Thermal Science & Engineering Vol. 14 No. 4 (2006)

Convection of Paramagnetic Fluid in a Cube Heated and Cooled
from Side Walls and Placed below a Superconducting Magnet*- Comparison between Experiment and Numerical ComputationsTomasz BEDNARZ†,‡, Elzbieta FORNALIK§, Toshio TAGAWA#, Hiroyuki OZOE‡
and Janusz S. SZMYD§
Abstract
The magnetic convection of paramagnetic fluid is studied in a strong magnetic field. The fluid in a cubic enclosureis heated from one vertical wall and cooled from the opposite one. The fluid is the 80% mass aqueous solution ofglycerol with 0.8 mol/kg concentration of gadolinium nitrate hexahydrate to make the working fluid paramagnetic. The
small amount of liquid crystal slurry is added to the fluid in order to visualize the temperature profiles in a verticalcross-section. This system is placed directly below the solenoid of the superconducting magnet which is oriented
vertically. The temperature of cold wall is constantly controlled by the water flowing from a thermostating bath. On theother hand, the hot wall is heated by a nichrome wire from a DC power supply. In the numerical computations, theconfiguration of the system is modeled to be as close as possible to the real system. The physical properties of theworking fluid are used to compute dimensionless parameters in the numerical model and the computations are carriedout for corresponding cases. Later, the numerical and experimental results are compared with each other.
Key Words:

Magnetic convection, Cubic enclosure, Numerical analysis, Visualization, Particle Image
Thermometry

Nomenclature
b
b0
B
C

ez
g
i
l
Nu
p
p0
P
Pr
Qcd
Qcv
Qloss

: magnetic induction[T]: (=μmi/l)referencemagnetic [T]
induction: (= b/b0) dimensionless magnetic[-]
induction[-]: (= 1+ 1/(βθ 0) ) dimensionless
momentum parameter forparamagnetic fluid: unit vector in the vertical direction[-]: gravitational acceleration[m/s2]: electric current in a coil[A]: length of the cubic enclosure[m]: (= Qcv/Qcd) Nusselt number[-]: pressure[Pa]: (= ρ 0α2/l2) reference pressure[Pa]: (= p/p0) dimensionless pressure[-]: (= ν/α) Prandtl number[-]: conduction heat flux[W]
: convection heat flux[W]: heat loss
[W]

Qncd

: (= Qcd – Qloss) net conduction heat
flux

Qncv

: (= Qcv – Qloss) net convection heat[W]flux
: position vector[m]: (= r /l) dimensionless position vector[-]: (=gβ(θh−θc)l3/(αν)) Rayleigh number[-]: tangential element of the coil[m]
: (= ds/l) dimensionless tangential[-]
element of the coil
: Time[s]: (= l2/α) reference time[s][-]
: (= (θ−θ 0)/(θh−θc)) dimensionlesstemperature: velocity vector[m/s]: (= α/l) reference velocity[m/s]: (= u/u0) dimensionless velocity[-]
vector: Dimensionless distance between[-]
center of a cube and center of thecoil (elevation): thermal diffusivity[m2/s]: thermal expansion coefficient
[K-1]

r
R
Ra
ds
dS
t
t0
T
u
u0
U
Zc

α
β

[W]

* Received: May 6, 2005, Editor: Hiroshi KAWAMURA†
School of Engineering, James Cook University (Townsville, Queensland 4811, AUSTRALIA)‡
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University (Kasuga Koen 6-1, Kasuga 816-8580, JAPAN)§
AGH Univeristy of Science and Technology (30 Mickiewicz Ave., 30-059 Krakow, POLAND)＃Tokyo Metropolitan University (Asahigaoka 6-6, Hino, Tokyo 191-0005, JAPAN)- 107 -

© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No. 4 (2006)

