A 3D numerical simulation of mixed convection of a magnetic nanofluid in the presence of non-uniform magnetic field in a vertical tube using two phase mixture model

Journal of Magnetism and Magnetic Materials 323 (2011) 1963–1972

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A 3D numerical simulation of mixed convection of a magnetic nanofluid in the presence of non-uniform magnetic field in a vertical tube using two phase mixture model Habib Aminfar a, Mousa Mohammadpourfard b, Yousef Narmani Kahnamouei a,n a b

Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran Department of Mechanical Engineering, Azarbaijan University of Tarbiat Moallem, Tabriz, P.O. Box 53751-71379, Iran

a r t i c l e i n f o

abstract

Article history: Received 10 December 2010 Received in revised form 14 February 2011 Available online 15 March 2011

In this paper, results of applying a non-uniform magnetic field on a ferrofluid (kerosene and 4 vol% Fe3O4 ) flow in a vertical tube have been reported. The hydrodynamics and thermal behavior of the flow are investigated numerically using the two phase mixture model and the control volume technique. Two positive and negative magnetic field gradients have been examined. Based on the obtained results the Nusselt number can be controlled externally using the magnetic field with different intensity and gradients. It is concluded that the magnetic field with negative gradient acts similar to Buoyancy force and augments the Nusselt number, while the magnetic field with positive gradient decreases it. Also with the negative gradient of the magnetic field, pumping power increases and vice versa for the positive gradient case. & 2011 Elsevier B.V. All rights reserved.

Keywords: Ferrofluid Mixed convection Non-uniform magnetic field Buoyancy force Mixture model

1. Introduction Magnetic nanofluid, also called ferrofluid, is a magnetic colloidal suspension consisting of carrier liquid and magnetic nanoparticles with a size range of 5–15 nm in diameter coated with a surfactant layer. The most often used magnetic material is single domain particles of magnetite, iron or cobalt; and the carrier liquids such as water or kerosene. The advantage of the ferrofluids is that the fluid flow and heat transfer may be controlled by an external magnetic field, which makes it applicable in various fields such as electronic packing, mechanical engineering, thermal engineering, aerospace and bioengineering [1–4]. Many investigations were carried out numerically and experimentally in the field of thermomagnetic convection of the ferrofluids in different geometries in the presence of an external magnetic field [5–16]. Results of these investigations show that the thermal behaviors of the magnetic fluids inside an enclosure are dominated by both the intensity of the external magnetic field and temperature gradient. Wrobel et al. [17] carried out an experimental and numerical analysis of a thermomagnetic convective flow of paramagnetic fluid in an annular enclosure with a round rod core and a cylindrical outer wall. Their results show that magnetizing force affects the

n

Corresponding author. Tel.:/fax: þ98 412 4327566. E-mail addresses: [email protected] (H. Aminfar), [email protected] (M. Mohammadpourfard), [email protected] (Y. Narmani Kahnamouei). 0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.02.039

heat transfer rate and that a strong magnetic field can control the magnetic convection of paramagnetic fluid. Sambamurthy et al. [18] presented a numerical analysis of a horizontal circular annulus with an inner heat source of square and circular shapes. They described the flow pattern with double or quadruple vortices. Lajvardi et al. [19] carried out an experimental work on the convective heat transfer of a ferrofluid flowing through a heated copper tube in the laminar regime in the presence of magnetic field and observed a significant enhancement on the heat transfer of ferrofluid by applying various orders of magnetic field. Kamiyama and Ishimoto [20], and Liu et al. [21], performed experiments to characterize boiling two-phase flow heat transfer and pool boiling, respectively, using water-based magnetic fluids. Their results show that the boiling heat transfer characteristics of the magnetic fluid are strongly influenced by the external magnetic field. Aihara et al. [22] simulated a two-dimensional flow of a magnetic fluid with 50% mass concentration of Mn–Zn ferrite particles in a horizontal circular tube using a single phase model and explained the controllability of convective heat transfer in the presence of non-uniform magnetic field. Ganguly et al. [23] simulated a two-dimensional pressure-driven flow of a magnetic fluid in a channel to investigate the influence of magnetic field created by a line-source dipole on the convective heat transfer. Xuan et al. [24] have developed the mesoscopic model to simulate the magnetic fluid flowing through a microchannel in the presence of a magnetic field gradient using lattice-Boltzmann method. Also Li and Xuan [25] carried out an experiment to investigate the heat transfer characteristics of magnetic fluid flow around a fine wire under the influence

