Journal of Magnetism and Magnetic Materials

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Developing the laminar MHD forced convection ﬂ ow of water/FMWNT carbon nanotubes in a microchannel imposed the uniform heat ﬂ ux

Arash Karimipour

^a

, Abdolmajid Taghipour

^a

, Amir Malvandi

^b,n

aDepartment of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

bDepartment of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran

a r t i c l e i n f o

Article history:

Received 12 February 2016 Received in revised form 11 June 2016

Accepted 23 June 2016 Available online 24 June 2016 Keywords:

FMWNT nanofluid Carbon nanotube Slipflow Magneticfield Developingflow

a b s t r a c t

This paper aims to investigate magneticfield and slip effects on developing laminar forced convection of nanofluids in the microchannels. A novel mixture of water and FMWNT carbon nanotubes is used as the workingfluid. To do this,fluidflow and heat transfer through a microchannel is simulated by a computer code in FORTRAN language. The mixture of FMWNT carbon nanotubes suspended in water is considered as the nanofluid. Slip velocity is supposed as the hydrodynamic boundary condition while the microchannel's lower wall is insulated and the top wall is under the effect of a constant heatflux. Moreover, theflowfield is subjected to a magneticfield with a constant strength. The results are presented as the velocity, temperature and Nusselt number profiles. It is observed that nanofluid composed of water and carbon nanotubes (FMWNT) can work well to increase the heat transfer rate along the microchannel walls. Furthermore, it is indicated that imposing the magneticfield is very effective at the thermally developing region. In contrast, the magneticfield effect at fully developed region is insignificant, especially at low values of Reynolds number.

1. Introduction

In the recent years, there has been increasing research on different ways of improving the heat transfer rate. In this regard, a new generation offluids with immense potential has been considered in industrial applications which are produced by the distribution of nanoscale particles in conventionalfluids. Thesefluids are known as nanofluids. The size of particles used in nanofluids varies from 1 nm to 100 nm. The particles are made from metals, such as copper, silver, or metal oxides. Thefluids that are com- monly used in heat transfer applications often have a low thermal conductivity. Since the nanoparticles have high thermal conductivity, the distribution of nanoparticles in the base fluid increases the thermal conductivity of the mixture. Therefore, the use of nanofluids to enhance heat transfer is a good option[1,2].

Increase in the heat transfer coefficient of nanofluids cannot be estimated by a conventional relationship. The heat transfer coefficient can be improved through changing the geometry or boundary conditions, or increasing the thermal conductivity of the fluid[3]. A number of studies have examined the heat transfer in different geometries as well as the effects of external factors on the heat transfer. Tahir and mital[4]studied the heat transfer in a

circular channel. Many studies have investigated the heat transfer in channels or around the obstacles of various geometries[5,6].

It is very important to investigate the conduction heat transfer coefficients of nanoscale particles and their dispersions influids in the microchannels. Cooling systems are one of the main concerns of the plants and industries that deal with heat transfer. In these cases, the use of advanced and optimized cooling systems is in- evitable. Tullius et al. [7] studies the cooling effects of microchannels in different geometries andfluids. The most common fluids used in microchannels were air and water whose conduction heat transfer rate is very low. On the other hand, in most cases, optimization of heat transfer systems is implemented though increasing their surface area, which increases the size of such devices. Therefore, to overcome this problem, new and effective cooling systems are required, where nanofluids have been introduced as a new approach[8].

Several models have been utilized to investigate the heat transfer in cavities and microchannels. The effects of microchannel geometry, Reynolds number, volume fraction, etc. on the thermal performance of the microchannels have also been investigated[9– 14]. In general, it seems that the nanoﬂuid-cooled microchannels can deal with the increased heat load produced by small electronic devices more efﬁciently[15–18].

The surface effects in the microscale are more signiﬁcant compared to the macro (metric) scale. For example, the non-slip condition, which is widely used in the macroscale, does not apply Contents lists available atScienceDirect

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Journal of Magnetism and Magnetic Materials

nCorresponding author.