λ
ν
θ
θ0
θc
θh
ρ
ρ0

[-]
: (=χ0b02/(μmρgl)) dimensionlessgamma parameter indicating magnetic
field strength: thermal conductivity[W/(m⋅K)]: kinematic viscosity[m2/s]: Temperature[K]: (= 293 K) reference temperature[K]: temperature of cooled wall[K]: temperature of heated wall[K]
: density[kg/m3]: reference density at temperature [kg/m3]τχχ0: (= t/t0) dimensionless time[-]
: mass magnetic susceptibility[m3/kg]: ref. volume magnetic susceptibility at [-]χm: (=χ⋅ρ) volume magnetic susceptibility

γ

θ0

θ0

[-]

Sub/Superscripts0: reference valuec: coldh: hot
cd: conduction
cv: convection

1

Introduction

Although the magnetic force has been known formany decades for ferromagnetic materials, the discoveryof high-temperature superconductors in 1986 has opened
the era on its study for general materials. With using a
superconducting magnet many new phenomena havebeen reported in various fields of science. The
super-computers gave an additional opportunity to solve
numerically for complex problem prediction. Wakayama
and coworkers have found both experimentally andnumerically many notable magneto-aero dynamic
phenomena which occur in a strong magnetic field. They
work for example on promotion of combustion [1],breath supporting [2], magnetic levitation, jet stream ofnitrogen gas into air [3] and so forth. Kitazawa’s group
have been also working on the phenomena associatedwith magnetic field. Among their many findings, theyreported for example novel separation method for the
feeble magnetic materials by the application ofmagneto-Archimedes levitation technique [4], enhancing
oxygen gas dissolution into water [5], etc.
Bai et al. [6] developed a model equation forconvection due to magnetic force and carried out
numerical computations for jet flow of nitrogen gas in adecreasing gradient of magnetic field for, so to speak, aWakayama jet. Huang et al. [7] carried out own
mathematical model describing convection ofparamagnetic and diamagnetic fluids and showed it
together with the stability analysis. Tagawa et al. [8]derived the numerical model equations for magnetic

convection using a method similar to the Boussinesq
approximation and carried out the numerical calculations.
Kaneda et al. [9] studied the magnetic enhancement forthe stably stratified air layer heated from above andcooled from below. The air was filled in a cubic
enclosure located among the four poles magnet. The
visualization with incense smoke showed that stagnant
state was disturbed in the presence of magnetic field. The
numerical analysis agreed well with the experiment tosupport governing equations. More information aboutmagnetic convection and its applications can be found inbook of Ozoe [10].
Krakov et al. [11,12] reported the study of magnetic
field on the natural convection of magnetic fluid. They
considered the system that the fluid is heated from below
and cooled from above in a square or cubic enclosure.
The effect of magnetic orientation [11] and the effect ofmagnetic strength [12] are studied. Their system isdifferent from ours in various aspects as follows. They
considered the artificial magnetic fluid with the base ofkerosene which is different from ours such as air, wateror present fluid (aqueous solution). They treated the 2D
convection in a square cavity heated from below which is
not stable in a gravity field. The 2D assumption for theflow presumes the same flow pattern in a horizontal
direction of a long channel with a square cross section.
However, the theoretical work by S.H.Davis [13] andour experimental and numerical works [14,15] show thatthe roll cells are oriented with their axes horizontal andperpendicular to the long channel axis. These cells are no
more 2D but 3D due to the rigid side walls on roll axes.
Furthermore, the theory by S.H.Davis [13] suggests thatthe preferred orientation of roll cell is not deterministic
in the cubic box heated from below. The numericalcomputation by Krakov et al. [12] also represents
non-deterministic flow modes with upward flow at one
corner and downward flow at three other corners, or viceversa. Due to these reasons, we will not compare thepresent work with those by Krakov et al. [11,12].
The present work concerns the magnetic convectionof paramagnetic fluid in a cubic enclosure heated andcooled from side walls. This system was placed under
the solenoid system of a superconducting magnet. The
convection was investigated under several strengths ofmagnetic inductions to see their effect in the experiment
as well as in the corresponding numerical computations.
2