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H. Aminfar et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 1963–1972

Nu Gr M Ms

Nomenclature

r cp

m k

b

! v T P ! v pf ! v dr

ap dp l D r

y z qw Re

density (kg/m3) specific heat (J/kg K) dynamic viscosity (kg/m s) conductivity (w/m K) thermal expansion coefficient (1/K) velocity (m/s) temperature (K) pressure (Pa) slip velocity vector (m/s) drift velocity vector (m/s) particle volume fraction magnetic particle diameter (m) vertical tube length (m) tube diameter (m) radial direction circumferential direction axial direction wall heat flux (W/m2) Reynolds number (rmvmD/mm)

m0 G g mp

m0 L

x

mB kB

Nusselt number (qwD/km(Tw  Tb)) Grashof number ðg bm qw r2m D4 =km m2m Þ Magnetization (A/m) saturation magnetization (A/m) magnetic field vector (A/m) magnetic field gradient (A/m2) gravitational acceleration (m/s2) particle magnetic moment (Am2) magnetic permeability in vacuum (4p  10  7 T m/A) Langevin function Langevin parameter Bohr magneton (9.27  10  24 Am2) Boltzmann constant (1.3806503  10  23 J/K)

Subscripts f p m 0

pertaining pertaining pertaining Pertaining

to to to to

base fluid magnetic particles mixture inlet conditions

analyzed. In the next section the basic theoretical formulation including the governing equations, boundary conditions and numerical method is presented. Results are discussed in Section 3. Finally, Section 4 contains some conclusions.

of uniform and non-uniform external magnetic fields. However, a full understanding of the magnetic field effects on the hydrothermal characteristics of the ferrofluid flow needs more researches. Above mentioned studies for the thermal behaviors were mainly focused on free convection of ferrofluids in enclosures. Literature concerning the hydrodynamic and heat transfer characteristics of ferrofluids in forced or mixed convection is relatively sparse. Also usually, the numerical simulations have often used a single phase model while ferrofluids are colloidal mixtures. The main aim of this paper is a 3D numerical investigation on the mixed convective heat transfer features of kerosene based ferrofluid flowing upward in a vertical circular tube under the influence of external non-uniform magnetic field using the two phase mixture model. The effects of the external magnetic field gradients on the hydrothermal behaviors of the ferrofluid flow are

2. Theoretical formulation 2.1. Governing equations Fig. 1 shows a schematic of the investigated problem. The physical properties of the fluid are assumed constant except for the density in the body force, which varies linearly with the temperature based on the Boussinesq’s model. In the present work, dissipation and pressure work are ignored. Considering

outlet

outlet g D

l 4

G=

dH dz

G=

l 2

dH dz

l 4

z

inlet

inlet

Fig. 1. Schematic of physical models with: (a) negative and (b) positive magnetic field gradients, and (c) used grid.

H. Aminfar et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 1963–1972

these assumptions the dimensional conservation equations for steady state condition are as follows: Continuity equation: , m mÞ ¼ 0

r:ðr n

ð1Þ

, dr,p

n

,

,

¼ n p n m

ð6Þ

are the mass-averaged velocity and drift velocity respectively and ap is the volume fraction of nanoparticles. Table 2 Properties of the studied fluid and particles.

