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Journal of Magnetism and Magnetic Materials 321 (2009) 3665–3670

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/jmmm

Experimental and numerical investigation of natural convection of magneticfluids in a cubic cavity

Hiroshi Yamaguchi a, Xiao-Dong Niu a,�, Xin-Rong Zhang b, Keisuke Yoshikawa a

a Energy Conversion Research Center, Department of Mechanical Engineering, Doshisha University, Kyoto 630-0321, Japanb Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing 100871, PR China

a r t i c l e i n f o

Article history:

Received 24 February 2009

Received in revised form

26 June 2009Available online 18 July 2009

Keywords:

Magnetic fluid

Natural convection

Lattice Boltzmann method

53/$ - see front matter & 2009 Elsevier B.V. A

016/j.jmmm.2009.07.013

esponding author.

ail address: xniu@mail.doshisha.ac.jp (X.-D. N

a b s t r a c t

In this article, natural convections of a magnetic fluid in a cubic cavity under a uniform magnetic field

are investigated experimentally and numerically. Results obtained from experiments and numerical

simulations reveal that the magnetic field and magnetization are influenced by temperature. There exist

relative larger magnetization and magnetic forces in the regions near the upper wall and center inside

the cavity than in the region near the bottom and side walls. A weak flow roll occurs inside cavity under

the magnetic force, and it brings the low temperature fluid downward in the center region, and streams

the high temperature fluid upward along the regions near the sidewalls. With the magnetic field

imposed, the heat transfer inside the cavity is enhanced significantly compared to that without the

magnetic field, and increasing the strength of the magnetic field the heat transfer is increased further.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Flow modes as well as heat transfer characteristics associatedwith natural convections of the magnetic fluids have been asubject of interest to engineers and scientists for many years[1–8]. Yamaguchi et al. [3,4] studied the heat transfer character-istics and flow behaviors of the natural convections of themagnetic fluid in a rectangular cavity under an imposed evenvertical magnetic field both experimentally and numerically. Theirstudy disclosed that the vertically imposed magnetic fielddestabilized the flow significantly, and at a critical state the flowmodes become substantially different from those without themagnetic field. The natural convections of the magnetic fluid in asquare Hele-Shaw cell were experimentally investigated with heattransfer measurements and liquid crystal thermography by Wenand Shu [5,6]. The results confirmed the findings of Yamaguchi etal. [3,4], and showed that a pair of symmetric counter-rotatingvortices exists in the first instability mode. In numericallystudying the works of Yamaguchi et al. [3,4], Krakov and Nikiforov[7] discovered that the angle between the directions of tempera-ture gradient and the magnetic field influences the convectionstructure and the intensity of heat flux. They also numericallystudied the influence of porous media and uniform magnetic fieldon thermal convection in magnetic fluids [8]. It was shown that, inporous square cavity, the competition between gravity convection

ll rights reserved.

iu).

and thermomagnetic convection can lead to a complicateddependence of the heat flux through the cavity on the magneticfield; increasing of magnetic field could both enhance and depressthe heat transfer.

In this article, three-dimensional natural convection flows ofthe magnetic fluid in a cubic cavity under an imposed uniformmagnetic field are further studied experimentally and numeri-cally. Particularly the effects of the magnetic field on the heattransfer characteristics and flow and temperature behaviors areinvestigated. In the experiment, a temperature sensitive magneticfluid is filled in the cubic cavity container, which is heated fromthe bottom wall and cooled at the upper wall while the side wallsare insulated. Heating is done by heat conduction from siliconcode heater from outside of the container. The vertical uniformmagnetic fields are applied at the container. The heat transfercharacteristics are obtained by measuring the temperaturedifference between the upper and bottom walls.

On the other hand, a simple lattice Boltzmann method (LBM)[9] is used as a numerical tool in the present study to investigatethe natural convection mechanism of the magnetic fluid in thecubic cavity. The LBM, as a mesoscopic particle method, has beenbecoming increasingly popular in many academic and engineeringfields in recent years [10]. The traditional CFD tools often needcomplicated time iteration and lengthy discretization procedureto solve a set of governing differential equations. For example, thewell established control volume method (CVM) [11] usually needsto divide the physical domain into a number of control cells wherethe variable of interest is located at the centroid of the cell, andthen to integrate the differential form of governing equations over

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H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3665–36703666

each cell with assumed interpolation profiles of the variation ofthe concerned variable between the cell centroids. Besides havingthe usually same second-order numerical accuracy as that of theCVM, the LBM has advantages of a set of simple explicit time-iteration algebraic lattice Boltzmann equations, easy coding,intrinsic parallelization, and clear physics compared to the CVM.A number of LBMs have been used successfully to study themagnetic fluid flows [12–18]. The present LBM [9] is formulatedbased on a derived scalar magnetic potential equation at the first-order time accuracy. By defining an effective velocity, which is afunction of the temperature gradient, a lattice Boltzmannequation is straightforwardly constructed in a similar fashion ofthermal lattice Boltzmann scheme. To ensure that the LBMsolution remains close to the derived scalar potential equation, apreconditioning parameter is introduced in the magnetic poten-tial equation. The present model has been verified in our earlierworks by comparing the experimental studies [9].

