FLOW OF IMMISCIBLE FERROFLUIDS IN A PLANAR...
Transcript of FLOW OF IMMISCIBLE FERROFLUIDS IN A PLANAR...
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FLOW OF IMMISCIBLE FERROFLUIDS IN A PLANAR GAP IN A ROTATING MAGNETIC FIELD
By
BHUMIKA SHRIKAR SULE
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2013
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© 2013 Bhumika Shrikar Sule
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To my parents Shrikar Sule and Neha Sule
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ACKNOWLEDGMENTS
I would like to express my sincere gratitude to my advisor, Dr. Carlos Rinaldi, for
his great support and advice. Thank you very much for always encouraging me to do
my best, and believing I could do it.
I would like to specially thank Postdoctoral Research Associate, Isaac Torres-
Diaz for his great help and support during my research. Also I would like to thank my
group members Rohan, Tapomoy, Ana, Lorena, Melissa and Maria for their invaluable
friendship and support.
I thank my family, especially my parents, for their motivation and support
throughout my graduate studies.
Finally I would like to thank the Chemical Engineering Department for giving me
the opportunity to study at University of Florida.
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TABLE OF CONTENTS page
ACKNOWLEDGMENTS .................................................................................................. 4
LIST OF TABLES ............................................................................................................ 7
LIST OF FIGURES .......................................................................................................... 8
ABSTRACT ................................................................................................................... 10
CHAPTER
1 INTRODUCTION .................................................................................................... 12
2 ANALYTICAL SOLUTION ....................................................................................... 16
2.1 Definition of the Problem ................................................................................... 16
2.2 Governing Equations ........................................................................................ 17 2.3 Magnetic Field Problem .................................................................................... 22 2.4 Zero Spin Viscosity Solution ............................................................................. 25
2.5 Non-zero Spin Viscosity Solution ...................................................................... 26 2.6 Relation between Pressure Gradients and Boundary Conditions ..................... 27
3 PREDICTED VELOCITY PROFILES ...................................................................... 31
3.1 Zero Spin Viscosity Case .................................................................................. 32
3.1.1 Zero Spin Viscosity without Pressure Gradient ....................................... 32 3.1.2 Zero Spin Viscosity with Pressure Gradient ............................................ 35
3.2 Non-zero Spin Viscosity Case .......................................................................... 38
3.2.1 Non-zero Spin Viscosity without Pressure Gradient ................................ 38 3.2.2 Non-zero Spin Viscosity with Pressure Gradient ..................................... 43
3.3 Comparison between Zero Spin Viscosity and Non-zero Spin Viscosity Cases .................................................................................................................. 44
4 CONCLUSIONS ..................................................................................................... 46
APPENDIX
A CONSTANTS EVALUATED ................................................................................... 48
A-1 Zero Spin Viscosity Case ................................................................................. 48 A-2 Non-zero Spin Viscosity Case .......................................................................... 49
B VELOCITY PROFILES FOR ZERO SPIN VISCOSITY CASE ................................ 87
C VELOCITY PROFILES FOR NON-ZERO SPIN VISCOSITY CASE ....................... 89
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C-1 Ferrofluid 1 – Ferrofluid A, Ferrofluid 2 – Ferrofluid C ...................................... 89
C-2 Ferrofluid 1 – Ferrofluid A, Ferrofluid 2 – Ferrofluid B ...................................... 93
LIST OF REFERENCES ............................................................................................... 95
BIOGRAPHICAL SKETCH ............................................................................................ 97
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LIST OF TABLES
Table page 3-1 Physical properties of ferrofluids ......................................................................... 31
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LIST OF FIGURES
Figure page
2-1 Schematic illustration for the flow of two immiscible ferrofluids of thickness 1L
and 2L between two parallel plates. The uniform rotating magnetic field is
generated by an imposed axial magnetic field zH and a transverse magnetic
flux density xB . ................................................................................................... 16
3-1 Body torque in ferrofluids at different magnetic field frequencies. ...................... 32
3-2 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for Ferrofluid A and Ferrofluid C. .................................... 33
3-3 Dimensional velocity profiles at different magnetic field amplitudes at a constant field frequency of 150 Hz for Ferrofluid A and Ferrofluid C. ................. 34
3-4 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for Ferrofluid A and Ferrofluid B. .................................... 35
3-5 Dimensional velocity profiles at different magnetic field amplitudes at a constant field frequency of 150 Hz for Ferrofluid A and Ferrofluid B. ................. 35
3-6 Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT and at different values of applied pressure gradient. ............................................................................................... 36
3-7 Effect of pressure gradient on translational velocity at a constant field frequency of 150 Hz and field amplitude of H = 2 mT. ........................................ 37
3-8 Non-dimensional velocity profiles at different values without the application of pressure gradient with ferrofluid/non-ferrofluid interface.. ............................... 38
3-9 Non-dimensional velocity profiles at different values without the application of pressure gradient for Ferrofluid A and Ferrofluid C.. ...................................... 39
3-10 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for ferrofluids with different spin viscosity values - Ferrofluid A and Ferrofluid C.. ............................................................................ 40
3-11 Dependence of direction of translational velocity when ferrofluids of different spin viscosity values are considered - Ferrofluid A and Ferrofluid C.. ................ 41
3-12 Dependence of direction of translational velocity when ferrofluids of similar spin viscosity values are considered - Ferrofluid A and Ferrofluid B. ................. 42
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3-13 Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for ferrofluids with similar spin viscosity values.. ............ 42
3-14 Effect of pressure gradient on translational velocity at a constant field frequency of 150 Hz and field amplitude of H = 2 mT for different ferrofluid combinations. ..................................................................................................... 43
3-15 Non-dimensional translational velocity profiles for zero spin viscosity and non-zero spin viscosity cases for different combination of ferrofluids.. ............... 45
B-1 Dimensional velocities at different field amplitudes at a field frequency of 150 Hz and pressure gradient of 0.1 Pa/m. ............................................................... 87
B-2 Effect of positive pressure gradient applied on spin velocity at a field frequency of 150 Hz and field amplitude of H = 2 mT.. ....................................... 87
B-3 Effect of negative pressure gradient applied on spin velocity at a field frequency of 150 Hz and field amplitude of H = 2 mT. ........................................ 88
B-4 Variation of dimensional spin velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient. .............. 88
C-1 Effect of pressure gradient applied on spin velocity profiles at a field frequency of 150 Hz and field amplitude of H = 2 mT. ........................................ 89
C-2 Dimensional velocities at different field amplitudes at a field frequency of 150 Hz and a pressure gradient of 0.1 Pa/m for Ferrofluid A and Ferrofluid C. ......... 90
C-3 Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for ferrofluids with widely different spin viscosity values – Ferrofluid A and Ferrofluid C. ............................................................................................. 91
C-4 Variation of dimensional spin velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for ferrofluids with widely different spin viscosity values – Ferrofluid A and Ferrofluid C. ........................................................................................................ 92
C-5 Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for similar spin viscosity ferrofluids – Ferrofluid A and Ferrofluid B. ..... 93
C-6 Variation of dimensional spin velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for similar spin viscosity ferrofluids – Ferrofluid A and Ferrofluid B.......................... 94
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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science
FLOW OF IMMISCIBLE FERROFLUIDS IN A PLANAR GAP IN A ROTATING
MAGNETIC FIELD
By
Bhumika Shrikar Sule
December 2013
Chair: Carlos Rinaldi Major: Chemical Engineering
We have obtained analytical solutions for the flow of two layers of immiscible
ferrofluids of different thickness between two parallel plates. The flow is mainly driven
by the generation of antisymmetric stresses and couple stresses in the ferrofluids due to
the application of a uniform rotating magnetic field. The translational velocity zv and
spin velocity y profiles were obtained for the zero spin viscosity and non-zero spin
viscosity cases and the effect of applied pressure gradient on the flow was studied. The
interfacial linear and internal angular momentum balance equations derived for the air-
ferrofluid interface case are extended for the case when there is a ferrofluid-ferrofluid
interface to obtain the velocity profiles. The magnitude of the translational velocity is
directly proportional to the frequency of the applied magnetic field and the square of the
magnetic field amplitude. The spin velocity is in the direction of the rotating magnetic
field and its direction remains the same at lower values of applied pressure gradient.