Properties of the working fluid

The 80% mass aqueous glycerol solution was
employed as a working fluid. This was mixed withcrystals of gadolinium nitrate hexahydrate to make itparamagnetic. The thermophysical properties of theworking fluid are listed in Table 1.
The density of the working fluid was measured with apycnometer (Brand 25ml) and the viscosity with the
Ostwald’s viscometer. The thermal expansion coefficient
was calculated from the measurements of the fluid
density at different temperatures and the properties with-108-

© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol. 14 No. 4 (2006)

Table 1

Properties of the working fluid at thetemperature of 298 K
Property
Value
Unit
m2/s
α*
1.01 × 10-7-3
K-1
0.52 × 10
β*W/(m⋅K)
0.397
λ
Pa⋅s
86.89 × 10-3μm2/s
5.9 × 10-5
ν
kg/m3
1463
ρ
m3/kg
23.094 × 10-8
χ
−
Pr
584

asterisk were estimated from [16]. The magnetic
susceptibility was measured with the Magnetic
Susceptibility Balance (MSB). Fig. 1 shows the mass
magnetic susceptibility χ for various concentrations ofgadolinium nitrate hexahydrate. The 80% massconcentration was employed for experiments.
3 Experimental system
Fig. 2 shows the experimental apparatus which wasassembled from five separate elements. Those included
two copper plates (one for cooling side, another one forheating side) with three holes in each to insert thethermocouples, the cubic cavity made of Plexiglas withthe hole to fill the working fluid and the cooling andheating chambers made of Plexiglas. The cooling copper
plate was cooled by the water flowing through thechamber pumped from a thermostating water bath while
the heating copper plate was heated by the nichrome wire
covered with a rubber and winded in the heating chamber.
The internal size of the cavity was chosen to be 0.032 m.
The working fluid in the cubic enclosure was filledusing a syringe and a thin needle. The front hole seen in

30

working fluid203

χ x 10 [m /kg]

25

-8

15
10
5
0
0.0

0.2

0.4

0.6

0.8

1.0

Cgado [mol/kg]

Fig. 1

Measured mass magnetic susceptibility plottedversus concentrations of Gd(NO3)3⋅6H2O. The
80% mass glycerol aqueous solution was
employed.

Fig. 2

Experimental apparatus

Fig. 2 was later closed by silicone resin to have noleakage of the fluid through it. The syringe was also very
useful to remove unwanted air bubbles which appeared
sometimes in the enclosure.
The experimental setup consisted of the experimental
apparatus, the heater control system, the constanttemperature bath and data acquisition system (NEC3100)
connected to a personal computer. The temperature of
the heated and cooled walls was measured by six T-type
thermocouples. Two other thermocouples were used tomonitor the temperature in the water bath and the room
temperature. The current and voltage of heating power
were regulated with two multimeters (Keithley 2000).
4

Experimental procedure

The heat transfer rates were measured with a 5 Tesla
superconducting magnet, however the strength ofmagnetic force was not sufficient to carry out thevisualization part. Therefore, visualization was done witha 10 Tesla superconducting magnet, with the enclosure
placed just outside of the bore.
4.1 Heat transfer rates
The experimental apparatus was well insulated withcotton-wool and with vinyl foil. Then, the enclosure wasplaced in the bore of the 5 Tesla superconducting magnet
in a distance of 0.07 m below from the solenoid center toensure minimal radial component of the magnetic force
whose value is proportional to ∇B 2 . Fig. 3(b) right halfshows that magnetic force vectors have almost uniform
vertical direction in the cubical enclosure.
The heating power and the temperature of coolingwater were set up. The temperatures of side walls weremonitored, and when a steady state was reached, they
were recorded. The steady state was attained usually in
2-3 hours. The magnetic field was applied to the systemand the magnetic induction was changed from 0 T to 5
Tesla

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© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No. 4 (2006)

(a)

(b)

Fig. 3 (a) Location of the cube in the bore of the 5 Tesla
superconducting magnet.
(b) Magnetic field
distribution (left half) with 5 T in the center of thesolenoid system and vectors of ∇B 2 (righthalf) .Center of the solenoid is located at z=0.
step by step. After every step the system was kept toreach the steady state.
The Nusselt number is defined as follows:Q(1)
Nu = ncv
Q ncd
In the present work, the method invented by Ozoe andChurchill [17] was used to estimate the net convection(Qncv) and net conduction heat fluxes (Qncd):
Qncv = Qcv − Qloss
Qncd = Qcd − Qloss