Momentum equation:

Property , ,

,

,

,

,

Value

Property

Value

rP

5200 kg/m3 670 J/kg K 6 w/m K 0.000013 1/K –

,

r:ðrm n m n m Þ ¼ rp þ r:ðmm rn m Þ þ r:ðaP rP n dr,p n dr,p Þ

rf

,

rm,0 ðTT0 Þbm g þ m0 ðM :rÞH

ð2Þ

cp,f kf

bf

Energy equation:

mf

,

1965

,

r:½ðaP rP cp,p n p þ ð1aP Þrf cp,f n f ÞT ¼ r:ðkm rTÞ

3

783 kg/m 2090 J/kg K 0.149 w/m K 0.001 1/K 0.0024 kg/m s

cp,p kP

bp –

ð3Þ

Volume fraction: ,

,

r:ðaP rP n m Þ ¼ r:ðaP rP n dr,p Þ

ð4Þ

where , m

,

n ¼

,

ap rP n p þ ð1aP Þrf n f rm

ð5Þ

Table 1 Grid independent test (Re ¼20); Gr¼ 10,000; G ¼  4  105; z/D ¼ 10; r/D ¼ 0.25. Node number r  y  z

vz/V0

T/T0

r-Directon

40  40  300 50  40  300 60  40  300

1.2468367 1.2476296 1.2484916

1.0036796 1.0036805 1.0036824

y-Directon

50  32  300 50  40  300 50  48  300

1.2490008 1.2476296 1.2473123

1.0037003 1.0036805 1.0036790

z-Directon

50  40  280 50  40  300 50  40  320

1.2480252 1.2476296 1.2475169

1.0036901 1.0036805 1.0036762

Fig. 2. Comparison of the Nusselt number with the experimental ones.

Fig. 3. Effects of the buoyancy force and magnetic force on the local Nusselt number for Re¼ 20 and 40.

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H. Aminfar et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 1963–1972

The slip velocity is defined as the velocity of a secondary phase (p) with respect to the velocity of the primary phase (f): , pf

,

,

n ¼ n p n f

ð7Þ

The drift velocity is related to the slip velocity , dr,p

n

,

¼ n pf 

ap rP , , ðn n Þ rm f p

ð8Þ

, pf

n ¼

rp d2p rp rf , m0 mp LðxÞ , gþ rH 3pmf dp 18mf rp

The last term in Eq. (2), is the effect of magnetic field, which is the so-called the Kelvin force density, derived from the stress of an electromagnetic field where M is the magnetization and is defined as [26]: M ¼ Ms LðxÞ ¼

Considering stokes drag coefficient and forces act on a single magnetic particle, the slip velocity is defined similar to Jafari et al. [15]:

ð10Þ

6ap mp ðcothðxÞ1=xÞ pd3p

ð11Þ

ð9Þ

The unit cell of the crystal structure of magnetite has a volume of about 730 A˚ 3 and contains 8 molecules Fe3O4 and each of them having a magnetic moment of 4mB [27]. Therefore

Fig. 4. Effects of the buoyancy force and magnetic force on the dimensionless axial velocity for Re ¼20 and 40.

Fig. 5. Effects of the buoyancy force and magnetic force on the dimensionless temperature for Re ¼ 20 and 40.

X, , , , F p ¼ 0-m0 mp LðxÞrH þ ðrp rf ÞVp g 3pmf dp n pf ¼ 0

H. Aminfar et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 1963–1972

the particle magnetic moment for the magnetite particles is obtained as: mp ¼

4mB pd3p 6  91:25  1030

m0 mp H kB T

Mixture thermal conductivity: km ¼ aP kp þð1aP Þkf

ð16Þ

ð13Þ

The mixture physical properties in the above equations are used as below: Mixture density:

rm ¼ aP rp þ ð1aP Þrf

ð15Þ

ð12Þ

also x is the Langevin parameter and is defined as [26]:

x¼

Mixture dynamic viscosity [1]:   5 mm ¼ 1 þ ap mf 2

1967

ð14Þ

Mixture thermal expansion coefficient [28]: "

bm ¼

# bp 1 1  þ b 1þ ð1aP Þrf =aP rP bf 1þ aP rP =ð1aP Þrf f

ð17Þ

2.2. Boundary conditions The above mentioned non-linear and coupled partial differential governing equations are subjected to the following boundary conditions:

 At the tube inlet (i.e., z¼0): vm,r ¼ vm, y ¼ 0;

vm,z ¼ V0 ;

T ¼ T0

ð18Þ

 At the fluid-wall interface (i.e., r ¼D/2) a uniform heat flux was applied: qw ¼ km

@T vm,r ¼ vm, y ¼ vm,z ¼ 0 @r

ð19Þ

 At the tube outlet (i.e., z ¼l): at the exit plain the pressure is assumed to be atmospheric pressure.

