magnetic and gravitational convection of air with a coil inclined around the x axis

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MAGNETIC AND GRAVITATIONALCONVECTION OF AIR WITH A COILINCLINED AROUND THE X AXISTomasz Bednarz a , Toshio Tagawa b , Masayuki Kaneda b , HiroyukiOzoe b & Janusz S. Szmyd ca Interdisciplinary Graduate School of Engineering Sciences ,Fukuoka, Japanb Institute for Materials Chemistry and Engineering , Fukuoka, Japanc AGH University of Science and Technology , Krakow, PolandPublished online: 17 Aug 2010.

To cite this article: Tomasz Bednarz , Toshio Tagawa , Masayuki Kaneda , Hiroyuki Ozoe & Janusz S.Szmyd (2004) MAGNETIC AND GRAVITATIONAL CONVECTION OF AIR WITH A COIL INCLINED AROUND THEX AXIS, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation andMethodology, 46:1, 99-113, DOI: 10.1080/10407780490457464

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MAGNETIC AND GRAVITATIONAL CONVECTION OF AIRWITH A COIL INCLINED AROUND THE X AXIS

Tomasz BednarzInterdisciplinary Graduate School of Engineering Sciences,Kyushu University, Fukuoka, Japan

Toshio Tagawa, Masayuki Kaneda, and Hiroyuki OzoeInstitute for Materials Chemistry and Engineering, Kyushu University,Fukuoka, Japan

Janusz S. SzmydAGH University of Science and Technology, Krakow, Poland

Air is filled in a cubic enclosure whose one vertical wall is isothermally heated and the

opposite one is cooled while the other four walls are thermally insulated. A large coil is

placed outside of this enclosure with the coil center coinciding with the cube center. An

electric current in the coil generates a magnetic field to affect the convection of air, because

the air contains oxygen whose magnetic susceptibility is exceptionally large among gases.

The coil is further inclined around the X axis, which is horizontal and perpendicular to the

hot and cold walls through the wall center. The heat transfer rate changes depending on the

inclination angle. This system is studied numerically for convection for the following

combination of parameters: Ra¼ 1.516104, 9.066104; Pr¼ 0.71; g¼ 07 100; xEuler¼ 0–

p=2, where g represents the strength of magnetic field and xEuler is the angle of inclination

of the coil. For example, at Ra¼ 1.516104 and g¼ 30, the average Nusselt number 2.535 at

xEuler¼ 0 increased to 2.823 at xEuler¼p=2. This study suggests that the coil inclination

affects the heat transfer rate extensively.

INTRODUCTION

In recent years, advances in superconducting-magnet technology have allowedthe generation of high magnetic fields such as 10 T in a room-temperature bore.Also, some new investigations allow developing a 20-T hybrid magnet withoutsupplying liquid helium [1]. Strong magnetic field gradients available from suchsuperconducting magnets can be used to produce magnetizing forces in weaklymagnetic materials.

Received 24 October 2003; accepted 6 February 2004.

Address correspondence to Prof. Hiroyuki Ozoe, Kyushu University, 6-1 Kasuga Koen, Kasuga,

Fukuoka 816-8580, Japan. E-mail: [email protected]

Numerical Heat Transfer, Part A, 46: 99–113, 2004

Copyright # Taylor & Francis Inc.

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780490457464

99

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Air convection usually accompanies heat transfer. In a terrestrial state thedensity difference in the fluid between hot and cold regions has been known to causeconvection. Furthermore, since the fluid has a magnetic susceptibility that varieswith temperature, another driving force also induces convective motion in a strongmagnetic field. Air contains 21 mol% oxygen gas and is known as a paramagneticsubstance. The paramagnetism of O2 originates from parallel spin of the two outer-shell electrons of an O2 molecule, and O2 gas has an exceptionally large positivevalue of magnetic susceptibility. As a result, oxygen gas is attracted toward astronger magnetic field.