The direction of translational velocity depends on the balance between the magnitudes
of vorticity, body torque, spin velocity and diffusion of internal angular momentum,
however at higher values of applied pressure gradient, the pressure gradient dominates
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the flow. This work shows the importance of surface stresses in driving flows in
ferrofluids.
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CHAPTER 1 INTRODUCTION
Ferrofluids are stable colloidal suspensions of single domain magnetic
nanoparticles in a non-magnetic carrier liquid such as water or oil.1 They are composed
of ferrimagnetic particles such as magnetite (Fe3O4), maghemite (γ-Fe2O3), or cobalt
ferrite (CoO· Fe2O3), and have diameters usually between 5-20 nm. There are a large
number of magnetic particles in a ferrofluid per unit volume due to the extremely small
size of the particles.2 The particles in a ferrofluid are randomly oriented and the fluid has
no net magnetization in the absence of an applied field. On application of a magnetic
field of moderate strength the individual dipole moments of the particles are aligned in
the direction of the field, resulting in a net magnetization in the ferrofluid.
The magnetic field energy in ferrofluids can be converted into motion without
using external mechanical parts. Ferrofluids have a variety of applications in
engineering, microfluidics, and biomedical fields. They are used in fluid seals, inertial
dampers, as heat transfer fluids in loudspeakers, and in stepper motors.3 Ferrofluids
also have applications in microfluidic pumps and valves4–6 and in microfluidic actuators
and devices where they can actuate flow.7,8 The ability to change the magnetic and
optical properties of ferrofluids have made it possible for ferrofluids to be used in
magneto-optic sensors,9 flow sensors,10 and in temperature sensors using thin
ferrofluid films.11
Flows can be generated in ferrofluids in response to magnetic fields, due to
generation of antisymmetric stresses, and because of couple stresses upon application
of a rotating magnetic field. The phenomenon of “spin up” flow was first reported by
Moskowitz and Rosensweig in 196712 for the flow of ferrofluid in a stationary cylindrical
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container subjected to a uniform rotating magnetic field. Various explanations have
been proposed for the phenomenon of spin-up flow. The spin diffusion theory by
Zaitsev and Shliomis,13 which assumes that the magnetic field is uniform in the ferrofluid
region and the fluid magnetization is proportional to the magnetic field, is based on the
structured continuum theory.14 Zaitsev and Shliomis predicted that the fluid would rotate
in rigid body motion and a thin boundary layer would form near the cylinder wall. The
spin diffusion theory includes the effect of spin viscosity, the dynamic coefficient in the
constitutive equation of the couple stress representing the short range transfer of
internal angular momentum.
Rosensweig et al.15 demonstrated that the observed spin-up flows in ferrofluids
were due to a magnetic field driven interfacial phenomenon, as they observed counter
rotation of fluid and field for a concave meniscus and co-rotation for a convex meniscus
in a capillary. Observations where the fluid co-rotates or counter rotates with the
magnetic field depending on the magnetic field frequency and amplitude were also
made by Brown and Horsnell16 and Kagan et al.17 Rosensweig et al.15 concluded that
surface stresses rather than volumetric stresses generated due to body couples in the
bulk of the fluid were responsible for the spin-up phenomenon. It was demonstrated
later by the experiments of Chaves et al.18 that the surface flow driven by the application
of the rotating magnetic field coexists with a bulk flow and can be suppressed by
covering the cylinder, eliminating the air-ferrofluid interface.
Krauss et al.5,19 performed experiments in a free surface geometry consisting of a
circular duct with square cross section with a pool of ferrofluid in contact with air, and in
which a uniform rotating magnetic field is applied with rotation axis parallel to the
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ferrofluid-air interface. This geometry was used to evaluate the existence of surface
flows in ferrofluids. Their experiments showed that the magnitude of flow increases with
the square of magnetic field amplitude, is proportional to the thickness of the ferrofluid
layer, and has a maximum at a fluid specific frequency. Their flow measurements were
consistent with the effects of magnetic stresses at the ferrofluid-air interface as
indicated by numerical calculations.
The equations governing the interfacial stress balance in ferrofluids have also
been recently derived by Rosensweig.20 However the analyses of Krauss et al.19 and of
Rosensweig20 did not consider the potential role of spin viscosity, related to the couple
stresses, in driving the flow in ferrofluids. Rinaldi and collaborators18,21–25 showed
qualitative agreement between experimental translational velocity profiles for cylindrical,
annular, and spherical geometries with the theoretical predictions of the spin diffusion
theory. The value of the spin viscosity estimated from experimental measurement18,22,25
was in the range of 10-12 to 10-8 kg m s-1, which is several orders of magnitude higher
than the value estimated by Zaitsev and Shliomis13 and Feng et al.26 on the basis of
dimensional arguments in the infinite dilution limit. These studies provide evidence for
the existence of couple stresses in ferrofluids and the role of spin viscosity in driving the
ferrofluid flow in the presence of rotating magnetic fields.
Past analyses for the flow driven by the application of a uniform rotating magnetic
field were made considering the presence of a single ferrofluid with uniform body torque
in the entire ferrofluid region and neglecting stresses at the air-ferrofluid interface.
Motivated by the results of Krauss et al.,19 we considered a ferrofluid-non-ferrofluid and
ferrofluid-ferrofluid interface to study the flows driven by surface stress balance. In this
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contribution, analytical solutions are obtained for the general case of flow of two layers
of immiscible ferrofluids between two plates driven by the application of a uniform
rotating magnetic field. The interfacial linear momentum balance and internal angular
momentum balance equations, including the effects of vortex viscosity, spin viscosity,
and surface tension derived by Chaves and Rinaldi27 are extended to the analysis of
two immiscible ferrofluids to obtain the translational and spin velocity profiles. The
existence of tangential stress and couple stress at the interface and the difference in the
magnitudes of body torques generated in the two ferrofluids is studied. Velocity profiles
are obtained for the zero spin viscosity and non-zero spin viscosity cases considering
different ferrofluid combinations and the effect of applied pressure gradient on the flows
is analyzed.
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CHAPTER 2 ANALYTICAL SOLUTION
2.1 Definition of the Problem
The flow geometry analyzed is illustrated in Figure 2-1. It consists of two layers of
immiscible ferrofluids of thickness 1L and 2L between two parallel horizontal plates. The
flow in the ferrofluids is induced by the application of a uniform rotating magnetic field.
The field is generated by an imposed uniform magnetic field zH and uniform
transverse magnetic flux density xB represented by the functions
,ˆ )(tj
zzfeHtH
(2-1)
tj
xxfeBtB
ˆ )( . (2-2)
In Equations (2-1) and (2-2), zH and xB are the complex amplitudes of magnetic field
and magnetic flux density respectively, which are independent of position, f is the
radian frequency and j is the imaginary number 12 j .
