Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)
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Exercise
>@
2
2
2
10
21
2
111
nMnM
ˆˆ
¸
¹
·
¨
©
§
PV
RR
p
(9)
Thus, at the interface, a pressure jump may occur due to the effect of
the surface tension and the difference of a squared magnetization (normal
magnetization component) of the media.
Equation (9) gives the idea that for the magnetic fluid in contact with
the atmosphere with the plane interface the pressure inside of the magnetic
fluid is less than that of atmospheric pressure by
2
2
0 n
M
P
. Furthermore,
in the case of only a tangential magnetic flux at the interface, there would
not be a pressure jump due to the magnetization difference.
Exercise 8.3.2 Effect of Mass Concentration on Thermomagnetic
Convection
In thermomagnetic convection, if the concentration of magnetic phase in a
magnetic fluid has any space dependency, include the terms due to the
non-uniformity of fluid components to the Rayleigh number
*
Ra defined
in Eq. (8.3.24).
Ans.
Let the density of the bulk magnetic fluid
U
be spatially non-uniform
and define
m
U
as the density of the magnetic phase in a magnetic fluid.
The concentration of the magnetic phase is thus written as
U
U
mm
n
(1)
m
n
is a function of spacial coordinates.
U
and
M
then have a functional
relationship for independent parameters;
T
,
H
and
m
n are as follows
m
nT ,
U
U
(2)
m
nHTMM ,,
(3)
so that, in differential forms
m
T
m
n
n
n
T
T
m
ේ
ේ
ේ
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
UU
U
(4)
'
535
m
HT
m
nTTn
n
n
M
H
H
M
T
T
M
M
mm
ේ
ේ
ේ
ේ
,
,,
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
(5)
The substitution of Eqs. (4) and (5) for Eq. (8.3.18) yields
0ේේ
ේේ
u
¸
¸
¹
·
¨
¨
©
§
u
¸
¸
¹
·
¨
¨
©
§
m
mm
nH
n
M
n
TH
T
M
T
gg
UU
(6)
For the existence of the critical phenomena, likewise discussed in Eq.
(8.3.20), the following conditions are to be met
H
T
M
T
ේ
w
w
w
w
g
U
and
H
n
M
n
mm
ේ
g
U
(7)
Therefore, when T
ේ or
m
nේ are parallel to g and Hේ , the convection
may set up under definite conditions. By definition of the Rayleigh number,
*
Ra given in Eq. (8.3.21), we can define the generalized Rayleigh number
for magnetodiffusion convection, considering
m
nේ and writing the neces-
sary condition on mechanical equilibrium similar to Eq. (8.3.20) as
m
mm
nH
n
M
n
u
¸
¸
¹
·
¨
¨
©
§
w
w
w
w
ේg
U
=0
(8)
Analogous to deriving Eq. (8.3.24), we can derive an expression for the
generalized Rayleigh number
*
Ra by writing
D
l
dz
dn
dz
dH
MRa
a
m
K
EEU
4
00
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
pn
g
*
(9)
where
n
E
and
p
E
are defined in such a way that
m
n
nw
w
U
U
E
0
1
and
m
p
n
M
M w
w
0
1
E
, Blums et al. (1997). Note that
D
is the diffusion coeffi-
cient of magnetic particles in a carrier liquid.
It is mentioned that under the non-isothermal conditions that the flow
instability is largely dependent upon by a thermoconvective mechanism,
rather than a mass transfer of colloidal particles. This is due to the reason
that the thermal relaxation time is considerably shorter than the time re-
quired for colloidal particles to establish an equilibrium concentration field
of
m
U
. This can be characterized by the Lewis number Le , which is de-
fined as
8 Magnetic Fluid and Flow 536
Problems
D
U
k
D
kcDLe
cp
0
(10)
where
D
k is the thermal diffusivity. The magnitude of
D
can be estimated
by Einstein’s formula for the diffusion coefficient
rTkD
B 0
6
S
K
(11)
where nm102 |r is the diameter of a magnetic particle. For a water base
magnetic fluid, Le may be calculated to give
4
7
11
1030
1051
10440
u
u
u
.
.
.
Le
(12)
Small Le shows that the mass diffusion is smaller than the thermal diffu-
sion, indicating that the relaxation is considerably shorter that the mass dif-
fusion time scale.
Problems
8.3-1 Consider a non-magnetic body of density
s
U
with a representative
radius of
R
. If the density
U
of a magnetic fluid is
3
1031 u. kg/m
3
,
and
2
1052
u .M Tesla and
5
108u H A/m
2
are kept constant,
what is the floating criterion of the nonmagnetic body when
0010.|R m is considered?
Ans.
>@
»
»
»
»
»
¼
º
«
«
«
«
«
¬
ª
3
3
kg/m33
5
4
0
)Eq. (8.3.13 From
.
s
s
RHM
U
SUU
g
8.3-2 Consider a single stage rotating shaft with a magnetic fluid seal, as in
Fig. 8.8, when a strong magnetic field of
3
2
10191 u .H kA/m and
0
1
|H is imposed at both sides of the seal. Estimate the seal pres-
sure p if the mean corresponding magnetization
S
M is
2
10024
u| .