Many recent articles have reported various interesting phenomena connectedwith the behavior of paramagnetic substances such as oxygen in a magnetic field.Wakayama and co-workers [2–5] are very active in this area. They have reportedmany innovative applications and many new findings, such as enhanced oxygensupply toward fuel cell poles using magnets, combustion sustaining in diffusionflames under microgravity, breathing support, etc. Also, Kitazawa’s group isworking on the phenomena associated with magnetizing force. They have reported [6]nonmechanical blowing of heated air in a strong magnetic gradient field, enhancingoxygen gas dissolution into water. By the application of the magneto-Archimedeslevitation technique [7], they realized a novel separation method for weakly magneticmaterials, that is, dia- and paramagnetic materials, etc. Braithwaite et al. [8] used a

NOMENCLATURE

b magnetic induction (bx, by, bz), T

b0 reference magnetic induction

(¼ mm i=l), T

B dimensionless magnetic induction,

b=b0 = (Bx,By,Bz)

Bmax maximum magnitude of nondimen-

sional magnetic induction vector

D=Dt ¼ ðq=qtþUq=qXþ Vq=qYþWq=qZÞD=Dt ¼ ðq=qtþ uq=qxþ vq=qyþ wq=qzÞ; 1=sg acceleration coefficient of gravity, m=s2

i electric current in a coil, A

l length of a cubic enclosure, m

p pressure, Pa

p0 reference pressure without convection

of gas (¼r0a2=l2), Pa

P dimensionless pressure (¼ p=p0)

Pr Prandtl number (¼ n=a)r position vector, m

R ¼ r=l

Ra ¼ gb(yhot7ycold)l3=(an)

ds tangential element of a coil, m

dS ds/l

t time, s

t0 reference time (¼ l2=a), sT dimensionless temperature

[¼ (y7y0)/(yhot7ycold)]u, v, w velocity components, m=s

u0 reference velocity (¼ a=l), m=s

u velocity vector [¼ (u, v, w)], m=s

U ¼ u=u0¼ (U, V, W)

xEuler ¼ inclination angle of the coil, rad

x0, y0, z0 ¼ l, m

a thermal diffusivity of air, m2=s

b thermal expansion coefficient, K71

g ¼ zwO2b20=mmgl

y temperature, K

z mass fraction of oxygen at some

reference state (air)

l thermal conductivity, W=m K

m viscosity, Pa s

mm magnetic permeability, H=m

n kinematic viscosity (¼m=r0), m2=s

r density, kg=m3

r0 reference density at temperature y0,kg=m3

wo2 magnetic susceptibility of oxygen,

m3=kg

t dimensionless time, (¼ t=t0)

H ¼ ðq=qX; q=qY; q=qZÞ orðq=qx; q=qy; q=qzÞ;dimensionless or 1=m

H2 ¼ q2=qX2 þ q2=qY2 þ q2=qZ2 or

q2=qx2 þ q2=qy2 þ q2=qz2;dimensionless or 1=m2

100 T. BEDNARZ ET AL.

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magnetic field both to enhance and suppress Rayleigh-Benard convection in asolution of gadolinium nitrate and showed that the effect depends on the relativeorientation of the magnetic field and temperature gradients. Huang et al. [9] studiedthermoconvection in a horizontal fluid layer by classical stability analysis. Tagawaand co-workers [10, 11] derived model equations for magnetizing convection using amethod similar to Boussinesq approximation and carried out numerical calculationswith supporting experiments.

In the present study, numerical calculations are carried out for the combinationof magnetizing and gravitational convection in a cubic enclosure with an electric coilinclined around the X axis.

SYSTEM CONSIDERED

Figure 1 shows the model system for the present calculation, subjected to amagnetizing force or with gravitational forces for the coil inclined around the X axis.The system consists of a cubic enclosure which is kept in a horizontal position and aone-turn electric circle coil which generates a magnetic field. The enclosure is iso-thermally heated from one vertical wall and cooled from the opposing one. The otherfour walls are thermally insulated. The electric coil is placed around the cubicenclosure and can be inclined with its plane oblique to the adiabatic side walls. Thedimensional parameters are as follows. The diameter of the electric coil is set as0.1 m and the length of the cubic enclosure as 0.032 m. The gravitational force actsin the minus Z direction. The natural convection of air in a cubic enclosure with theinclined coil is considered herein.