E-mail address:[email protected](A. Malvandi).

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to microchannels, where the slip boundary condition should be used along the wall. Also in micro and nanoscales, the particle- based models should be used such as the Lattice Boltzmann Method or molecular dynamics[19–23]. Raisi et al.[24]numerically investigated the heat transfer in microchannels under the boundary conditions of presence or absence of slip velocity. They reported the effects of the use of nanofluid on the heat transfer rate of a microchannel in the slipflow regime. Numerous studies have been conducted on nanofluid heat transfer in microchannels under different boundary conditions such as constant temperature and constant heatflux. Some have also studied the slip velocity boundary condition[25–29].

In the recent years,flow and heat transfer at different macroscale sections subjected to magneticfield has been a topic of in- terest to the researchers. In this case, a force called the Lorentz force affects the flow subjected to the magnetic field [30–33].

Aminossadati et al.[34]studied the effects of magneticﬁeld on a microchannel under constantﬂux. They neglected the effects of slip velocity along the microchannel walls.

The important slip velocity boundary condition has been ig- nored when studying theflow and heat transfer in microchannels subjected to magneticfields (MHD flow). No extensive research has been conducted on the effects of the magneticfields, the slip velocity boundary condition, and forced heat transfer simultaneously. Therefore, in the present work, all these conditions were simultaneously considered. This study attempted to investigate the effects on magneticfield on the slip velocity of thefluid mo- lecules adjacent to the wall for thefirst time. The working nanofluids used in the study was a mixture of water (as the basefluid) and functionalized multi-walled carbon nanotubes (FMWNT)[35]

as the suspended nanoparticles. More new works can be referred which are developing the nanofluidflow and heat transfer and its properties at different geometries and conditions[36–39]. Unlike the conventional nanofluids of water and metal particles, the using nanofluid yielded a desirable increase in the thermal conductivity coefficient of the mixture by slightly increasing in the volume fraction of the particles. To the best of the authors’knowledge, no study has been conducted on this subject and all the presented

outcomes are novel and original.

2. Problem statement

Fig. 1 shows a schematic of the studied microchannel. The width and the length of the microchannel arehandlwhere the dimensionless height and length are H¼h/h¼1 and L¼l/h¼30, respectively. The microchannel was divided into two distinct parts.

Thefirst part, which includes the inlet, was insulated in both top and bottom walls. In the second part, the upper wall was under the constant heat flux q″₀ and a uniform magnetic field with the strength of B0. The slip velocity boundary condition was considered along the walls of the microchannel.

The cold nanofluid mixture composed of water and FMWNT carbon nanotube particles entered the microchannel and was heated under the heat flux, and ultimately, existed from the other side. Given the very low volume fraction of the nanoparticles, the nanofluid was considered as Newtonian fluid. The flow was assumed laminar and incompressible. The effects of changing various parameters such as volume fraction of nanoparticles (

ϕ

¼0.1, 0.2%), Reynolds number (Re¼10, 100), the dimensionless slip coefﬁcient (

β

^*¼0, 0.04, 0.08), and the Hartmann number (Ha¼0, 20, 40) on the motion and thermal properties of the ﬂuid ﬂow were studied.

3. Mathematical formulation 3.1. Governing equations

The two-dimensional form of the Navier–Stokes equations (continuity, momentum, and energy) considering the effect of a magneticﬁeld with strength ofB₀is as follows:

∂

∂ + ∂

∂ =

( ) u

x v y 0

1 l Microchannel length (m)

L Dimensionless microchannel length Nu Nusselt number

p Pressure (Pa)

P Dimensionless pressure Pr Prandtl number q″₀ Constant heatﬂux Re Reynolds number T Temperature (K)

Tc The temperature of the cold inﬂow nanoﬂuid (K) u Horizontal velocity (m s¹)

uc Inlet nanoﬂuid velocity (m s¹) U Dimensionless horizontal velocity us Slip velocity of nanoﬂuid (m s¹) Us Dimensionless slip velocity v Vertical velocity (m s¹) V Dimensionless vertical velocity