2.3. Numerical method

Fig. 6. Effect of the various magnetic field gradients on the local Nusselt number for Re ¼20 and 40.

The set of 3D coupled non-linear differential equations were discretized with the control volume technique. For the convective and diffusive terms a second order upwind method was used while the SIMPLEC procedure was introduced for the velocity– pressure coupling. A structured non-uniform grid has been used to discretize the computational domain (see Fig. 1c). It is finer near the tube entrance, magnetic field maximum and near the wall where the velocity and temperature gradients are large. Several different grid distributions have been examined to ensure that the calculated results are grid independent. The used grid for the present calculations consisted of 50, 40 and 300 nodes, respectively, in the radial, circumferential and axial directions. As shown in Table 1, increasing the grid numbers does not change significantly the dimensionless velocity and temperature at the mentioned point in the table. It should be noted that the reasoning of the very fine grid in axial direction is the axial gradient of magnetic field and its effects on the flow variables. In order to demonstrate the validity and also precision of the model and the numerical procedure, a comparison with the previously published experimental work using water–Al2O3 nanofluid in the absence of magnetic field has been done. Fig. 2 indicates the comparison of the present results for the Nusselt number with the experimental ones of Kim et al. [29] in a horizontal tube. As seen there is very good agreement between them.

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3. Results and discussion The results are presented for kerosene based ferrofluid consisting 4 vol% Fe3O4 particles with 10 nm mean diameter (spherical shape). Physical model is a straight circular tube with the length of 200 mm and diameter of 10 mm (see Fig. 1). Grashof number for the presented results is 10,000. For this value the bulk temperature variation is low enough for reliability of the Boussinesq approximation in the range of applied Reynolds numbers. The applied magnetic field is considered to be uniform in radial and circumferential direction and varying linearly in the z-direction, which starts from zero at the z¼L/4 and finishes with maximum value at the z ¼3L/4 for the positive gradient case and vice versa for the negative gradient case (see Fig. 1). Although the magnetic field affects the viscosity and thermal conductivity of the ferrofluid, these effects are assumed negligible, because of its necessity for the experimental correlations and study only the Kelvin force (i.e., magnetic force) effects on the hydrothermal behaviors. Properties of the studied nanofluid are presented in Table 2. Fig. 3 shows the effects of the buoyancy force and magnetic force on the local Nusselt number for the two Reynolds numbers. It can be seen that both the buoyancy force and the magnetic force with negative gradient enhance the Nusselt number and these enhancements are more considerable for low Reynolds numbers. Figs. 4 and 5 indicate the dimensionless axial velocity and temperature profile for these cases at z/D ¼10. Buoyancy force near the wall is greater than the centerline and accelerates

near wall fluid and causes wall temperature decrease and consequently Nusselt number enhancement with respect to no buoyancy case. Magnetic field with negative gradient exerts a downward body force similar to gravitational force. This braking force decelerates fluid flow near the centerline and accelerates near the wall and causes wall temperature decrease. As seen in Figs. 3–5, the presence of external magnetic field with negative gradient has the effects similar to buoyancy effects although these two forces are in opposite directions. The external magnetic field effects with different positive and negative gradient values on the local Nusselt number have been presented in Fig. 6. For the positive gradient case with starting the magnetic field by zero value at z/D ¼5, Nusselt number starts to decrease compared to no magnetic field case. Decrease in the Nusselt number continues as long as the magnetic field exists. By the sudden elimination of magnetic field at z/D ¼15, Nusselt number recovers rapidly to the values of no magnetic field case. For the negative magnetic field gradient the behavior is vice versa. Also these effects increase with increase in the slope of the magnetic field variation. Fig. 7a shows the axial variation of velocity profile for the no magnetic field case. As it is seen from this figure, developed velocity profile is flat because of the downward gravitational force. As shown in Fig. 7b, when the ferrofluid passes over the section of the magnetic field with G 40, fluid accelerates along the tube due to Kelvin force exertion concurrent with fluid flow. As the fluid passes from z/D ¼15, the accelerating Kelvin force reduces from maximum value to zero suddenly and velocity

Fig. 7. Variation of velocity profiles along the tube for Re ¼ 20: (a) no magnetic field, (b) G ¼ 4Eþ 5 and (c) G ¼  4Eþ5.