MODEL EQUATIONS

According to Tagawa et al. [11], the model equations for natural convection ofair with an inclined electric coil are given in dimensionless form as follows:

Continuity equation:

= �U ¼ 0 ð1Þ

Momentum equation in non gravity field:

DU

Dt¼ �=Pþ Pr H2U� gRaPrT=B2 ð2aÞ

Momentum equation in a gravity field:

DU

Dt¼ �=Pþ Pr H2UþRa Pr T �g=B2 þ

0

0

1

0BB@

1CCA

2664

3775 ð2bÞ

MAGNETIC AND GRAVITATIONAL CONVECTION OF AIR 101

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Energy equation:

DT

Dt¼ H2T ð3Þ

Magnetic field distribution was calculated using Biot-Savart’s law:

B ¼ 1

4p

Icoil

dS� R

R3ð4Þ

The dimensionless variables and parameters are defined as follows:

X ¼ x

x02Y ¼ y

y0Z ¼ z

z0U ¼ u

u0V ¼ v

v0W ¼ w

w0

t ¼ t

t0P ¼ p

p0B ¼ b

b0T ¼ y� y0

yh � ycx0 ¼ y0 ¼ z0 ¼ l

Figure 1. Schema of system considered. Coil center is located in the center of the cubic enclosure and

inclined around the X axis.


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u0 ¼ v0 ¼ w0 ¼al

t0 ¼l2

ab0 ¼

mmil

p0 ¼r0a

2

l2

Pr ¼ na

Ra ¼ gbðyh � ycÞl3an

g ¼ w0b02z

mmgl

The boundary conditions are

At t< 0, the initial condition is a heat conduction state,

U ¼ V ¼ W ¼ 0 T ¼ �X ð�0:5 � X � 0:5ÞAt all cubic enclosure walls: U¼V¼W¼ 0At the vertical wall (at X¼ 70.5): T¼ 0.5At the vertical wall (at X¼ 0.5): T¼ 70.5At the vertical adiabatic walls (at Y¼ 70.5, 0.5): qT=qY ¼ 0At the horizontal adiabatic walls (at Z¼ 70.5, 0.5): qT=qZ ¼ 0

These dimensionless equations were approximated by finite-difference equations andsolved numerically with the HSMAC (Highly Simplified Marker and Cell) [12, 13]scheme for a staggered mesh system. The number of meshes is 40640640.

COMPUTED RESULTS

The system shown in Figure 1 was studied numerically for convection for thefollowing combination of parameters: Ra¼ 1.516104, 9.066104; Pr¼ 0.71; g¼ 0–100; xEuler¼ 0–p=2. Dimensional equivalence for air at Ra¼ 1.516104 and g¼ 30 isyh7yc¼ 5 K and bmax¼ b0Bmax¼ 1.930 T, where Bmax¼ 0.32 for the system shownin Figure 1. Diameter of the coil was set as 0.1 m and the length of the cube as0.032 m. These sample values are only for a reference value and in dimensionlessform the length of the cube was taken as 1.0 and diameter of the coil as 3.125. Table 1lists magnetic susceptibilities of some paramagnetic and diamagnetic materials atroom temperature. The magnetizing and the gravitational forces act independentlyon the air in the cubic enclosure. Computations were carried out first for nongravitycases (g¼ 0) with the convection driven solely by the magnetizing force, and latergravitational force was added.

Table 1. Magnetic susceptibilities of some paramagnetic and diamagnetic

materials at room temperature [14]

Material Magnetic susceptibility

Aluminum 2.361075

Copper 70.9861075

Diamond 72.261075

Tungsten 6.861075

Hydrogen (1 atm) 70.2161078

Oxygen (1 atm) 209.061078

Nitrogen (1 atm) 70.5061078


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Computed Results without Gravitational Force

For g¼ 0 (no magnetic field), only gravitational natural convection occurs andthe average Nusselt number for this case is 2.388 at Ra¼ 1.516104. For g¼ 0 (spacecondition), Ra becomes zero and g becomes infinity, and finite product g Ra has tobe used to describe nongravity cases.

Table 2 lists the computed average Nusselt numbers for a set of g Ra values atPr¼ 0.71 and for inclination angle xEuler from 0 to p=2. The average Nusselt num-bers are plotted versus angle of inclination in Figure 2. The average Nusselt numberincreases with g Ra for each inclination angle. The range of inclination angle is from0 to p=2 rad. At g¼ 0, only the magnetizing force acts and the resulted convection issymmetrical in terms of the angle at xEuler¼ p=4. This means that for correspondingnegative values of angle xEuler, the same values of average Nusselt number will beobtained. As seen, the coil inclination changes the average Nusselt number and alocal maximum is obtained at xEuler¼ p=4 for every case. Local minima are obtainedat xEuler¼ 0, p=2, etc. Values of the average Nusselt number appear periodically withperiod p=2 in the inclination angle. Symmetry property in the average heat transferrate is observed in terms of xEuler¼ 0, p=4, p=2, 3p=4, etc.