The rest of the paper is organized as follows. In Section 2,experimental apparatus and measurements are briefly introduced.A short description of the lattice Boltzmann method [9], used inthe present study is given in Section 3. Section 4 is devoted toresults and discussion of the present study. A conclusion is givenin Section 5.

2. Experimental apparatus

An experiment is conducted to obtain the heat transfercharacteristics of the magnetic fluid in the cubic cavity container.

Fig. 1. Experimental apparatus (left

Table 1Cavity dimension, fluid properties used in present study.

Scale length of cavity L (mm) 5

Density r0 (kg/m3) 1397�103

Viscosity Z (Pa s) 1.680�10�3

Thermal conductivity l (W/(m K) 1.750�10�1

Saturation magnetization Ms (A/m) 4.10�105

Permeability of vacuum m0 (H/m) 4p�10�7

Fig. 1 depicts a schematic diagram of the experimental apparatusused in the present study. The uniform magnetic field is imposedto the test section B including a cubic cavity container R by theelectromagnet A vertically. The bottom of the container R isheated through the copper block by a silicon code heater whilethe upper wall of the cell is cooled through a copper block whoseupper surface is opened to the ambient air. The heat transfer rateto and from the container is obtained by measuring thetemperature gradient in the copper blocks. A representativetemperature T0 inside of the container R is also measured. Thetest magnetic fluid used in the experiment is made of Mn–Znferrite in alkyl-naphthalene base fluid, whose magnetization M ata representative temperature T0 and magnetic field H0 can beapproximated as a linear magnet [19]. Illustrations of otherdevices used in the present experiment can be referred in Fig. 1,and the experimental conditions of the test section and testmagnetic fluid properties are given in Table 1.

3. Numerical method

In the theory of the magnetic fluids, as the flow under theinfluence of the magnetic field, they undergo magnetic force. Themagnetic hydrodynamics can be described by the followinggoverning equations [9,20,21]:

@trþrr0u ¼ 0; ð1Þ

) and test section (right).

Specific heat Cp (J/kg K) 4.453�101

Expansion coefficient b (1/K) 6.90�10�4

Curie temperature Tc (K) 477.35

Reference temperature T0 (K) 298.15

Magnetization rate w0 0.2650

Gravitational acceleration g (m/s2) 9.8

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x

y

u=0, T=Tu

u=0xT=0

u=0 yT=0

u=0 yT=0 u=0xT=0

u=0, T=TbH=H0

H=H0

z

g

Fig. 2. Sketch of the thermal magnetic nature convection flow condition in the

cubic cavity.

H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3665–3670 3667

@tr0uþrr0uu ¼ �rp

þ Zr2uþw0m0H2

2ðTc � T0ÞrT � r0bðT � T0Þg; ð2Þ

@tT þ u � rT ¼lðTc � T0Þ

½rCPðTc � T0Þ þ m0w0H2Þ�r

2T

�m0w0HTðDH=DtÞ

½rCPðTc � T0Þ þ m0w0H2�; ð3Þ

@tfþ guT � rf ¼ g 1þw0ðT � T0Þ

Tc � T0

� �r

2f: ð4Þ

In the above equations, r0 is the constant density, u the velocity,uT ¼ (w0/Tc�T0)rT the nominal thermal velocity, p the pressure,and T the temperature; Z the dynamical viscosity, m0 the magneticpermeability of vacuum, b the expansion coefficient under theBoussinesq approximation; H the modulus of the magneticintensity H and f the scalar potential and rf ¼ H; l thecoefficient of thermal conductivity of the fluid, Cp the specificheat at constant pressure, g the preconditioning parameter usedto ensure that the following LBM solution remains close to thederived scalar potential equation.