Figure 2-1. Schematic illustration for the flow of two immiscible ferrofluids of thickness
1L and 2L between two parallel plates. The uniform rotating magnetic field is
generated by an imposed axial magnetic field zH and a transverse magnetic
flux density xB .
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The thickness of the ferrofluids, 1L and 2L , is small compared to the length and
width of the plate, thus the vectors of translational velocity v and spin velocity ω are
unidirectional and dependent only on the x coordinate,
zz xvx iv )()( , .)()( yy xx iω (2-3)
The flow direction zv and the direction of spin of the particles y depend on the
direction of rotation of the magnetic field.27 For the purpose of this analysis, we assume
the magnetic field to be rotating in the counterclockwise y direction.
2.2 Governing Equations
The ferrohydrodynamic equations governing the flow as given by Rosensweig1
are
0v , (2-4)
vωvHMgv 2
2
12
p
Dt
Do , (2-5)
ωωvHMω 2'
2
14
o
Dt
DI , (2-6)
)(1
eqt
MMMωMvM
. (2-7)
Equation (2-4) is the equation of continuity for an incompressible fluid, Equation (2-5)
represents the conservation of linear momentum, Equation (2-6) is the internal angular
momentum equation and Equation (2-7) is the magnetization relaxation equation. In
these equations, v is the local mass average velocity, ω is the spin velocity vector,
is the fluid density, g is the gravitational acceleration, o is the permeability of free-
space 17104 mHo , p is the fluid pressure, is the suspension-scale shear
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viscosity, is the vortex viscosity, ' is the coefficient of spin viscosity, I is the moment
of inertia density of the suspension, is the relaxation time, M is the magnetization
vector of the suspension, and H is the magnetic field vector. The second term on the
right hand side of the linear momentum equation, Equation (2-5), represents the
magnetic body force due to field inhomogeneities and the fourth represents the
antisymmetric component of the Cauchy stress which occurs when there is difference
between the rate of rotation of the particles and half the local vorticity of the flow.28 In
the internal angular momentum balance equation, Equation (2-6), the first tem on the
right hand side represents the external body torque acting on the ferrofluid whenever
the local magnetization is not aligned to the applied field, the second represents the
interchange of momentum between internal angular and macroscopic linear forms and
the third term represents the diffusion of internal angular momentum between
contiguous material elements.28 The first term on the right hand side of Equation (2-7) is
the spin magnetization coupling term and the second term represents the orientational
diffusion of M towards an equilibrium value.
The Langevin relation1 gives the equilibrium magnetization eqM for a
superparamagnetic ferrofluid as
Tk
HVML
M B
cdo
d
eq
,
1coth
H
H
H
HM, (2-8)
where L is the Langevin function, is the Langevin parameter which is a measure
of the relative magnitudes of magnetic and thermal energy, dM is the domain
magnetization of the magnetic nanoparticles, Bk is Boltzmann’s constant, T is the
absolute temperature and cV is the volume of magnetic cores. It is commonly assumed
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that the magnetic relaxation can simultaneously occur by the Brownian and Néel
mechanisms,
,111
NB (2-9)
where B is the Brownian relaxation time and N is the Néel relaxation time.
The vortex viscosity in the dilute limit for monodisperse nanoparticle suspensions in
Newtonian fluids is given as22
oh 5.1 , (2-10)
where o is the shear viscosity of the suspending fluid and h is the hydrodynamic
volume fraction of suspended particles.
Maxwell’s equations in the magnetoquasistatic limit29 are
0H , 0HM . (2-11)
The interfacial boundary conditions for the magnetic field, the continuity of the normal
component of magnetic induction and the jump in the tangential magnetic field due to
surface currents, are
0MHMHn 21 , (2-12)
KHHn 21 , (2-13)
in which n is a unit vector, locally normal to the interface and pointing from phase 2 to 1
and K is the surface current density.
Scaling of the governing equations - The problem is solved for the two
ferrofluids separately using a regular perturbation expansion to decouple the
ferrohydrodynamic equations and to obtain the magnetic field and flow field solutions.
The ferrohydrodynamic equations are considered under the assumptions of
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incompressible flow, low Reynolds number and small amplitude of the magnetic field in
order to formulate the regular perturbation problem. The perturbation parameter is
defined as
1
1
2
1,
Hio (2-14)
and the conditions 1 ,1 21 ff which usually apply in ferrofluids are used as
shown by Chaves et al.22 The following scaled variables are introduced
,~
,~
,~
,~
,~
,~ 2
21
12
21
1
1,
22
1,
11
HHHHHH ooii
BB
BB
HH
HH
MM
MM (2-15)
,~~
,~~
,L~
,~ ,~
1
2
1,1
212
1
2
1,1
1111
1 fiofio
fH
PP
H
PP
L
xxtt
(2-16)
1
2
1,
212
1
2
1,
111
11
2
1,
212
11
2
1,
111 ~
~ ,~~ ,~
~ ,~~
fiofiofiofio HHLHLH
ωω
ωω
vv
vv (2-17)
as given by Chaves et al.22 The subscript ‘1’ and ‘2’ refer to the properties of ferrofluid 1
and ferrofluid 2 respectively and i is the initial magnetic susceptibility. The resulting
scaled variables are assumed to be of order unity and they are denoted by an over-tilde.
The details of the scales for pressure, spin and linear velocity are given by Chaves et
al.22P is the dynamic pressure30 obtained by taking 111 gpP and
.222 gpP
The equilibrium magnetization eqM in Equation (2-8) reduces to the form
,~
,~
22,211,1 HMHM HH ieqieq (2-18)
by expanding Equation (2-8) in powers of and taking the terms with power zero in ,
as shown by Chaves et al.22
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Similarly, using the scaled variables given in Equations (2-15), (2-16) and (2-17), the
dimensionless scaled ferrohydrodynamic equations for the zeroth order problem of the
regular perturbation expansion for ferrofluid 1 and ferrofluid 2 are
,~~ ,~~
21 0v0v (2-19)
,~~~~2~~~~~~ 1
2
1
11
1
1111
11
1 vωHM0
e
f
P (2-20)
,~~~~2~~~~~~ 2
2
1
22
1
2222
11
1 vωHM0
e
f
P (2-21)
,~~4~4~~2
~~~1
1
2
2
1
11111
1
ωωvHM0
ef
(2-22)
,~~4~4~~2
~~~1
2
2
2
11
212
1
22
1
222
1
ωωvHM0
ef
(2-23)
,~~
~
~~
111
1 MHM
tf (2-24)
,~~
~
~~
22
1,
2,22 MH
M
i
i
ft
(2-25)
,~
,~
21 0H0H (2-26)
.~
,~
21 0B0B (2-27)
In Equations (2-15) to (2-17) and Equations (2-19) to (2-27), f is the frequency of the
applied magnetic field and the dimensionless frequency if defined as
2211
~ ,
~ ffff (2-28)
Additionally, the effective viscosity of the ferrofluids is
,222111 , ee (2-29)
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and the dimensionless parameter is defined as
.4
11
1
2
112
e
L (2-30)
2.3 Magnetic Field Problem
Under the assumption that the fluid layers are infinitely long in the y and z
directions, Maxwell’s equations reduce to the form
,~
~~~
,~
~~~ 2
21
1 0iH0iH
y
zy
z
x
H
x
H (2-31)
,~
~~~
,~
~~~ 2
21
1 0B0B
x
B
x
B xx (2-32)
which shows that the components 2121
~,
~,
~,
~xxzz BBHH are independent of x~ . However, the
magnetization generated by the imposed fields makes the components 2121
~,
~,
~,
~zzxx BBHH ,
potentially dependent on the x - coordinate.31 We assume the vectors
212121
~,
~,
~,
~,
~,
~BBMMHH have the functional form
, )ˆ)~(ˆ( )~
,~(~
, )ˆ)~(ˆ( )~
,~(~ ~
222
~
111
tj
zzxx
tj
zzxx eHxHtxeHxHtx iiHiiH (2-33)
, ) )~(ˆˆ( )~
,~(~
, ) )~(ˆˆ( )~
,~(~ ~
222
~
111
tj
zzxx
tj
zzxx exBBtxexBBtx iiBiiB (2-34)
, ) )~(ˆ )~(ˆ( )~
,~(~
, ) )~(ˆ )~(ˆ( )~
,~(~ ~
222
~
111
tj
zzxx
tj
zzxx exMxMtxexMxMtx iiMiiM (2-35)
where the symbol (^) is used to denote the dimensionless complex components.