Figure 3 shows perspective views of magnetic buoyancy force vectors in threehorizontal planes at Z¼ 70.35, 0, and 0.35 at Pr¼ 0.71, g Ra¼ 4.536105, g¼ 0, forfour different values of inclination angle, xEuler¼ 0, p=6, p=4, and p=2. Figures 4 and5 show corresponding isothermal surfaces and long-time streak lines for the sameconditions.

When the coil is in a horizontal position (xEuler¼ 0), Figure 3a shows themagnetic buoyancy force near the left-hand-side hot wall acting to move the hot airalong the hot wall toward the top and bottom adiabatic walls from the mid-heightplane. Isothermal contours in Figure 4a suggest the flow along the top ceiling andbottom floor toward the right-hand side cold wall. Near the cold wall, due to thestrong attracting force, the cold air is going downward and upward to the mid-heightcoil plane and returning flow appears to be stronger along the adiabatic side walls atY¼ 70.5 and Y¼ 0.5, as seen in Figure 4a. The average Nusselt number is 2.471 forair in this magnetizing force field. Figures 3d, 4d, and 5d show corresponding plots atinclination angle xEuler¼ p=2. The flow structure and isotherms look to be quite

Table 2. Computed average Nusselt numbers on the hot wall at Pr¼ 0.71, without a gravity field, with a

coil inclined around the X axis, for various values of g Ra

Average Nusselt number, Nuave

Inclination

angle, xEuler

g Ra¼1.516104

g Ra¼1.516105

g Ra¼4.536105

g Ra¼9.066105

g Ra¼1.516106

g Ra¼2.726106

0 1.008 1.479 2.471 3.396 4.228 5.385

p=12 1.012 1.571 2.605 3.533 4.357 5.485

p=6 1.020 1.736 2.826 3.756 4.573 5.682

p=4 1.024 1.807 2.918 3.852 4.674 5.792

p=3 1.020 1.736 2.826 3.756 4.573 5.682

5p=12 1.012 1.571 2.605 3.533 4.357 5.485

p=2 1.008 1.479 2.471 3.396 4.228 5.385


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different, but they are just p=2 rad inclined from (a) since this is a nongravity field.Mirror or symmetry property was also observed for other inclination angles. Forexample, the heat transfer rates at xEuler¼ p=6 and xEuler¼ p=3 are the same as seenin Table 2.

When the coil is inclined from a horizontal position around the X axis by p=4rad (xEuler¼ p=4), the magnetic buoyancy force distribution is twisted over the hotand cold walls (Figure 3c). In the upper part of the cubic enclosure (Z¼ 0.35), nearthe left-hand side hot wall, the magnetic buoyancy force acts to repel the hot airfrom the coil. Especially strong repulsing flow occurs from the back adiabatic wall,due to the close distance to the coil, and horizontal flow results as seen in Figures 4cand 5c. Near the hot wall, close to the front adiabatic wall, the force is acting in thepositive Z direction, so the hot air is going upward and drives a strong flow of hot airalong the top horizontal edge (Z¼ 0.5, Y¼ 70.5) toward the right-hand-side coldwall as seen in Figure 4c. Returning flow near the cold wall is horizontal along thetop ceiling wall and vertical along the front side adiabatic wall due to the attractingcharacter of the force toward the stronger magnetic field. It generates horizontal andvertical rolls as seen in Figure 5c. The same character of flow can be observed alongthe back side and bottom adiabatic walls, due to the symmetrical distribution of themagnetizing force with respect to the coil plane. In this way four separate roll cellsappear over four adiabatic side walls as seen in Figure 5c. Symmetrical ‘‘twisted’’

Figure 2. Average Nusselt numbers versus angle of inclination at Pr¼ 0.71 without a gravity field.


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distribution of temperature can be seen in Figure 4c. The cold fluid has largermagnetic susceptibility due to the Curie’s law, and near the cold wall proceeds to thetwo edges (Y¼Z ¼ 70.5 and Y¼Z¼ 0.5) closest to the coil, where the magneticfield is stronger. On the other hand, the hot air (smaller magnetic susceptibility) isrepelled to opposite edges (Y¼ 70.5, Z¼ 0.5 and Y¼ 0.5, Z¼ 70.5), where themagnetic field is relatively weaker. The average Nusselt number attains its maximumvalue of 2.918 for this case.

