)y(X) ,~ ( 2 ( -
[
(44) ( ( 2 2 - 1) Off
a ) iT°
T/eft
a_7 '
(45)
where the effective viscosity is (46) To compare with previously published results of the effective viscosity [4], we consider a (~+~)
In the small spin-velocity limit of (34), this equation can be rewritten as
,7-d~- (¢%--~) ~ ay (¢+ ~) ~o
(58)
(57) Note that by integrating (56) over the duct cross-section
M. Zahn, D.R. Greer/Journal of Magnetism and Magnetic Materials 149 (1995) 165-173
with ~z zero at the two walls, C is related to the spinvelocity as (59)
C=ffJ0 wy d$. The flow solutions are then
+ (e v~ - 1)(1 - e -'e) + (e-V~ _ 1)(1 - eV)],
(60) = D [ 1 + (e Z , _ l ) e V , ~ + ( l _ e - / ) e 2 sinh y
t
vii,
]
(61)
where D=
References
2¢2S+2~Yg 1 - a ( s T + ~ ) ' S=I
~'=
(eV+e-~-
2)
3, sinh y ~+ ~ - ~ _
/~'.
We use the solution method of Section 4.1. to numerically solve for the dimensionless flow velocity ?~z and f~y versus position ~. To emphasize magnetic field effects on the flow profile we generally considered small pressure gradients. Fig. 5 shows how the flow profiles approach the zero spin-viscosity limits of Section 3 as ~ ' ~ 0 . Acknowledgements: This work has been supported by NSF Grant No. ECS-9220638 and by the NSF Research Experiences for Undergraduates Program. Stimulating conversations with Dr R.E. Rosensweig of Exxon Research and Engineering Co. are greatly appreciated.
spin-velocity
[2.~(e v - 1)(e -z' - 1)
3'( ~ + ~)sinh 3'
&v(~)
Note that 3' = 0 when ~/en in (46) is zero and that y can become purely imaginary for large enough a , although our approximate solutions are only valid for &y