The magnetization equation for the two ferrofluids, Equations (2-24) and (2-25),
under these assumptions reduces to the components
,~~
~
~~
,~~
~
~~
22
1,
2,2211
11 xx
i
ixfxx
xf MH
t
MMH
t
M
(2-36)
23
.~~
~
~~
,~~
~
~~
22
1,
2,2211
11 zz
i
izfzz
zf MH
t
MMH
t
M
(2-37)
Introducing Equations (2-33) and (2-35) in the Equations (2-36) and (2-37), the complex
components of the magnetization are obtained as
,)
~1(
)~(ˆ)~(ˆ ,
)~
1(
)~(ˆ)~(ˆ
2
2
1,
2,
2
1
11
f
x
i
i
x
f
xx
j
xHxM
j
xHxM
(2-38)
.)
~1(
ˆ)~(ˆ ,
)~
1(
ˆ)~(ˆ
2
2
1,
2,
2
1
11
f
z
i
i
z
f
zz
j
HxM
j
HxM
(2-39)
To obtain the complex components of the magnetic field vector, we introduce Equations
(2-33) to (2-35) and Equation (2-38) in the magnetic flux density vector,
,~~~
,~~~
221,2111,1 HMBHMB ii (2-40)
resulting in
,)
~1(
ˆ)~
1()~(ˆ ,
)~
1(
ˆ)~
1()~(ˆ
22,
22
2
11,
11
1
fi
xf
x
fi
xf
xj
BjxH
j
BjxH
(2-41)
.)
~1(
)~(ˆ)~
1(ˆ ,
)~
1(
)~(ˆ)~
1(ˆ
212,
22
2
11,
11
1
fi
zf
z
fi
zf
zj
xBjH
j
xBjH
(2-42)
Substituting Equation (2-41) in Equation (2-38) yields,
.)
~1(
ˆ)~(ˆ ,
)~
1(
ˆ)~(ˆ
22,
2
1,
2,
2
11,
11
fi
x
i
i
x
fi
xx
j
BxM
j
BxM
(2-43)
Equations (2-41) and (2-43) show that 2121ˆ ,ˆ ,ˆ ,ˆ
xxxx MMHH are uniform for the zeroth
order analysis. From the equations of magnetic field and magnetization, we conclude
that the magnetic field is uniform, therefore the magnetic body force in both the
ferrofluids is zero,
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.~~~
,~~~
2211 0HM0HM (2-44)
The y component of the body couple is given by
,~~~~~
,~~~
xkzkzkxkykkkk HMHMlHMl (2-45)
,ˆ ˆˆ ˆ~ ~~~~tj
xk
tj
zk
tj
zk
tj
xkyk eHeMeHeMl (2-46)
where the subscript (k) is used to indicate ferrofluid 1 or ferrofluid 2. Equation (2-46) is
simplified by using the relation of the real part of the product of two complex functions
)(tA and )(tB given by
)~~~~(
4
1)
~~~~(4
1)()(
~2**
~2** tjtj ebaebababatBtA (2-47)
where the superscript (*) denotes the complex conjugate. Thus the expression for the
y component of the body torque is obtained as
)~
21( )~
1( 2
~~ )
~)(
~1(
~~ )
~~1(
~~
2
1
2
1,1,
2
1
1
*
1111,
*
1111,
2
11,1
1
fiif
xzffixzfifif
y
bhjjjbhjjl
(2-48)
.)
~21( )
~1( 2
~~ )
~)(
~1(
~~ )
~~1(
~~
2
2
2
2,2,
2
2
2
*
2222,
*
2222,
2
22,2
1,
2,
2
fiif
xzffixzfifif
i
i
y
bhjjjbhjjl
(2-49)
To generate a uniform rotating magnetic field in the counterclockwise direction, the
imposed z- component of the magnetic field and the x - component of the magnetic
induction are taken as jBH xz ˆ,1ˆ ,27 which gives the magnetic body couple in the
ferrofluids as
,~21
~1
~1
~~
2
1
2
1,1,
2
1
2
11,1
1
fiif
fif
yl
(2-50)
25
.~21
~1
~1
~~
2
2
2
2,2,
2
2
2
22,2
1,
2,
2
fiif
fif
i
i
yl
(2-51)
The above expressions for the body couple indicate that the body couple is constant
within each ferrofluid.
2.4 Zero Spin Viscosity Solution
In this case the spin viscosity in both the ferrofluids is taken to be zero
0 ,0 21 , that is, couple stresses are neglected. This condition reduces the linear
momentum balance and internal angular momentum balance equations, Equations (2-
20) to (2-23) in component form, to
,0~
~
~
~
2~
~
1
1
1
1
2
1
2
1
1
zd
Pd
xd
d
xd
vd yze
(2-52)
,0~
~
~
~
2~
~
2
2
1
2
2
2
2
1
2
zd
Pd
xd
d
xd
vd yze
(2-53)
,0~
~~4~
~2
1
1
11
f
y
yz
l
xd
vd (2-54)
.0~
~~4~
~2
1
2
1
1
22
1
2
f
y
yz
l
xd
vd
(2-55)
Differentiating Equations (2-54) and (2-55) and substituting the result in Equations (2-
52) and (2-53) respectively and integrating the resulting differential equations, we find
that the translational velocity profiles are given by
,~
2
~
~
~~
21
2
1
1 cxcx
zd
Pdvz
(2-56)
.~
2
~
~
~~
43
2
22
12 cxc
x
zd
Pdvz
(2-57)
26
Equations (2-56) and (2-57) are differentiated with respect to x~ and the result is
substituted in Equations (2-54) and (2-55) respectively to get the spin velocity profiles
as
,2
~4
~
2
~
~
~~ 1
1
1
1
1
clx
zd
Pd
f
y
y
(2-58)
.2
~4
~
2
~
~
~~ 3
1
2
2
1
22
12
clx
zd
Pd
f
y
y
(2-59)
In the above equations, 4321 ,,, cccc are constants which depend on the properties of the
ferrofluids, the applied magnetic field and the relevant boundary conditions.