S
M Tesla.
Ans.
>
@
Pa10784
4
u.
'
537
8.3-3 Considering the constitutive equation given in Eq. (8.2.9), derive Eq.
(8.3.1) by describing the conditions to derive Eq. (8.3.1).
8.3-4 Give reasons why the curve for 0zH in Fig. 8.10 shifts to left.
Nomenclature
A
: Pseudovector, vector of tensor
B
: magnetic induction vector
b
: heat generation per unit mass
c
: couple stress tensor
D
: diffusion coefficient
d
: diameter
f
: body couple per unit mass
0
f
: Larmor frequency
g
: body force
g
: gravity acceleration
Gr
: Grashof number
H
: magnetic field vector
I
: unit tensor
I
: moment of inertia
K
: Pyromagnetic coefficient, characteristic parameter of
nonlinearity or anisotropy constant
c
k
: thermal conductivity
B
k
: Boltzmann constant
D
k
: thermal diffusivity
l
: characteristic length
Le
: Lewis number
M
: magnetization vector
S
M
: saturation magnetization
m
: magnetic moment
N
: number of particle per unit volume
Nu
: Nusselt number
n
ˆ
: normal unit vector
p
: pressure
p
: total pressure
em
P
: electromagnetic energy per unit volume
0
p
: hydrostatic pressure
8 Magnetic Fluid and Flow 538
Nomenclature
P
r
: Prandtl number
q
: heat transfer rate
q
: heat flux vector
21
RRR ,,
: representative radius
Ra
: Rayleigh number
*
Ra
: generalized Rayleigh number
m
R
: magnetic Rayleigh number
s
: intrinsic angular momentum vector
s
: specific entropy, surfactant thickness
T
: total stress tensor
T
: temperature
0
T
: reference temperature
u
: velocity vector
u
: internal energy of system
m
u
: Internal energy of magnetic fluid
V
: volume
zy
x
,,
: Cartesian coordinates system
zr ,,
T
: cylindrical coordinates system
I
T
,,r
: spherical coordinates system
D
: diffraction angle
m
E
: relative pyromagnetic coefficient
T
E
: coefficient of thermal expansion
İ
: polyadic alternator
]
: Langevin argument, vortex viscosity
K
: viscosity of bulk magnetic fluid
0
K
: base (carrier) liquid viscosity
a
K
: apparent viscosity
r
K
: rotational (additional) viscosity
K
c
: shear coefficient of spin viscosity
U
: density of bulk magnetic fluid
s
U
: particle material density
0
P
: permeability of free space
21
P
P
P
P
,,,
s
: magnetic permeability
O
c
: bulk coefficient of spin viscosity
I
: void fraction
c
I
: critical volume fraction
m
I
: void fraction of magnetic material
539
v
I
: void fraction of magnetic material with surfactant layer
V
: (coefficient of) surface tension
IJ
: stress tensor
B
W
: Brownian diffusion time constant
N
W
: Néel relaxation time
s
W
: rotational relaxation time
A
WW
,
//
: characteristic relaxation time
X
: kinematic viscosity
U
K
X
F
: magnetic susceptibility
Y
: mass of a particle
Ȧ
: vorticity vector
p
Ȧ
: spin angular velocity
: fluid particle rotation rate vector, angular velocity
Bibliography
With the appearance of a magnetic fluid, the conceptual treatment of ferro-
hydrodynamics was first introduced by
1.
J.L. Neuringer and R.E. Rosensweig, Ferrohydrodynamics, Phys. Fluid
7 (12), 1964.
An excellent and rather authoritative overview on ferrohydrodynamics is
found in the text, of which students who wish to study this field of science
should be aware.
A great deal of thermodynamical treatment on the subject and its engineer-
ing applications are found in
V.G. Bashtovoy, B.M. Berkovsky and A.N. Vislovich, Introduction to
Thermodynamics of Magnetic Fluids, Hemisphere Publication Corpo-
ration, New York, 1988.
:
8 Magnetic Fluid and Flow
2.
R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press,
Cambride, MA, 1985. Republished from Dover Publications, Inc., 1997.
5.
3.
R.E. Rosensweig, An Introduction to Ferrohydrodynamics, Chemical
Engineering Communication, 67, 1–18, 1988.
4.
E. Blums, A. Cebers and M.M. Maiorov, Magnetic Fluids, Walter de
Gryuter & Co., Berlin – New York, 416, 1997.
540
Bibliography
Recent developments in ferrohydrodynamics is well documented via vari-
ous topics in the academic field of magnetic fluid, in
S. Odenbach (edited), Magnetoviscous Effects in Ferrofluids, Springer,
New York, 2002.
Flow phenomena, specific data, correlations and approximations that are
referred to in individual topics are presented in
W.F. Brown, Thermal fluctuation of single domain particle, Phys. Rev.,
130 (5), 1963.
L. Néel, Effect of thermal fluctuation on the magnetization of small
particles, C. R. Acad. Sci. Paris, 1949.
11.