4.2 Visualization
The visualization of the temperature profiles was donewith the 10 Tesla superconducting magnet. Very small
amount of thermochromic liquid crystal slurry(KWN-2025) was added into the working fluid.
Normally clean or slightly milky in appearance, theliquid crystals reflect definite colors at specifictemperature and specific viewing angle. Therefore, themiddle vertical cross-section of the cubic enclosure was
illuminated by a white light generated by a halogen lamp.
Thin slit film was placed at the film location of the
projector so that slit light can be obtained for a long
distance.
The red color is at 292.2 K red and blue color at 296.7
K. The experimental photographs were taken by a digital
camera (EOS 10D + lens EF 70-200L) with an exposure
time of 2 s.
Fig. 4 shows the photograph of the enclosure in theexperiment under the bore of 10 Tesla superconducting
magnet. The illuminating light was projected from above
through the bore. The white light was reflected by the
mirror placed above the magnet. Since the cubic
enclosure was heated and cooled from side walls, thevertical mid-plane of the fluid was illuminated. The
projector’s focus was adjusted to keep sharp and thinlight sheet coming through the cube’s verticalcross-section. The camera was located away about 3 mfrom the magnet.

(2)
(3)

where Qcd is the total heat supply during the conduction
experiment and Qcv the one during the convection
experiment. Qloss was assumed to depend only on thetemperature of heated wall but not on the heat transfer
mode inside the enclosure and was estimated fromconduction experiment by the following equation:
Qloss = Qcd − Aλ (θ h − θ c ) / l(4)where Aλ (θh−θc)/l term represents the Fourier’s law andA = l2 is a conduction area of the cross-section. Heat loss
(Qloss) can be linearly approximated for various heating
rates.
Having relations (2), (3) and (4), the equation (1) canbe rewritten in the form:
Q − Qloss(5)
Nu = cvlλ (θ h − θ c )where the total heat supply is given by a product of
current and voltage of the heater supply.

Fig. 4

Photograph of the enclosure position in
the visualization part of the experiment

5 Numerical analysis
The numerical computations were based on the modelfor the magnetic convection of paramagnetic fluid
derived by Bai et al. [6] and subsequently with
Boussinesq approximation by Tagawa et al. [8]. It
employs the fact, that the magnetic susceptibility ofparamagnetic substance is inversely proportional to its
absolute temperature (Curie’s law). Therefore, if thetemperature difference occurs in the paramagnetic fluid,
the magnetic buoyancy force is generated in the presenceof magnetic field. Some of our previous computations
can be found in [18-20].

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Thermal Science & Engineering Vol. 14 No. 4 (2006)

The governing equations in non-dimensional form aredefined as follows:(6)∇ ⋅U = 0
DU
C⎡⎤= −∇P + Pr ∇ 2U + Ra Pr T ⎢e z − γ ∇B 2 ⎥
Dτ
2
⎣
⎦

(7)

DT
= ∇ 2TDτ

(8)

6.1 Heat transfer rates
The Nusselt number was estimated from equation (5).
Fig. 5 shows corresponding, vertical cross-section of themodeled system for the numerical computations. Axial
dimension in this figure is non-dimensionalized.
Fig. 5 shows also the forces acting on the fluid. The
arrow symbols indicate the relative force balance in

Computational conditions are as follows:
Pr = 100; C = 1+1/(βθ0); Ra = 28040⋅Δθ; γ = 0.586b02.
It should be mentioned that the flow and heat transfer
characteristics are almost independent of the Pr number
for higher Pr (Pr > 10) numbers fluids. Therefore inertia
can be neglected and the Ra number is unique parameterto characterize the buoyant convection in the absence ofmagnetic field. That is why Pr = 100 was chosen in the
numerical computations to get faster the final solution.
In order to compute the distribution of magnetic field,
the Biot-Savart’s law was used:B=dS × R1∫
4π multi −coil R 3(9)

The boundary conditions for this system are as
follows:
U=V=W=0
T = 0.5
T = -0.5∂T/∂Y = 0
∂T/∂Z = 0

at all walls of the cube,at X = -0.5,
at X = 0.5,
at Y = -0.5, 0.5,
at Z = -0.5, 0.5.