2.5 Non-zero Spin Viscosity Solution
By considering the existence of spin viscosity, the linear momentum balance
equations reduce to the component forms given in Equations (2-52) and (2-53) and the
internal angular momentum balance equations, Equations (2-22) and (2-23) reduce to
the component forms given by
,0~
~~4~
~2~
~4
1
1
11
2
1
2
2
1
1
f
y
yzy
e
l
xd
vd
xd
d
(2-60)
.0~
~~4~
~2~
~4
1
2
2
1
22
1
2
2
2
2
2
11
21
f
y
yzy
e
l
xd
vd
xd
d
(2-61)
We integrate Equations (2-52) and (2-53) once and introduce the result in Equations (2-
60) and (2-61) respectively to yield a non-homogeneous differential equation for the y
component of the spin velocity, with the solution
,2
~4
~
)~cosh()~sinh(2
~
~
~
)~(~ 1
11
11
43
1
1
ClxCxC
x
zd
Pdx
f
ye
y
(2-62)
27
.2
~4
~
)~cosh()~sinh(2
~
~
~
)~(~
2
12
212
122
76
22
12
ClxQCxQC
x
zd
Pdx
f
ye
y
(2-63)
The translational velocity profiles are determined by substituting Equations (2-62) and
(2-63) in Equations (2-52) and (2-53), and integrating to obtain
,~~
2
~~)~sinh(
2)~cosh(
2
2
~
~
~
)~(~5
11
11
14
1
13
1
12
1
1 Cxl
xCxCxCx
zd
Pdxv
f
y
ee
z
(2-64)
.~~
2
~~)~sinh(
2)~cosh(
2
2
~
~
~
)~(~8
12
21
2
2
17
2
26
2
22
22
12 Cx
lxCxQC
QxQC
Q
x
zd
Pdxv
f
y
ee
z
(2-65)
In the above equations, 2
1
'
2
'
1
2
1
1
2
1
2
e
eQ and 87654321 ,,,,,,, CCCCCCCC are
constants which depend on the properties of the ferrofluids, the applied magnetic field
and the relevant boundary conditions.
2.6 Relation between Pressure Gradients and Boundary Conditions
The total linear and angular momentum balance is required to determine the
relation between pressure gradients in the two ferrofluids and to obtain the boundary
conditions at the fluid-fluid interface.
The total interfacial linear momentum balance at the fluid-fluid interface is derived
in27 as
,)( 2
1)(2)(
2
2
1
2
12120nHHnTTn HHB ons Η (2-66)
where , ,2
2
2
2
2
2
1
2
1
2
1
2
tntn HHHHHH n is the unit normal, Η is the radius of
curvature, and is the surface tension. The interface is considered flat and the unit
28
normal is in the x direction. Using the scaled variables as defined in Equation (2-15),
the normal component of the interfacial stress balance is given by the equation
.)~~
(2
1)
~~(
~)(
1
2
2
2
122
12 0TT
HHHHBH
xxx
o
xxxx
(2-67)
The Cauchy stress tensor in ferrofluids is given by the relation1
.2 )( ωvεIvvvIT t p (2-68)
Using the normal component of the Cauchy stress tensor in Equation (2-68) and
substituting the expressions for magnetic field and magnetic flux density from Equations
(2-33), (2-34) and (2-41) and considering jBH xz ˆ,1ˆ , Equation (2-67) is solved to
obtain the pressure relation between the two ferrofluids as
,0 ]2sin[]2cos[ ]2sin[]2cos[
~1
~14
1)(2
2
2
2
2
2,22
2
1
2
1
2
1,11
2
2
2
2,
2
2,
2
1
2
1,
2
1,
2
12
ba
tbta
ba
tbta
H
pp ii
fi
i
fi
i
o
(2-69)
where
.~
2~
2 ,~
21 ,~
2~
2 ,~
21 22,22
2
2
2
2,2,211,11
2
1
2
1,1,1 fiffiififfii baba
(2-70)
From the above relation, it is seen that the static pressure difference between the two
fluids depends only on the properties of the ferrofluids and the frequency of the
magnetic field. The pressure difference is independent of the z - direction, which means
that
dz
dp has the same constant value for both fluids.30 As given by the relation
zk
kk
gdz
dp
dz
dP
, and considering gravity acting perpendicular to the flow 0zg
29
we obtain that the dynamical pressure gradient
dz
dP also has the same constant value
throughout both ferrofluids, such that
.21
dz
dP
dz
dP (2-71)
Scaled boundary conditions - To evaluate the constants in the zero spin
viscosity case, we consider the no slip condition for translational velocity at the plates
given by
.L
L ;0)1~(~ ,0)0~(~
1
221
xvxv zz (2-72)
The boundary conditions at the interface include the continuity of tangential component
of translational velocity
)1~(~)1~(~21 xvxv zz , (2-73)
and the tangential component of total linear momentum balance, Equation (2-66),
),1~(~2~
~)1~(~2~
~
22
1~
2211
1~
11
xxd
vdx
xd
vdy
x
zey
x
ze (2-74)
the detailed derivation for Equation (2-74) is given by Chaves and Rinaldi.27
In order to evaluate the constants in the non-zero spin viscosity case, we use the
boundary conditions for the spin velocities in addition to the boundary conditions given
in Equations (2-72) to (2-74).
The no slip condition for the spin velocity at the walls is given by
,0)1~(~ ,0)0~(~21 xx yy (2-75)
we consider the boundary condition for spin velocities at the interface as
),1~(~)1~(~21 xx yy (2-76)
30
analogous to the continuity of tangential velocity at the fluid-fluid interface. The
interfacial internal angular momentum balance, derived in detail by Chaves and
Rinaldi,27 is extended to the case of two ferrofluids and the boundary condition at the
interface is given by
.~
~
~
~
1~
2
2
1~
1
1
x
y
x
y
xd
d
xd
d
(2-77)
Using the boundary conditions mentioned, the constants for the zero spin viscosity and
non-zero spin viscosity cases were evaluated using Mathematica® and the velocity
profiles are plotted in Chapter 3. The constants are given in Appendix A.
31
CHAPTER 3 PREDICTED VELOCITY PROFILES
To calculate the translational and spin velocity profiles, we considered ferrofluids
with representative properties as ferrofluid 1 and ferrofluid 2. The properties of the
representative ferrofluids are summarized in Table 3-1. The thickness of ferrofluid 1 is
considered to be 2 mm and the aspect ratio
1
2
L
L is taken to be 2. To show the
dependence of flow profiles on properties of ferrofluids, we consider Ferrofluid A and
Ferrofluid B with similar properties, and Ferrofluid C with properties different as
compared to Ferrofluid A and Ferrofluid B. Ferrofluid C has a spin viscosity which is two
orders of magnitude higher than that of Ferrofluid A and Ferrofluid B.
Table 3-1. Physical properties of ferrofluids
Physical Properties Ferrofluid A Ferrofluid B Ferrofluid C
(kg/m3) 1080 1030 1030
(m Pa s) 5.1 4.5 1.03
o (m Pa s) 1.64 1.64 1.02
(s) 1.7 x 10-6 1.9x 10-6 1.67 x 10-5
i 0.9 1.2 0.1
0.05 0.04 0.002
(m Pa s) 0.123 0.0984 0.00306 ' (kg m/s) 5.6 x 10-10 6 x 10-10 3.6 x 10-8
In Figure 3-1, non-dimensional values of torque generated in the ferrofluids are
plotted against the frequency of the applied magnetic field. The body torque depends on
the properties of the ferrofluid, such as initial magnetic susceptibility and relaxation time,
and on the direction of the rotating magnetic field. For counterclockwise rotating
magnetic field, positive torque is generated in each of the ferrofluids. The torque
32
generated is constant within each ferrofluid. Figure 3-1 shows that the magnitude of
torque increases with increase in the frequency of the magnetic field. It is also seen that
the difference between the magnitudes of torques in the ferrofluids increases with
increase in the magnetic field frequency.