H. Aminfar et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 1963–1972

1969

profiles return to the no magnetic field case (see Fig. 7a). When G o0 (see Fig. 7c), the Kelvin force acts as a braking force starting from z/D ¼5 with maximum value. In this case both of the gravitational force and Kelvin force are in the opposite direction of fluid flow and cause more deceleration of the fluid near the centerline. Because of this reason, peaks of the velocity profiles move to the wall vicinity with respect to other cases. After the z/D ¼5, the braking Kelvin force starts to reduce linearly with magnetic field intensity. By decreasing this force the velocity profile starts to recover gradually to the no magnetic field case and completely recovers after the Kelvin force reaches zero at z/D ¼15. Fig. 8 presents the velocity profile deformation due to various magnetic field gradients for two different Reynolds numbers. These profiles are plotted at z/D ¼10 because in this section the

intensity of the magnetic field and consequently Kelvin force magnitude are the same for corresponding negative and positive magnetic field gradients and comparison of these profiles is reasonable. As seen in Fig. 8, when G 40, upward Kelvin force causes the fluid in vicinity of the centerline to accelerate and the peak of velocity profile increases. When G o0, downward Kelvin force cause the fluid in vicinity of the centerline to decelerate and the near wall mass flow rate increases. Also increase in slope of the magnetic field gradients increases these effects. Comparison of the results for the two presented Reynolds numbers in this figure shows that for higher Reynolds numbers it needs high gradients and therefore intensive magnetic fields to affect the flow field. Fig. 9 indicates the diametrical temperature distribution at z/D ¼10 with various magnetic field gradients for two different

Fig. 8. Effects of the various magnetic field gradients on the dimensionless axial velocity for Re ¼20 and 40.

Fig. 9. Effects of the various magnetic field gradients on the dimensionless (a) temperature for Re ¼20 and 40.

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H. Aminfar et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 1963–1972

Reynolds numbers. As mentioned before when G40, mass flow rate and consequently velocity gradient in r-direction decreases near the wall. Since this causes the reduction in heat diffusion toward the centerline, wall temperature increases with regard to no magnetic field case. On the other hand for G o0, the wall temperature decreases compared to no magnetic field because of the increment in heat diffusion. The axial distribution of static pressure and Kelvin force along the tube with various magnetic field gradients for two different Reynolds numbers has been presented in Figs. 10 and 11. It should be noted that the values of these figures are gage pressure. For the positive gradient cases (i.e., G 40), the upward Kelvin force acts as a pumping power and sucks the ferrofluid from the tube inlet. Therefore, the static pressure for these cases is lower than outlet pressure and gage pressure is negative. With starting

Fig. 10. Effects of the various magnetic field gradients on the pressure distribution along the flow.

the magnetic field from zero at z/D ¼5, the static pressure starts to increase with increase in Kelvin force until the atmospheric pressure is reached at the outlet. For no magnetic field case (i.e., G¼ 0), the pressure drop is very low due to short length of the tube and low Reynolds numbers and its curve cannot be seen beside the other curves. For the negative gradient cases (i.e., G o0), because of the Kelvin force exerting against the fluid flow, it needs a stronger pumping power at the inlet to overcome this force. For these cases suddenly starting of the magnetic field with maximum value at z/D ¼5 causes a discontinuity in the pressure distribution at this position. After this position, the static pressure starts to decrease with decrease in the Kelvin force until the atmospheric pressure is reached at the outlet. Fig. 12 shows the ratio of the Nusselt numbers, Nu/Nu0, where the subscript 0 refers to no magnetic field case. These results are obtained at z/D ¼10. As seen while the Reynolds number is reduced, the magnetic field becomes more effective and for high

Fig. 11. Kelvin force distribution along the flow corresponding to Fig. 10.