The initial condition is at a conduction state :U = V = W = 0, T = −X where −0.5 ≤ X ≤ 0.5.
The average Nusselt number was computed on the hotwall from the following definition:
0.5

∫ ∫
Nu =
∫ ∫

0.5

− 0.5 − 0.5
0.5
0.5− 0.5 − 0.5(∂T / ∂X )cvX = −0.5 dYdZ(∂T / ∂X )cd
X = −0.5

(10)

dYdZ

The above equations were approximated with thefinite difference equations. The HSMAC method wasused to iterate mutually the pressure and velocity field
and the number of meshes was chosen to be 40x40x40.
The grid dependency test for the same model equations
was carried out in our previous work [19] and has
showed that the effect of grid size was slight comparing
with the mesh 30x30x30.

Fig. 5

Schematic view of the cross-section of thesystem with 5 Tesla magnet for the comparisonof Nu numbers

terms of those at the average temperature. The gravity
force is always working downwards but net flow
becomes upward for hot fluid and downwards for cold
fluid in terms of the average temperature fluid. It is thesame for the magnetic force. Thus, for this configuration,
the magnetic buoyancy force Fm acts in the opposite
direction against the gravitational buoyancy force Fg.
Near the heated wall (on the right-hand side), Fm acts
downward while near the cooled wall (left-hand side)
upward. The gravitational buoyancy force acts in the
opposite direction as seen in Fig. 5.
Table 2 gathers experimental conditions and theircorresponding non-dimensional parameters for heattransfer experiment with 5 Tesla super-conducting
magnet.
Fig. 6 shows the comparison between experimental
and numerical results. The Nusselt numbers are plottedversus the maximum induction. As shown in Fig. 5, themagnetic and gravitational buoyancy forces act in the

6 Results
The numerical computations were carried out for thedimensionless conditions of the experiments. At firstheat transfer rates are compared with those obtained for 5
Tesla superconducting magnet and then the visualization
experiment with the 10 Tesla magnet .

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Table 2

b0
[T]
0
1
2
3
4
5

Experimental data and its non-dimensional
values for heat transfer experiment with
5 Tesla super-conducting magnet

γ [−]

Temp.
hot

Temp.
cold

Temp.
difference

0
0.59
2.34
5.27
9.37
14.64

22.54
23.56
22.13
21.60
21.31
21.13

17.67
17.79
17.71
17.75
17.76
17.74

4.87
5.77
4.42
3.85
3.55
3.39

Rayleigh
number
1.38×105
1.62×105
1.24×105
1.08×105
0.99×105
0.95×105© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No. 4 (2006)

Table 3 Experimental data and its non-dimensionalparameters for visualization experiment with
0 Tesla super-conducting magnet

12

Nusselt number [-]experimental results
10

b0
[T]
0
5
6
8

8
6
computational results
4

γ [−]

Temp.
hot

Temp.
cold

Temp.
difference

0
14.64
21.09
37.49

20.81
21.14
21.21
20.54

17.90
17.96
17.96
17.96

2.91
3.18
3.25
2.58

Rayleigh
number
8.16×104
8.92×104
9.11×104
7.23×104

2
0

1

2

3

4

5

Magnetic induction in the center of the solenoid [T]Fig. 6Comparison of the Nusselt numbers at
Ra0 = 1.38×105. ( See Table 2 for details.)