For the inclination angle xEuler¼ p=6, the general character of convection isvery similar to the case at p=4. The inclined coil is located lower, near the frontadiabatic wall, and the magnetic buoyancy force generates horizontal rolls along thetop ceiling and bottom floor as seen in Figure 5b. The vertical rolls near the adiabaticfront wall are smaller in comparison to those at xEuler¼ p=4. Near the left-hand-sidehot wall the magnetic field acts to repel hot air upward near Y¼ 70.5. The fluidreaches the corner and is moving along the edge toward the cold wall. Near thecold wall, the magnetic field acts to attract the colder air toward the coil along the

Figure 3. Vectors of magnetic buoyancy force at g Ra¼ 4.536105, Pr¼ 0.71, without a gravity field.

Inclination angle of the coil, xEuler: (a) 0; (b) p=6; (c) p=4; (d) p=2.


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horizontal edges, and finally returning flow appears to be strongest at the front wallat the elevation corresponding to the coil plane, as can be seen in Figure 4b at thelower part of the front adiabatic wall at about Z¼ 7 0.3. Calculated averageNusselt number is 2.826.

These results clarify the effect of magnetizing force in the cube with the inclinedcoil around the X axis. The additional effect of gravity force will be now consideredfor the same system.

Computed Results with Gravitational Field

The standard acceleration due to gravity, g¼ 9.806 m=s2 is added in the minusZ coordinate for the system shown in Figure 1. Table 3 lists computed averageNusselt numbers at g¼ 10, 30, 60, and 100 at Ra¼ 1.516104 and g¼ 30 atRa¼ 9.066104. As can be seen, the increase in the Rayleigh number from 1.516104

to 9.066104 at g¼ 30 increases the average Nusselt numbers for each inclination

Figure 4. Isothermal surfaces at g Ra¼ 4.536105, Pr¼ 0.71, without a gravity field. Inclination angle of

the coil, xEuler: (a) 0; (b) p=6; (c) p=4; (d) p=2.


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Figure 5. Long-time streak lines at g Ra¼ 4.536105, Pr¼ 0.71, without a gravity field. Inclination angle

of the coil, xEuler: (a) 0; (b) p=6; (c) p=4; (d) p=2.

Table 3. Computed average Nusselt numbers on the hot wall at Pr¼ 0.71, with a coil inclined around the

X axis, for various values of Ra and g

Average Nusselt number, Nuave

g¼ 0 g¼ 10 g¼ 30 g¼ 60 g¼ 100

Inclination

angle, xEuler

Ra¼1.516104

Ra¼1.516104

Ra¼1.516104

Ra¼9.066104

Ra¼1.516104

Ra¼1.516104

0 2.349 2.535 5.394 3.396 4.227

p=12 2.352 2.671 5.496 3.542 4.360

p=6 2.368 2.831 5.645 3.750 4.570

p=4 2.388 2.392 2.913 5.764 3.848 4.672

p=3 2.420 2.831 5.657 3.746 4.569

5p=12 2.452 2.796 5.622 3.535 4.349

p=2 2.467 2.823 5.715 3.474 4.252


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angle xEuler. This suggests that convection is generated, and the heat transfer rateincreases with the increase in the temperature difference. The increase in the mag-netic strength from g¼ 10 to g¼ 100 at Ra¼ 1.516104 also increases the averageNusselt numbers. The effect of parameter g at constant Ra¼ 1.516104 is shown inFigure 6, where average Nusselt numbers are plotted versus the inclination anglexEuler. A solid line represents the case at g¼ 0 (no magnetic field), where puregravitational convection takes place and the average Nusselt number is 2.388. Asseen at g¼ 10, the heat transfer rate can be lower than that for pure gravitationalconvection between xEuler¼ 0 and p=4, and the heat transfer rate is increasingmonotonically with xEuler. For larger g, a local maximum is observed at the in-clination angle xEuler¼ p=4 and there is no symmetry with respect to xEuler¼ p=4.Symmetrical property in the average Nusselt numbers was obtained with respect toangles xEuler¼ 0, p=2, p, etc., i.e., the period is p rad.

Figure 7a shows the combination of magnetic and gravitational buoyancyforce vectors at xEuler¼ 0, Ra¼ 1.516104, and g¼ 30. The combination forces areacting almost in the same direction in the upper part of the cubic enclosure, andlarger magnitudes of resultant force and accelerated flow can be expected. Figure 8ashows temperature distribution at xEuler¼ 0. Comparison with figures for the non-gravity case show the effect of gravitational force. Computed average Nussseltnumber is 2.535 and is slightly higher than that for the nongravity case. When

Figure 6. Average Nusselt numbers versus angle of inclination at Pr¼ 0.71, Ra¼ 1.516104, and g¼ 0, 10,

30, 60, and 100.