In terms of the lattice Boltzmann theory, Eqs. (1)–(4) can besolved by the following lattice Boltzmann scheme by employingthree distribution functions for the velocity, thermal and magneticfield [9]:

faðrþ nadt ; t þ dtÞ � faðr; tÞ ¼ �faðr; tÞ � f eq

a ðr; tÞ

tf

þwaðtf � 0:5Þdt

tf c2s

F � ðna � uÞ; ð5Þ

gaðrþ nadt ; t þ dtÞ � gaðr; tÞ ¼ �gaðr; tÞ � geq

a ðr; tÞ

tgþwaSdt ; ð6Þ

haðrþ nadt ; t þ dtÞ � haðr; tÞ ¼ �haðr; tÞ � heq

a ðr; tÞ

th; ð7Þ

with respective equilibrium distribution functions

f eqa ðr; tÞ ¼ wa rþ r0

na � uc2

s

þ1

2c2s

ðna � uÞ2

c2s

� u2Þ�g;

"(ð8Þ

geqa ðr; tÞ ¼ waT 1þ

na � uc2

s

þ1

2c2s

ðna � uÞ2

c2s

� u2Þg;

(ð9Þ

heqa ðr; tÞ ¼ waf 1þ

na � guT

c2s

þ1

2c2s

ðna � guT Þ2

c2s

� ðguT Þ2Þg;

(ð10Þ

and the force F and source S

F ¼w0m0H2

2ðTc � T0ÞrT � r0bðT � T0Þg; ð11Þ

S ¼ �m0w0HTðDH=DtÞ

½rCPðTc � T0Þ þ m0w0H2�; ð12Þ

where r ¼ r(x,y,z) is the spatial vector. The relaxation parametersin Eqs. (5)–(7) are given by tf ¼ (Z/r0cs

2dt)+0.5,

tg ¼D

ðDþ 2Þ

lðTc � T0Þ

½rCPðTc � T0Þ þ m0w0H2Þ�c2s dtþ 0:5:

and

th ¼g

c2s dt

1þw0ðT � T0Þ

Tc � T0

� �þ 0:5;

respectively. The sound speed cs, the weight coefficient wa and thediscrete velocity na used in the above LBM scheme can be referredto the discrete velocity model of the D3Q19 for three-dimension(3D) [22]. The density, velocity, and temperature are calculated by

r ¼Xa

fa

ru ¼Xa

faxa þ 0:5dtF

T ¼Xa

ga

f ¼Xa

ha: ð13Þ

A number of previous researches [9,22] have shown, by theChapman–Enskog analysis, the lattice Boltzmann Eqs. (5)–(7) areable to recover the macroscopic Eqs. (1)–(4), respectively.

The LBM simulations performed in this study are in line withthe experimental conditions listed in Table 1. All the simulationsare based on the uniform grid of 41�41�41. The grid-indepen-dent study of the simulations is neglected in this article and canbe referred in our earlier work [9] if interested. The parametersused in all simulations are the Rayleigh number Ra ¼ (r0gbDTL3)/kZ and the magnetic Rayleigh number Ram ¼ (m0H0MsL

2)/kZ,which correspond the value of the imposed magnetic field H0

and temperature difference DT ¼ Tb�Tu of the upper and bottomwalls, and k ¼ l/r0CP is the thermal diffusivity.

The boundary conditions are referred in Fig. 2. For themagnetic field, the magnetic potential boundary conditions areset by the magnetic intensity, which are described as follows:

@f@xjx¼0;L ¼ 0;

@f@yjy¼0;L ¼ 0;

@f@zjz¼0;L ¼

H0

ð1þ wÞ ; ð14Þ

where w ¼ w0(Tc�T)/(Tc�T0). For the distribution functions of fa, gaand ha on the boundaries, the non-equilibrium bounce backboundary conditions [9] are employed when the macroscopicvelocity, temperature and scalar potential on the walls arecalculated, and they are given as follows:

fa ðr; tÞ � f eqa ðr; tÞ ¼ faðr; tÞ � f eq

a ðr; tÞ; ð15Þ

ga ðr; tÞ � geqa ðr; tÞ ¼ �½gaðr; tÞ � geq

a ðr; tÞ�; ð16Þ

ha ðr; tÞ � heqa ðr; tÞ ¼ �½haðr; tÞ � heq

a ðr; tÞ�; ð17Þ

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3.5Exp. Ram = 0

H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3665–36703668

respectively. a in Eqs. (14)–(16) denotes directions of theunknown distribution function, and a ¼ �a.

XY

Z

Fig. 4. The calculated typical magnetic intensity (H) field (color plotted against H)

in y–z plane of x ¼ 0.5 in the cubic cavity for Ra ¼ 5000 and Ram ¼ 1.00�108.