opposite direction and when the strength of magnetic
field is weak, they are compensating each other. The
Nusselt number have its minimum around bmax = 1 Tesla,both in the numerical and experimental analyses. For
larger bmax, the Nusselt number increases with increasing
magnetic induction. Therefore, it can be said that whenthe magnetic field strength increases, the magnetic
buoyancy force becomes dominant and the convection isenhanced.
6.2 Visualization
In the visualization part, the 10 Tesla superconducting
magnet was used. The enclosure could be then placed
just outside of the bore of the magnet to make thevisualization procedure possible. Fig. 7 shows thecorresponding cross-section view of the system for thenumerical computations.
Table 3 gathers experimental conditions and theircorresponding non-dimensional parameters for visualiza-tion experiment with 10 Tesla super-conducting magnet.
Fig. 8 shows numerical results in the form of resultant
force vectors and longtime streak lines for four cases
which correspond to the experiment at 0, 5, 6 and 8 Teslaof the magnetic induction respectively. Fig. 9 shows thecorresponding vertical cross-section of the temperaturefield. In Fig. 9, the left-hand side column shows theexperimental results with the isolines obtained with PIT(Particle Image Thermometry) method [21] and theright-hand side column shows temperature distributions
computed numerically.
At 0 Tesla the usual gravitational convection occurred.
The force vectors near the hot part of the enclosure direct
upward, while the force vectors near the cold part of the
cube direct downward. When the magnetic field is
imposed, the magnetic buoyancy force is expected tocancel or overcome the gravity effect.
When the magnetic induction is 6 Tesla, the numericalcomputation shows two rolls in a vertical cross-section(Fig. 8 c). At this case, the balance between gravitational
buoyancy force and magnetic buoyancy force is attained.
Fig. 9 shows corresponding isotherms. In the experiment
the upper and lower rolls were not in the same size sinceeven small difference in the magnetic strength affected
the final profile of steady state. At 8 Tesla, the magnetic
force becomes dominant, and the convection is reversedin comparison to that at 0 Tesla.
Comparison between experimental and numericalisotherms shows good agreement and proves thecorrectness of the applied numerical model.
7 Conclusions

Fig. 7 Schematic view of the cross-section of thesystem with the 10 Tesla magnet for thecomparison of temperature distributions.

The heat transfer rates of magnetic convections in a
cubic enclosure placed below the magnet coil were
measured with 5 Tesla magnet for 80% massconcentration of aqueous glycerol solution with 0.8
mol/kg of gadolinium nitrate hexahydrate. The
isothermal profiles were also pictured with liquid crystal
slurry dispersed in the fluid. The visualization
experiments were carried out with 10 Tesla
superconducting magnet. These results were employed tocompare with the numerical computations for the almost
same conditions. Both the average Nusselt numbers andthe isothermal profiles represented good correspondence
to each other to assure the model equations and thenumerical approach.

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Thermal Science & Engineering Vol. 14 No. 4 (2006)

ForcesTemperature distributions:computations
experimentLongtime streak-lines(a) 0 T(a) 0 T(b) 5 T(b) 5 T(c) 6 T(c) 6 T(d) 8 T

Fig. 8(d) 8 T

Fig. 9

Numerical results computed at four different
strengths of magnetic induction. Left column:
force vectors, right: longtime streak lines

Comparison of the isotherms obtained in theexperiment and the numerical computations
at four different strengths of magnetic induction

Acknowledgments
This research was partially supported by European
Commission (project Dev-CPPS, No. FP6-002968).
References[1] Wakayama, N. I., “Magnetic promotion of
combustion in diffusion flames”, Combustion and
Flame, 93 (1993), 207-214.
[2] Wakayama, M., and Wakayama, N. I., “Magnetic
acceleration of inhaled and exhaled flows inbreathing”, Japanese J. Applied Physics, 39 (2000),
262-264.

[3] Wakayama, N. I., “Effect of decreasing magnetic
field on the flow of nitrogen gas”, Chem. Phys.
Letters, 185 (1991), 449-451.
[4] Hirota, N., Ikezoe, Y., Uetake, H., Kaihatsu, T.,
Takayama, T., and Kitazawa, K., “MagnetoArchimedes levitation and its application”, RIKEN
Rev., 44 (2002), 159-161.
[5] Ikezoe, Y., Hirota, N., Sakihama, T., Mogi, K.,
Uetake, H., Homma, T., Nakagawa, J., Sugawara,
H., and Kitazawa, K., “Acceleration effect of therate of dissolution of oxygen in a magnetic field”, J.
of Jap. Inst. of Applied Magnetics, 22 (1998),
821-824.