Figure 3-1. Body torque in ferrofluids at different magnetic field frequencies.
3.1 Zero Spin Viscosity Case
To plot the velocity profiles for the zero spin viscosity case, we have taken
ferrofluid 1 as Ferrofluid A and ferrofluid 2 as Ferrofluid C or Ferrofluid B, and set 0'
for both.
3.1.1 Zero Spin Viscosity without Pressure Gradient
Figures 3-2 and 3-3 show the predicted dimensional translational and spin
velocity profiles for several frequencies and amplitudes of the applied magnetic field for
Ferrofluid A and Ferrofluid C. It is seen that the translational velocity varies linearly with
distance from the plates in both the ferrofluids and the spin velocity is constant within
each ferrofluid. The spin velocity is in the positive y- direction, that is, in the direction of
rotating magnetic field. The magnitude of the body torque generated in ferrofluid 2 is
33
higher than in ferrofluid 1, thus driving the translational velocity in the positive z-direction
in both ferrofluids. It was also seen that if the torque generated in ferrofluid 1 is higher
than that generated in ferrofluid 2, the translational velocity is in the negative z-direction.
The existence of translational velocity in the zero spin viscosity case is in contrast with
the prediction of no flow in planar, cylindrical, annular, and spherical geometries in the
absence of a free surface.18,21–25 These cases concluded that flow in ferrofluids is due to
the presence of couple stresses, related to the diffusive transport of internal angular
momentum which are neglected in the zero spin viscosity case. In the case of a
ferrofluid-non-ferrofluid interface or ferrofluid-ferrofluid interface, there is an imbalance
of the tangential component of the antisymmetric stress tensor, which drives the flow.
Also, as seen from Figure 3-1, the magnitude of the body torques generated in the two
ferrofluids is different, driving the flow in ferrofluids.
Figure 3-2. Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for Ferrofluid A and Ferrofluid C. A) Translational velocity profiles, B) spin velocity profiles
A B
34
Figure 3-3. Dimensional velocity profiles at different magnetic field amplitudes at a constant field frequency of 150 Hz for Ferrofluid A and Ferrofluid C. A) Translational velocity profiles, B) spin velocity profiles.
The magnitude of the translational and spin velocities increases with increasing
frequency of the applied magnetic field at a constant field amplitude as seen in Figure 3-
2. This is because the difference between the magnitudes of torques generated in the
two ferrofluids increases with the field frequency as shown in Figure 3-1. Figure 3-3
shows that the magnitude of dimensional velocities increases with the square of the
magnetic field amplitude at a constant field frequency.
The predicted dimensional translational and spin velocity profiles at different field
frequencies and field amplitudes for Ferrofluid A and Ferrofluid B are plotted in Figures
3-4 and 3-5. As the properties of Ferrofluid A and Ferrofluid B are similar, the difference
between the magnitude of torques generated in the two ferrofluids is small, as seen
from Figure 3-1, and thus the magnitude of translational velocity obtained is less
compared to the case when Ferrofluid A and Ferrofluid C are considered. It is therefore
concluded that for flows to be significant, properties of the two immiscible ferrofluids
should be widely different.
A B
35
Figure 3-4. Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for Ferrofluid A and Ferrofluid B. A) Translational velocity profiles, B) spin velocity profiles
Figure 3-5. Dimensional velocity profiles at different magnetic field amplitudes at a constant field frequency of 150 Hz for Ferrofluid A and Ferrofluid B. A) Translational velocity profiles, B) spin velocity profiles
3.1.2 Zero Spin Viscosity with Pressure Gradient
The application of a pressure gradient causes the translational velocity to acquire
a parabolic shape, as seen in Figure 3-6. The direction of velocity depends on the
magnitude and direction of the applied pressure gradient. A positive pressure gradient
drives the flow in the negative z-direction. Figures 3-6 show that this effect is seen the
most at lower values of applied field frequency. As noted, at a higher magnitude of
36
positive pressure gradient, the pressure gradient dominates the flow and the
translational velocity is in the negative z-direction in both ferrofluids. The application of a
negative pressure gradient causes the ferrofluids to flow in the positive z-direction as
seen in Figures 3-6 C, D, that is, in the same direction as the flow generated by the
rotating magnetic field for the particular properties of Ferrofluid A and Ferrofluid C.
Thus the negative pressure gradient assists the flow and the magnitude of velocity is
higher for a particular field frequency with a higher pressure gradient applied. Also it is
seen that at higher applied pressure gradient values, the dependence of velocity
magnitude on the field frequency reduces. Figures 3-7 A, B clearly show the effect of
applied pressure gradient on the magnitude and direction of translational velocity, as
explained before.
Figure 3-6. Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT and at different values of applied pressure gradient. A) (dP/dz) = 0.1 Pa/m, B) (dP/dz) = 1 Pa/m, C) (dP/dz) = -0.1 Pa/m, D) (dP/dz) = -1 Pa/m
A B
37
Figure 3-6. Continued
Figure 3-7. Effect of pressure gradient on translational velocity at a constant field frequency of 150 Hz and field amplitude of H = 2 mT. A) Application of negative pressure gradient, B) application of positive pressure gradient.
The spin velocity varies linearly with x when a pressure gradient is applied. Low
values of applied pressure gradient have minimal effect on the magnitude of spin
velocity and the spin velocity remains in the direction of the rotating magnetic field
(positive y-direction). At higher values of pressure gradient, there is an effect on the
magnitude of spin velocity. The graphs showing the effect of pressure gradient on spin
velocity are given in Appendix B. Also the graphs for the variation of translational and
C D
A B
38
spin velocity profiles with field amplitude at a constant field frequency of 150 Hz and
pressure gradient of 0.1 Pa/m are given in Appendix B.
3.2 Non-zero Spin Viscosity Case
3.2.1 Non-zero Spin Viscosity without Pressure Gradient
Using the appropriate solutions, we plot the non-dimensional velocity graphs for
the case of ferrofluid -non-ferrofluid interface. We consider ferrofluid 1 as Ferrofluid A
and the non-ferrofluid as water. As seen from Figure 3-8, the translational velocity is
not linearly varying with x and the solution tends to the zero spin viscosity case as
. Figure 3-8 A shows that the magnitude of flow increases with increasing ,
which is opposite to what is observed in spin-up flow in a cylindrical container.
Increasing spin viscosity acts to resist rotation of the particles, leading to a lower spin
velocity and concomitantly, a decreased mismatch in the interfacial momentum balance.
This is evident in Figure 3-8 B, where the spin velocity is not constant and tends to form
a boundary layer close to the wall at high values of , similar to the case of spin-up flow
in a cylindrical container.
Figure 3-8. Non-dimensional velocity profiles at different values without the application of pressure gradient with ferrofluid/non-ferrofluid interface. A) Translational velocity profiles, B) spin velocity profiles.
A B
39
Figure 3-9. Non-dimensional velocity profiles at different values without the application of pressure gradient for Ferrofluid A and Ferrofluid C. A) Translational velocity profiles, B) spin velocity profiles.