1

1.5

2

2.5

3

0.0 2.0 4.0 6.0 8.0 10.0

Exp. Ram=1.00e8Exp. Ram=1.25e8Cal. Ram=0Cal. Ram=1.00e8Cal. Ram=1.25e8

Ra

Nu

(×103)

Fig. 3. Dependence of the average Nusselt numbers of the Rayleigh numbers at

three magnetic Rayleigh numbers of 0, 1.00�108 and 1.25�108.

4. Results and discussion

Fig. 3 displays the dependence of the average Nusselt numbers(Nu ¼

R R(�qT/qZ)z ¼ 0,Ldxdy) on the Rayleigh numbers at three

magnetic Rayleigh numbers of 0, 1.00�108 and 1.25�108.Experimental and numerical results are all plotted in this figure.Seen from Fig. 3, for Ram ¼ 0, corresponding no imposed magneticfield, it is found that the natural convection occurs above a criticalRayleigh number of RacE3400. This observation is agreed withthe earlier studies [23,24]. Moreover, the calculated results are inwell agreement with the experimental data. The quantitativedifference between the experimental data and calculated resultsexists, and the difference becomes relatively larger when thestrength of the magnetic field is increased. This would be due tothe reasons that, at higher Rayleigh number, the non-linearity forthe magnetization may appear, and the effect of the internaldegree of particle rotation and chaining of particles [3,4] becomesstrong. However, the general trend of the results depicted in Fig. 3shows close similarity, indicating that the heat transfer rateincreases when the magnetic field strength increases. Thisobservation can be explained more clearly by the followingnumerical results, which depict details of the magnetic and flowfield.

Fig. 4 shows the calculated typical magnetic intensity field iny–z plane of x ¼ 0.5 in the cubic cavity for Ra ¼ 5000 andRam ¼ 1.00�108. The magnetic intensity H can be evaluatedfrom definition of H ¼ rf by using the second order central finitedifferencing. As shown in Fig. 4, the magnetic intensity vectors areparallel to each other and are aligned with the positive z-coordinate. With the boundary condition of Eq. (13), theamplitude H is varied slightly from a large value at the bottomwall to a small value at the upper wall. However, the differencebetween the maximum (0.209) and the minimum (0.193) of H isquite small and less than 3%. As the magnetization M iseverywhere parallel to the magnetic field H in the flow field, thecontours of the modulus of the magnetization are only plotted inx–y plane of z ¼ 0.5 and y–z plane of x ¼ 0.5 in the cubic cavity forflows of Ra ¼ 5000 at Ram ¼ 1.00�108 and Ram ¼ 1.25�108,respectively (Fig. 5). The modulus of M is calculated directly byM ¼ w0H(Tc�T)/(Tc�T0). As shown in Fig. 5, due to the temperaturedependence of the magnetization, the modulus of themagnetization M is shown to be varied in the cavity. Largemagnetization is found in the regions near the upper wall andcenter of the cavity, and low magnetization neighbored thebottom and side isolated walls. These findings can be attributed tothe results of the non-uniform distribution of the temperaturecaused by the natural convection of the magnetic fluid insidecubic cavity (see Fig. 7), where the temperature in the regionsnear the upper wall and center of the cavity is lower than that inthe regions near the bottom and side isolated walls. Since themagnetic force, Fmagnetic ¼ MrH, is dependent of themagnetization, it is clearly learnt that the magnetic force islarger in the region near the upper wall and center of the cavitythan in the region near the bottom and side walls.

With the knowledge of the magnetic force, the velocity vectorsof Ra ¼ 5000 inside the cavity at Ram ¼ 1.00�108 (left) and1.25�108 (right) are plotted in Fig. 6. As shown in Fig. 6, a weakflow roll occurs in side cavity under the magnetic forces. The mainstream goes downward in the center region with relative largevelocities and the side flow streams upward with relative smallvelocities. Figs. 7(a) and (b) show the mid-plane flow patterns andtemperature fields of Ra ¼ 5000 inside the cavity at

Ram ¼ 1.00�108 and 1.25�108, respectively. Corresponding tothe observations in Figs. 5 and 6, the low temperature fluid isbrought downward in the center region with relative rapid speeds,and the high temperature fluid is brought upward from regionsnear the sidewalls with relative slow speeds. Comparing Figs. 7(a)with (b), it can be concluded that the larger the magnetic Rayleighnumber, the faster the heated transportation in the cavity.