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xEuler¼ p=2 the average Nusselt number is 2.823 and the flow mode is not the sameas that at xEuler¼ 0, due to the additional gravitational buoyancy force. Figure 9ashows long-time streak lines for this case. As seen in the figure, large rolls are

Figure 7. Vectors of Ra Pr T��gHB2 þ 0; 0; 1ð ÞT

�force at g¼ 30, Ra¼ 1.516104, Pr¼ 0.71. Inclination

angle of the coil, xEuler: (a) 0; (b) p=6; (c) p=4; (d) p=3; (e) p=2.

Figure 8. Isothermal contours at g¼ 30, Ra¼ 1.516104, Pr¼ 0.71. Inclination angle of the coil, xEuler:

(a) 0; (b) p=6; (c) p=4; (d) p=3; (e) p=2.


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generated in the upper part of the cube and the cold air along the cold wall is goingdownward while the hot air is going upward along the left-hand-side hot wall. Nearthe bottom floor, small convection rolls in the opposite direction are generated.

At xEuler ¼ p=2, the shape of the isothermal contours as seen in Figure 8e isopposite that of Figure 8a near the upper adiabatic wall. The detailed view of theflow mode as seen in Figure 9e shows that the hot air goes to the cold wall along thetop adiabatic wall, not through the top central but along the adiabatic top corner(Y ¼ 70.5, 0.5), and the air arriving at the cold wall descends to the core of theenclosure. Then it approaches the bottom of the hot wall and again ascends alongthe hot wall. This mode of flow appears to have resulted in the isothermal contoursas seen in Figure 8e, which is quite different from Figure 8a at xEuler¼ 0. In the lowerpart of the enclosure, the tilted roll cells approaching from the hot wall to thecold wall are along the adiabatic side at mid-height. This mode is similar to that ofFigure 8a.

When the inclination angle is xEuler¼ p=4 at g¼ 30 and Ra¼ 1.516104, theflow mode is similar to that for the nongravity case. Comparing resultant forcevectors (Figure 7c) with those at g¼ 0 (Figure 3c), the influence of the gravitationalforce can be observed. The hot air near the top wall is repelled by the coil near thetop of the left-hand-side hot wall and the cold air near the right-hand-side cold wallproduces an anticlockwise convection as seen from the top. Finally, a large hor-izontal convection roll is generated. Due to the coil position and the repelling andattracting behavior, horizontal rolls are generated near the bottom floor and topceiling as seen in Figure 9c. Temperature distribution is shown in Figure 8c andlooks to be similar to that for the nongravity case.

Figure 9. Long-time streak lines at g¼ 30, Ra¼ 1.516104, Pr¼ 0.71. Inclination angle of the coil, xEuler:

(a) 0; (b) p=6; (c) p=4; (d) p=3; (e) p=2.


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At g¼ 30, gravitational buoyancy force is much weaker than the magneticbuoyancy force. This means that magnetic convection is stronger and the flow modeis similar to that in pure magnetic convection. The differences in the average Nusseltnumbers and the flow modes between those at xEuler¼ 0 and p=2 represent the effectof gravity. Comparing average Nusselt numbers for corresponding conditions ingravity and nongravity fields, more differences can be observed at weaker magneticfields.

CONCLUSIONS

Thermal convection was studied numerically for the air in a cubic enclosurewith an inclined coil around the X axis in the presence or absence of a gravity field.First a nongravity case was considered and later gravitational force was added to seehow convection changes by the effect of a gravity field. For the nongravity case, atg Ra¼ 1.516104, the heat transfer rate was 1.02 or less and convection was weak.Increasing the parameter g Ra, convection became stronger and stronger. In thegravity field, Nu¼ 2.388 at Ra¼ 1.516104 without a magnetic field g¼ 0. Itincreases to Nu¼ 4.227 at g¼ 100 at xEuler¼ 0. At xEuler¼ p=4, Nu¼ 4.672 atg¼ 100. It was shown that the character of convection and heat transfer rate dependstrongly on the inclination of the coil around the enclosure.

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magnetic and gravitational convection of air with a coil inclined around the x axis

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