Figure 3-9 shows the non-dimensional translational and spin velocity profiles at
different values of considering both as ferrofluids. It is seen that the magnitude of the
translational velocity increases with increasing value, as seen in the ferrofluid-non-
ferrofluid case, however the behavior observed in case of spin velocity is different. Also
it was observed that the solution tends to infinity at higher values of , so the non-
dimensional velocity plots are made only at lower values. Interestingly the solution
does not tend to the zero spin viscosity solution as . This may be due to the spin
velocity being required to satisfy a boundary condition at the interface which is taken to
be analogous to the condition of continuity of tangential velocity at the fluid-fluid
interface. Hence, appears to lead to a singular problem.
From Figure 3-10 it is seen that the magnitude of dimensional velocity increases
with increase in frequency of the applied magnetic field, as observed for the zero spin
viscosity case. The spin velocity is in the positive y-direction in both ferrofluids, however
the translational velocity is in the negative z-direction, opposite to that observed when
the spin viscosity is zero.
A B
40
Figure 3-10. Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for ferrofluids with different spin viscosity values - Ferrofluid A and Ferrofluid C. A) Translational velocity profiles, B) spin velocity profiles.
The flow reversing in the non-zero spin viscosity case is explained by the internal
angular momentum balance equations,
,~~2~2~2
~
~
~
1
2
2
1
11
1
11y
e
y
f
yzl
xd
vd
(3-1)
.~~2~2~2
~
~
~
2
2
'
1
'
2
2
1
2
1
12
1
2
2
12y
e
y
f
yzl
xd
vd
(3-2)
We consider two cases – different spin viscosity values in both ferrofluids and
similar spin viscosity values in both ferrofluids. Figure 3-11 shows non-dimensional plots
of translational velocity, slope of the velocity, spin velocity and second derivative of spin
velocity for Ferrofluid A and Ferrofluid C. Ferrofluid C has a spin viscosity value two
orders of magnitude higher than Ferrofluid A. From Figure 3-11, it is seen that the
translational velocity is in the negative Z-direction in both ferrofluids except very close to
the wall in Ferrofluid A where the velocity is in the positive z-direction. This region is not
seen clearly due to the magnitude of velocity in the order of 10-6. The spin velocity is
B A
41
close to zero near each wall to satisfy the boundary condition and the body couple
dominates the diffusion term in Equations (3-1) and (3-2). In order to balance the
positive torque generated by the counter clockwise rotating field, the slope of the
velocity is positive and the vorticity is negative very close to the wall in Ferrofluid A and
in the entire region in Ferrofluid C. This implies a positive velocity very close to the wall
in Ferrofluid A and negative velocity in the entire Ferrofluid C region. In the region away
from the wall in Ferrofluid A, the spin velocity and the spin diffusion dominate the torque
and the velocity has a negative slope as shown in Figure 3-11 A, thus the translational
velocity is along the negative z-axis.
Figure 3-11. Dependence of direction of translational velocity when ferrofluids of
different spin viscosity values are considered - Ferrofluid A and Ferrofluid C. A) Non-dimensional translational velocity and slope of velocity, B) non-dimensional spin velocity and second order derivative of spin velocity.
When ferrofluids with similar spin viscosity values are considered, as shown in
Figure 3-12, the body couples dominate the flow near the walls and the slope of
velocity is positive near the walls, resulting in translational velocity in the positive z-
direction in Ferrofluid A and in the negative z-direction in Ferrofluid B. In the region near
the interface, the magnitude of spin velocity increases as seen in Figure 3-12 B, so the
slope of the velocity becomes negative. This is shown by the decrease in magnitude of
A B
42
translational velocity after the peak in Ferrofluid A and the flow going in the negative z-
direction in Ferrofluid B.
Figure 3-12. Dependence of direction of translational velocity when ferrofluids of similar
spin viscosity values are considered - Ferrofluid A and Ferrofluid B. A) Non-dimensional translational velocity and slope of velocity, B) Non-dimensional spin velocity and second order derivative of spin.
Figure 3-13. Dimensional velocity profiles at different field frequencies at a constant field amplitude of H = 2 mT for ferrofluids with similar spin viscosity values. A) Translational velocity for Ferrofluid A and Ferrofluid B, B) spin velocity for Ferrofluid A and Ferrofluid B.
A B
A B
43
Plots for dimensional velocity profiles for Ferrofluid A and Ferrofluid B are shown
in Figure 3-13 at different values of field frequencies. They show the same behavior as
in the zero spin viscosity case.
3.2.2 Non-zero Spin Viscosity with Pressure Gradient
To show the effect of applied pressure gradient on the translational velocity
profiles for the non-zero spin viscosity case, we have plotted graphs at different
pressure gradient values at a field frequency of 150 Hz and field amplitude of 2 mT as
shown in Figures 3-14. The nature of the graphs is similar to those observed for the
zero spin viscosity case, and it is clearly seen that the flow is in the direction opposite
the pressure gradient at higher values of applied pressure gradient.
Figure 3-14. Effect of pressure gradient on translational velocity at a constant field frequency of 150 Hz and field amplitude of H = 2 mT for different ferrofluid combinations. A) Positive pressure gradient for Ferrofluid A and Ferrofluid C, B) negative pressure gradient for Ferrofluid A and Ferrofluid C, C) positive pressure gradient for Ferrofluid A and Ferrofluid B, D) negative pressure gradient for Ferrofluid A and Ferrofluid B.
A B
44
Figure 3-14. Continued
As in the zero spin viscosity case, the spin velocity remains in the positive y-
direction at low values of applied pressure gradient and the plots are given in Appendix
C. The plots for dimensional velocities at various field frequencies and field amplitudes
for different ferrofluid combinations and pressure gradient values are given in Appendix
C and they show similar behavior as observed for the zero spin viscosity case.
3.3 Comparison between Zero Spin Viscosity and Non-zero Spin Viscosity Cases
Figure 3-15 shows the non-dimensional translational velocity profiles for zero
spin viscosity and non-zero spin viscosity cases for Ferrofluid A and Ferrofluid C, and
Ferrofluid A and Ferrofluid B combinations. Comparing the velocity profiles in the zero
spin viscosity and non-zero spin viscosity cases, it is seen that the flow direction
changes when the effect of spin viscosity is considered. Such a stark qualitative change
in the flow profile could be used as an experimental verification of the existence of spin
viscosity, neglected in the analyses by Rosensweig et al.15 and Krauss et al.19 As such,
experiments where two immiscible ferrofluids are in contact and subjected to a rotating
magnetic field, in the manner of Krauss et al.,19 could test the existence of the spin
C D
45
viscosity and complement experiments carried out with a single ferrofluid in a cylindrical
geometry.22
Figure 3-15. Non-dimensional translational velocity profiles for zero spin viscosity and
non-zero spin viscosity cases for different combination of ferrofluids. A) Ferrofluid A and Ferrofluid C, B) Ferrofluid A and Ferrofluid B.
An interesting feature of the solutions obtained here is that the non-zero spin
viscosity solution does not tend to the zero spin viscosity case as . This
discrepancy may be due to the boundary condition of continuity of spin velocity at the
ferrofluid-ferrofluid interface, which at the moment is ad hoc. As such, experimental
measurements of the velocity profile could also be used to test whether this boundary
condition is correct. When a ferrofluid-non-ferrofluid interface was considered, as shown
in Figure 3-8, the condition of continuity of spin velocity at the interface was not
required, and the solution tends to the zero spin viscosity case as .