- 113 -

© 2006 The Heat Transfer Society of Japan

Thermal Science & Engineering Vol.14 No. 4 (2006)[6] Bai, B., Yabe, A., Qi, J.W., Wakayama N.I.,
Quantitative analysis of air convection caused bymagnetic-fluid coupling, AIAA Journal, 37:12(1999), 1538-1543.
[7] Huang, J., Gray, D.D., and Edwards, B.F.,“Thermoconvective instability of paramagnetic
fluids in a nonuniform magnetic field”, Phys. Rev. E,
57 (1998), 5564-5571.
[8] Tagawa, T., Shigemitsu, R., and Ozoe, H.,“Magnetizing force modeled and numericallysolved for natural convection of air in a cubic
enclosure: effect of the direction of the magnetic
field”, Int.J.Heat Mass Transfer, 45 (2002),
267-277.
[9] Kaneda, M., Tagawa, T., and Ozoe, H., “Convection
induced by cusp-shaped magnetic field for air in a
cube heated from above and cooled from below”, J.
Heat Transfer, 124 (2002), 17-25.
[10] Ozoe, H., “Magnetic Convection”, Imperial College
Press, ISBN 1-86094-578-3 (2005).
[11] Krakov, M.S., Nikiforov, I.V., “To the influence of
uniform magnetic field on thermomagnetic
convection in square cavity, Journal of Magnetismand Magetic Materials, 252(2002)209-211.
[12] Krakov, M.S., Nikiforov, I.V., Reks, A.G.,“Influence of the uniform magnetic field on naturalconvection in cubic enclosure: experiment andnumerical simulation”, Journal of Magnetism andMagnetic Materials, 289 (2005), 272-274.
[13] Davis, S.H., “Convection in a box: linear theory”, J.
Fluid Mechanics, 30 (1967), 465-478.
[14] Ozoe, H., Sayama, H., Churchill, S.W., “Naturalconvection in an inclined square channel”, Int. J.
Heat Mass Transfer, 17 (1974), 401-406.

[15] Ozoe, H., Yamamoto, K., Sayama, H., Churchill,
S.W., “Natural circulation in an inclined rectangularchannel heated on one side and cooled on the
opposing side,” Int. J. Heat Mass Transfer, 17(1974), 1209-1217.
[16] Lide, D.R., “Handbook of Chemistry and Physics”,
Chemical Rubber Company Press, Baca Raton, FL,
2001-2002.
[17] Ozoe, H., Churchill, S.W., “Hydrodynamic stability
and natural convection in Newtonian andnon-Newtonian fluids heated from below”, AIChE
Symposium Series Heat Transfer, 69 (1973),
126-133.
[18] Bednarz, T., Tagawa, T., Kaneda, M., Ozoe, H.,
Szmyd, J.S., “Magnetic and gravitational
convection of air with a coil inclined around the Xaxis”, Numerical Heat Transfer, 46 (2004), 99-113.
[19] Bednarz, T., Tagawa, T., Kaneda, M., Ozoe, H.,
Szmyd, J.S., “Convection of air in a cubic enclosure
with an electric coil inclined in generalorientations”, Fluid Dynamics Research, 36 (2005),
91-106.
[20] Bednarz, T., Tagawa, T., Kaneda, M., Ozoe, H.,
Szmyd, J.S., “Numerical study of joint
magnetisation and gravitational convection of air ina cubic enclosure with an inclined electric coil,”
Progress in Computational Fluid Dynamics, 5(2005), No. 3/4/5.
[21] Kowalewski, T., “Experimental methods forquantitative analysis of thermally driven flow,
experimental benchmarks for code validations”, 5lectures from Advanced School on Phase Changewith Convection – modeling and validation:
Coordinated by Prof. Kowalewski and Prof. Gobin,
CISM, Udine, Italy, 2002.

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