H. Aminfar et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 1963–1972



1971

pumping power, because the magnetic field acts as a magnetic pump. The magnetic field is more effective in low Reynolds numbers and for high Reynolds numbers it needs intensive magnetic fields with high gradients.

References

Fig. 12. Local Nusselt number ratio vs. Reynolds number.

Reynolds numbers it needs stronger magnetic fields with intensive gradients, or more magnetic nanoparticles concentration (i.e. ap in Eq. (11)) to increase the Kelvin force. Also it can be seen that the decrement of the Nusselt number due to positive Kelvin force (corresponding to positive gradient of the magnetic field) is more than the increment of the Nusselt number due to negative Kelvin force (corresponding to negative gradient of the magnetic field). For example in Re ¼20 for G ¼6Eþ5 the Nusselt number decrement is 40.6% while for G ¼ 6E þ5 the Nusselt number increment is 35.3%.

4. Conclusion This article describes, the 3D numerical study of developing laminar mixed convection of a ferrofluid consisting of kerosene and Fe3O4 nanoparticles in the presence of non-uniform magnetic fields. The two phase mixture model and the control volume technique have been used to study the flow. The following conclusions were obtained:

 For the upward flow of a ferrofluid in a vertical tube, although







the buoyancy force and downward Kelvin body force (corresponding non-uniform magnetic field with negative gradient in axial direction) are in opposite directions, both of them have similar effects on the heat transfer coefficient and augment the Nusselt number. The flow field is influenced by the magnetic field with axial gradient. For the positive gradient, the flow is accelerated near the tube centerline and for the negative gradient is decelerated. Nusselt number is increased in the negative gradients of magnetic field but it decreases in the positive gradients. It should be noted that the gradient and intensity of magnetic field determine the intensity of increase or decrease in the Nusselt number. Therefore, it can be used to control the heat transfer coefficient. Using the magnetic field with negative gradient needs a high pumping power with regard to no magnetic field case. In contrast, the positive gradient cases need a lower or no