A B
46
CHAPTER 4 CONCLUSIONS
In this work we have obtained analytical solutions for the translational and spin
velocity profiles for the flow of two immiscible ferrofluids driven by the application of a
uniform rotating magnetic field, extending past work of modeling the flow of ferrofluid
with a ferrofluid-air interface. The analysis predicts the existence of translational velocity
in the zero spin viscosity case. This is in contrast with analyses that predict no flow
when the spin viscosity is neglected for the bulk flow of ferrofluid in a cylindrical
container. It therefore underscores the importance of surface stresses responsible for
flow in ferrofluids as implied by Rosensweig et al.15 The interfacial linear and internal
angular momentum balance conditions show that a flow can be generated by an
imbalance of the tangential component of the antisymmetric stress tensor at the fluid-
fluid interface in addition to the couple stresses.
The analyses using different combinations of ferrofluids in the zero spin viscosity
and non-zero spin viscosity cases show that for flow to be significant, the properties of
the ferrofluids must be significantly different. The magnitude of the dimensional
translational and spin velocity in the two ferrofluids increases with the frequency of the
applied magnetic field and also increases with the square of the magnetic field
amplitude, as seen from the velocity profiles plotted for the different cases. The spin
velocity is in the direction of rotating magnetic field, y direction, and low values of
applied pressure gradient have minimal effect on the spin velocity. The direction of
translational velocity depends on the balance between the terms in the internal angular
momentum balance equation. At higher values of applied pressure gradient, the
47
pressure gradient dominates the flow generated by the rotating magnetic field and the
ferrofluids flow in the direction of applied pressure gradient.
This work of modeling the flow of ferrofluids in a planar channel has potential
applications in microfluidic and nanofluidic devices. Ferrofluids are being considered as
components in microfluidic devices where they could actuate flow. The parameters of
the rotating magnetic field and the magnitude and direction of the pressure gradient can
be varied to get the desired flow magnitude and direction.
48
APPENDIX A CONSTANTS EVALUATED
The constants for zero spin viscosity and non- zero spin viscosity cases are
evaluated using Mathematica®.
A-1 Zero Spin Viscosity Case
,02 c
,
~ 2
~~
~
~
2
11 ~
~
2~
~
21f1
1y21
21
1
2212
22
1
221
1
1
3
yll
zd
Pd
zd
Pd
zd
Pdc
,c
2
1 ~
~
2~
~
~
~
2
13
2
2
1
22
1
21
1
zd
Pd
zd
Pd
zd
Pdc
.1
2
1 ~
~
3
2
2
1
2
4
c
zd
Pdc
49
A-2 Non-zero Spin Viscosity Case
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
APPENDIX B VELOCITY PROFILES FOR ZERO SPIN VISCOSITY CASE
Figure B-1. Dimensional velocities at different field amplitudes at a field frequency of 150 Hz and pressure gradient of 0.1 Pa/m. A) Translational velocity profiles, B) Spin velocity profiles
The magnitude of velocitiy increases with increasing magnetic field amplitude at
a constant field frequency and pressure gradient value as shown in Figure B-1.
Figure B-2. Effect of positive pressure gradient applied on spin velocity at a field
frequency of 150 Hz and field amplitude of H = 2 mT. A) Spin velocity of ferrofluid 1, B) spin velocity of ferrofluid 2.
From Figure B-2, it is seen that the magnitude of spin velocity increases with
increasing value of positive pressure gradient applied in the entire ferrofluid 1region and
in ferrofluid 2 upto a certain point after which the magnitude of spin velocity decreases.
A B
A B
88
Figure B-3 shows that the magnitude of spin velocity decreases with increasing
magnitude of negative pressure gradient applied in entire ferrofluid 1 region and in
ferrofluid 2 upto a certain point after which the magnitude of spin velocity increases.
Figure B-3. Effect of negative pressure gradient applied on spin velocity at a field frequency of 150 Hz and field amplitude of H = 2 mT. A) Spin velocity of ferrofluid 1, B) spin velocity of ferrofluid 2.
The effect of field frequency on spin velocity profiles at different values of applied
pressure gradient is given in Figure B-4.
Figure B-4. Variation of dimensional spin velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient. A) (dP/dz) = 0.1 Pa/m, B) (dP/dz) = -0.1 Pa/m
B A
A B
89
APPENDIX C VELOCITY PROFILES FOR NON-ZERO SPIN VISCOSITY CASE
C-1 Ferrofluid 1 – Ferrofluid A, Ferrofluid 2 – Ferrofluid C
As seen in Figure C-1, the magnitude of spin velocity increases with increasing
value of positive pressure gradient applied and decreases with increasing value of
negative pressure gradient applied in both ferrofluids for the non-zero spin viscosity
case.
Figure C-1. Effect of pressure gradient applied on spin velocity profiles at a field
frequency of 150 Hz and field amplitude of H = 2 mT. A) Positive pressure gradient values, B) negative pressure gradient values.
A B
90
Figure C-2 shows that the magnitude of translational and spin velocity increases
with increasing magnetic field amplitude similar to the zero spin viscosity case
Figure C-2. Dimensional velocities at different field amplitudes at a field frequency of
150 Hz and a pressure gradient of 0.1 Pa/m for Ferrofluid A and Ferrofluid C. A) Translational velocity profiles, B) Spin velocity profiles
.
A B
91
Graphs showing the effect of field frequency on translational and spin velocity
profiles at different values of applied pressure gradient are given in Figures C-3 and C-4
for Ferrofluid A and Ferrofluid C combination.
Figure C-3. Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for ferrofluids with widely different spin viscosity values – Ferrofluid A and Ferrofluid C. A) (dP/dz) = 0.1 Pa/m, B) (dP/dz) = 1 Pa/m, C) (dP/dz) = -0.1 Pa/m, D) (dP/dz) = -1 Pa/m
A B
C D
92
Figure C-4. Variation of dimensional spin velocity with field frequency at a constant field
amplitude of H = 2mT at different values of applied pressure gradient for ferrofluids with widely different spin viscosity values – Ferrofluid A and Ferrofluid C. A) (dP/dz) = 0.1 Pa/m, B) (dP/dz) = 1 Pa/m, C) (dP/dz) = -0.1 Pa/m, D) (dP/dz) = -1 Pa/m
A B
C D
93
C-2 Ferrofluid 1 – Ferrofluid A, Ferrofluid 2 – Ferrofluid B
Graphs showing the effect of field frequency on translational and spin velocity
profiles at different values of applied pressure gradient are given in Figures C-5 and C-6
for Ferrofluid A and Ferrofluid B combination.
Figure C-5. Variation of dimensional translational velocity with field frequency at a constant field amplitude of H = 2mT at different values of applied pressure gradient for similar spin viscosity ferrofluids – Ferrofluid A and Ferrofluid B. A) (dP/dz) = 0.1 Pa/m, B) (dP/dz) = 1 Pa/m, C) (dP/dz) = -0.1 Pa/m, D) (dP/dz) = -1 Pa/m
A B
C D
94
Figure C-6. Variation of dimensional spin velocity with field frequency at a constant field
amplitude of H = 2mT at different values of applied pressure gradient for similar spin viscosity ferrofluids – Ferrofluid A and Ferrofluid B. A) (dP/dz) = 0.1 Pa/m, B) (dP/dz) = 1 Pa/m, C) (dP/dz) = -0.1 Pa/m, D) (dP/dz) = -1 Pa/m
A B
C D
95
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BIOGRAPHICAL SKETCH
Bhumika Sule received the degree of Bachelor of Chemical Engineering from
Institute of Chemical Technology (formerly UDCT), Mumbai, India in May 2012. In fall
2012, she joined University of Florida to pursue a Master of Science degree in chemical
engineering.