A. Einstein, On the movement of small particles suspended in a sta-
tionary liquid demanded by the molecular kinetic theory of heat, An-
naler der physic 17, 1906, which is also found in an English translation,
R. Furth, Einstein Investigation on the Theory of the Brownian Move-
ment, Dover, New York, 1956.
12.
M.A. Martsenyuk, Yu. L, Raikher and M.I. Shliomis, On the kinetics
of magnetization of suspensions of ferromagnetic particles, Sov. Phys.
JETP, 38 (2), 1974.
13.
L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media,
(2nd Edition, 1984) Pergamon Press, Oxford, 1960.
14.
M.I. Shliomis, Effective viscosity of magnetic fluid suspensions, So-
viet Phys. JETP, 341 (6),1972.
15.
L.D. Landau and E.M. Lifshitz, Fluid Mechanics, (2nd Edition, 1987)
Pergamon Press, 1959.
16.
M.I. Shliomis, Convective instability of magnetized ferrofluid, Lecture
Notes in Physics, Springer-Verlag, New York, 2001.
17.
K. Gotoh and M. Yamada, Thermal convection in a horizontal layer of
magnetic fluids, J. Phys. Soc., 51 (9), 1982.
18.
L. Schwab, U. Hildebrandt and K. Stierstadt, Magnetic Benard con-
vection, J. Magn. Magn. Mater., 39 (1–2), 1983.
8.
9.
10.
B.M. Berkovsky, V.F. Medvedev and M.S. Krakov, Magnetic Fluid,
Engineering Application, Oxford University Press, Oxford, 1993.
6.
7.
R.E. Rosensweig, Basic equations for magnetic fluids with internal
Ferrofluids (Edited by S. Odenbach), Springer, 2002.
541
rotations,
19. G. Kronkalns, M. Maiorov and E. Blums, Preparation and Properties of
Temperature-Sensitive Magnetic Fluids, Magnetohydrodynamics, 33,
92–96, 1997.
20.
G.Z. Gershuni and E.M. Zhukhovitskii, Convective Stability of Incom-
pressible Fluids, Kepter Publishing House Jerusalem Ltd. (translated
from Russian), Jerusalem, 1976.
542 8 Magnetic fluid and flow
Appendix
Appendix A
Table A.1 Physical properties of water
>@
C
$
T
>
@
3
mkg
U
>@
sPa
0
K
>@
sm
2
X
>@
kPa
vapor
p
>@
mN
V
>@
GPaK
0 1000
3
10751
u.
6
10751
u.
6110. 07560. 022.
10 1000
3
10301
u.
6
10301
u.
231.
07420. 102.
20 998
3
10021
u.
6
10021
u.
342
. 07280. 182.
30 996
4
10008
u.
6
10038
u.
244.
07120. 252.
40 992
4
10516
u.
7
10566
u.
387.
06960. 282.
50 988
4
10415
u.
7
10485
u.
312. 06790. 292.
60 984
4
10604
u.
7
10674
u.
919.
06620. 282.
70 978
4
10024
u.
7
10114
u.
231.
06440. 252.
80 971
4
10503
u.
7
10603
u.
447. 06260. 202.
90 965
4
10113
u.
7
10223
u.
170.
06080. 142.
100 958
4
10822
u.
7
10942
u.
3101.
05890. 072.
543
Appendix
Table A.2 Typical physical Properties of some common liquids at atm1 and
C20
o
Liquid
>
@
3
mkg
U
>@
sPa
0
K
>@
kPa
vapor
p
>@
mN
V
Ammonia
829
4
10202
u.
910 02130.
Benzene
879
4
10516
u.
110. 02890.
Ethanol
7887
3
10201
u.
755. 02280.
Glycerine
1258 491.
5
1041
u.
06330.
Kerosene
819
3
10921
u.
113. 02770.
Methanol
788
4
10985
u.
413. 02260.
Appendix B
Appendix B-1 Vector Tensor Operations
Write a vector u as a sum
i
i
i
u
¦
e
ˆ
and a tensor T as a sum
ij
i
ji
T
¦
ee
ˆˆ
,
where
ji
ee
ˆˆ
is the unit dyad. Note that the unit vectors
i
e
ˆ
are defined to give
vectors and there are the scalar products
ji
ee
ˆˆ
and vector products
ji
ee
ˆˆ
u . A third kind of product can be formed with the unit vector,
namely the dyadic product
ji
ee
ˆˆ
, where the products
ji
ee
ˆˆ
are the second
order tensor.
i
e
ˆ
and
j
e
ˆ
are of unit magnitude, so that the products
ji
ee
ˆˆ
are
treated as unit dyads.
Based upon the dot and cross products of unit vector, which are per-
formed by the geometrical definitions, the analogous operations for the
unit dyads are defined by relating them to the operations for unit vectors
iljklikjlkji
G
G
eeeeeeee
ˆˆˆˆˆˆ
:
ˆˆ
(B.1-1)
k
jikjikji
G
eeeeeee
ˆˆˆˆˆˆˆ
(B.1-2)
k
j
ikjikji
eeeeeee
ˆˆˆˆˆˆˆ
G
(B.1-3)
544