5. Conclusions

In this article, three-dimensional natural convections of themagnetic fluid in a cubic cavity under the uniform magnetic fieldare investigated experimentally and numerically. Results obtainedfrom experiments and numerical simulations reveal that themagnetic field and the magnetization are influenced by tempera-ture. Due to the non-uniform distribution of the temperaturecaused by the natural convection of the magnetic flow inside thecavity, the magnetization and magnetic force in the regions nearthe upper wall and center inside the cavity are larger than those inthe region near the side and bottom walls. A weak flow roll occursinside the cavity under the magnetic forces, and it brings the low

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XY

Z

0.2090.2070.2050.2030.2010.1990.1970.1950.193

XY

Z

0.2090.2070.2050.2030.2010.1990.1970.1950.193

XY

Z

0.2090.2070.2050.2030.2010.1990.1970.1950.193

XY

Z

0.2090.2070.2050.2030.2010.1990.1970.1950.193

Fig. 5. The contours of modulus of the magnetization M in x–y plane of z ¼ 0.5 and y–z plane of x ¼ 0.5 in the cubic cavity for flows of Ra ¼ 5000 at (a) Ram ¼ 1.00�108 and

(b) Ram ¼ 1.25�108, respectively.

XY

Z

1.0000

XY

Z

1.0000Fig. 6. The velocity vectors (color plotted against temperature field) of Ra ¼ 5000 inside the cavity at Ram ¼ 1.00�108 (left) and 1.25�108 (right), respectively.

H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3665–3670 3669

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Fig. 7. The temperature contours and velocity vectors of flows of Ra ¼ 5000 inside the cavity at two mid-planes at Ram ¼ 1.00�108 and 1.25�108, respectively.

H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3665–36703670

temperature fluid downward in the center region, and streams thehigh temperature fluid upward from regions near the sidewalls.With the magnetic field imposed, the heat transfer inside thecavity is enhanced significantly compared to that without themagnetic field. Moreover, increasing the strength of the magneticfield, the heat transfer is increased further.

Acknowledgement

This work was supported by a grant-in-aid for ScientificResearch (C) from the Ministry of Education, Culture, Sports,Science and Technology, Japan.

References

[1] B.M. Berkovsky, Magnetic Fluids Engineering Applications, Oxford UniversityPress, New York, 1993.

[2] B.A. Finlayson, J. Fluid Mech. 40 (1970) 753.[3] H. Yamaguchi, I. Kobori, Y. Uehata, K. Shimada, J. Magn. Magn. Mater. 201

(1999) 264.[4] H. Yamaguchi, I. Kobori, Y. Uehata, J. Thermophys. Heat Transfer 13 (1999)

501.

[5] C.Y. Wen, W.-P. Shu, J. Magn. Magn. Mater. 252C (2002) 206.[6] C.Y. Wen, W.-P. Shu, J. Magn. Magn. Mater. 289 (2005) 299.[7] M.S. Krakov, I.V. Nikiforov, J. Magn. Magn. Mater. 252 (2002) 209.[8] M.S. Krakov, I.V. Nikiforov, A.G. Reks, J. Magn. Magn. Mater. 289 (2005) 272.[9] X.D. Niu, H. Yamaguch, K. Yoshikawa, A Lattice Boltzmann model for

simulating Thermo-sensitive Ferrofluids, Phys. Rev. E 79 (4) (2009) 046713-1–046713-6.

[10] S. Chen, G.D. Doolen, Annu. Rev. Fluid Mech. 30 (1998) 329–364.[11] H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid

Dynamics: The Finite Control Volume Method, Longman Scientific &Technical, Essex, UK, 1995.

[12] S. Chen, H. Chen, D.O. Martinez, W.H. Matthaeus, Phys. Rev. Lett. 67 (1991)3776–3779.

[13] S. Succi, M. Vergassola, R. Benzi, Phys. Rev. A 34 (1991) 4521–4524.[14] D.O. Martinez, S. Chen, W.H. Matthaeus, Phys. Plasmas 1 (1994) 1850–1867.[15] M. Hirabayashi, Y. Chen, H. Ohashi, Phys. Rev. Lett. 87 (2001) 178301.[16] V. Sofonea, W.G. Frueh, Euro. Phys. J. B 20 (2001) 141–149.[17] P.J. Dellar, J. Comput. Phys. 179 (2002) 95–126.[18] P.J. Dellar, J. of Stat. Phys. 121 (1/2) (2005) 105–118.[19] B.A. Finlayson, J. Fluid Mech. 40 (part 4) (1970) 753.[20] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, London,

1985.[21] H. Yamaguchi, Engineering Fluid Mechanics, Springer, Netherlands, 2008.[22] D.A. Wolf-Gladrow, Lattice-Gas Cellula Automata and Lattice Boltzmann

Models, Springer, Berlin, 2000.[23] W.L. Heitz, J.W. Westwater, ASME J. Heat Mass Transfer C-93 (1971) 188–196.[24] I. Catton, Int. J. Heat Mass Transfer 15 (1972) 665–672.