[1] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, London, 1985. [2] R. Hiegeister, W. Andra, N. Buske, R. Hergt, I. Hilger, U. Richter, W. Kaiser, Application of magnetite ferrofluids for hyperthermia, Journal of Magnetism and Magnetic Materials 201 (1999) 420–422. [3] K. Nakatsuka, B. Jeyadevan, S. Neveu, H. Koganezawa, The magnetic fluid for heat transfer applications, Journal of Magnetism and Magnetic Materials 252 (2002) 360–362. [4] S. Shuchi, K. Sakatani, H. Yamaguchi, An application of a binary mixture of magnetic fluid for heat transport devices, Journal of Magnetism and Magnetic Materials 289 (2005) 257–259. [5] H. Kikura, T. Sawada, T. Tanahashi, Natural convection of a magnetic fluid in a cubic enclosure, Journal of Magnetism and Magnetic Materials 122 (1993) 315–318. [6] H. Yamaguchi, I. Kobori, Y. Uehata, K. Shimada, Natural convection of magnetic fluid in a rectangular box, Journal of Magnetism and Magnetic Materials 201 (1999) 264–267. [7] C. Tangthieng, B.A. Finlayson, J. Maulbetsch, T. Cader, Finite element model of magnetoconvection of a ferrofluid, Journal of Magnetism and Magnetic Materials 201 (1999) 252–255. [8] C.Y. Wen, C.Y. Chen, S.F. Yang, Flow visualization of natural convection of magnetic fluid in a rectangular Hele–Shaw cell, Journal of Magnetism and Magnetic Materials 252C (2002) 206–208. [9] M.S. Krakov, I.V. Nikiforov, To the influence of uniform magnetic field on thermomagnetic convection in square cavity, Journal of Magnetism and Magnetic Materials 252 (2002) 209–211. [10] H. Yamaguchi, Z. Zhang, S. Shuchi, K. Shimada, Heat transfer characteristics of magnetic fluid in a partitioned rectangular box, Journal of Magnetism and Magnetic Materials 252 (2002) 203–205. [11] S.M. Snyder, T. Cader, B.A. Finlayson, Finite element model of magnetoconvection of a ferrofluid, Journal of Magnetism and Magnetic Materials 262 (2003) 269–279. [12] R. Ganguly, S. Sen, I.K. Puri, Thermomagnetic convection in a square enclosure using a line-dipole, Physics of Fluids 16 (2004) 2228–2236. [13] C.Y. Wen, W.P. Su, Natural convection of magnetic fluid in a rectangular Hele–Shaw cell, Journal of Magnetism and Magnetic Materials 289 (2005) 209–302. [14] T. Tagawa, A. Ujihara, H. Ozoe, Average heat transfer rates measured in two different temperature ranges for magnetic convection of horizontal water layer heated from below, International Journal of Heat Mass Transfer 49 (2006) 3555–3560. [15] A. Jafari, T. Tynjala, S.M. Mousavi, P. Sarkomaa, Simulation of heat transfer in a ferrofluid using computational fluid dynamics technique, International Journal of Heat and Fluid Flow 29 (2008) 1197–1202. [16] A. Bozhko, G. Putin, Thermomagnetic convection as a tool for heat and mass transfer control in nanosize materials under microgravity conditions, Microgravity Science and Technology 21 (2009) 89–93. [17] W. Wrobel, E. Fornalik-Wajs, J.S. Szmyd, Experimental and numerical analysis of thermo-magnetic convection in a vertical annular enclosure, International Journal of Heat and Fluid Flow, doi:10.1016/j.ijheatfluidflow.2010.05.012. [18] N.B. Sambamurthy, A. Shaija, G.S.V.L. Narasimham, M.V. Krishna Murthy, Laminar conjugate natural convection in horizontal annuli, International Journal of Heat and Fluid Flow 29 (2008) 1347–1359. [19] M. Lajvardi, J. Moghimi-Rad, I. Hadi, A. Gavili, T. Dallal, F. Zabihi, J. Sabbaghzadeh, Experimental investigation for enhanced ferrofluid heat transfer under magnetic field effect, Journal of Magnetism and Magnetic Materials 322 (2010) 3508–3513. [20] S. Kamiyama, J. Ishimoto, Boiling two-phase flows of magnetic fluid in a nonuniform magnetic field, Journal of Magnetism and Magnetic Materials 149 (1995) 125–131. [21] J. Liu, J. Gu, Z. Lian, H. Liu, Experiments and mechanism analysis of pool boiling enhancement with water-based magnetic fluid, Heat Mass Transfer 41 (2004) 170–175. [22] T. Aihara, J. Kim, K. Okuyama, Controllability of convective heat transfer of magnetic fluid in a circular tube, Journal of Magnetism and Magnetic Materials 122 (1993) 297–300. [23] R. Ganguly, S. Sen, I.K. Puri, Heat transfer augmentation using a magnetic fluid under the influence of a line dipole, Journal of Magnetism and Magnetic Materials 271 (2004) 63–73. [24] Y. Xuan, Q. Li, M. Ye, Investigations of convective heat transfer in ferrofluid microflows using lattice-Boltzmann approach, International Journal of Thermal Science 46 (2007) 105–111.

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[25] Q. Li, Y. Xuan, Experimental investigation on heat transfer characteristics of magnetic fluid flow around a fine wire under the influence of an external magnetic field, Experimental Thermal and Fluid Science 33 (2009) 591–596. [26] H. Yamaguchi, Engineering Fluid Mechanics, Springer Science, Netherlands, 2008. [27] Ch. Kittel, Introduction to Solid State Physics, John Wiley & Sons, New York, 1967.

[28] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two dimensional enclosure utilizing nanofluids, International Journal of Heat Mass Transfer 46 (2003) 3639–3653. [29] D. Kim, Y. Kwon, Y. Cho, C. Li, S. Cheong, Y. Hwang, J. Lee, D. Hong, S. Moon, Convective heat transfer characteristics of nanofluids under laminar and turbulent flow conditions, Current Applied Physics 9 (2009) 119–123.