β

^{Slip coef}ﬁcient (m)

β

^*¼

β

^/h Dimensionless slip coefﬁcient

α

¼k/

ρ

^cpThermal diffusivity

ϕ

Volume fraction of nanoparticles

μ

Dynamic viscosity (Ns m²)

θ

Dimensionless temperature

ρ

Density (kg m³)

υ

Kinematic viscosity (m²s¹) Super- and sub-scripts

f ﬂuid

m mean value

nf nanoﬂuid

s solid

x value in theX-direction

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ρ υ σ ρ

∂

∂ + ∂

∂ = − ∂

∂ + ∂

∂ −

( )

⎛

⎝⎜ ⎞

⎠⎟ u u

x v u y

p x

u x

u y

B u 1

2

nf nf

nf nf 2

2 2

2

02

ρ υ

∂

∂ + ∂

∂ = − ∂

∂ + ∂

∂ ( )

⎛

⎝⎜ ⎞

⎠⎟ u v

x v v y

p y

v x

v y 1

3

nf nf

2 2

∂ α

∂ + ∂

∂ = ∂

∂ + ∂

∂ ( )

⎛

⎝⎜ ⎞

⎠⎟ u T

x v T y

T x

T

y 4

nf 2

2 2

2

In the Eq.(2),snfrepresents the electrical conductivity of na- noﬂuid. The Eqs.(1)–(4)should be written in the dimensionless form:

∂

∂ +∂

∂ =

( ) U

X V Y 0

5

∂

∂ + ∂

∂ = − ∂

∂ + ∂

∂ +∂

∂ −

( )

⎛

⎝⎜ ⎞

⎠⎟ U U

X V U Y

P X

U X

U

Y U

1 Re

Ha

Re 6

2 2

2

∂

∂ + ∂

∂ = − ∂

∂ + ∂

∂ +∂

∂ ( )

⎛

⎝⎜ ⎞

⎠⎟ U V

X V V Y

P Y

V X

V Y 1

Re 7

2 2

θ θ θ θ

∂

∂ + ∂

∂ = ∂

∂ + ∂

∂ ( )

⎛

⎝⎜ ⎞

⎠⎟ U X V

Y X Y

1

PrRe 8

2 2

In Eqs.(5)–(8), the dimensionless numbers are all written with respect to the properties of the nanoﬂuid, i.e.,Re= ^ρ^{nf c}_μ^{u h}

nf , Pr=^υ_α^nf

nf,

=B h(^σ_μ ) Ha ₀ ^nf^0.5

nf . The following dimensionless parameters are also used in Eqs.(5–8).

ρ

θ Δ Δ

= = = = =

= −

= ″

( ) X x

h Y y

h U u

u V v

u

p u T T

T

q h K

, , , , P ,

, T

9

c c nf c

c

nf

2

0

The thermo-physical properties of FMWNT are presented in Table 1; which implies the MMWNT-Water nanoﬂuid properties are determined according to the results of an empirical research work by Amrollahi and Rashidi[35]. This approach increases the adaption of present work results with those of experimental ones.

So in spite of previous papers concerned nanofluidflow, there are no utilized correlations for nanofluid properties at present article.

However for the electrical conductivity the Maxwell's model is used as follows[36,41]:

σ σ

σ σ ϕ

σ σ σ σ ϕ

= + ( − )

( + ) − ( − ) ( )

1 3 / 1

/ 2 / 1 10

nf f

s f

s f s f

wheresnf,sfandssrepresent the electrical conductivity of nano- ﬂuid,ﬂuid and solid nanoparticles, respectively.

3.2. Boundary conditions

It is clear that the non-slip boundary condition should be used at the macroscopic level. However, the slip boundary condition must be used in the slip ﬂow regime in a microchannel which indicates the presence of slippage (slip velocity) in theﬂuid particles on the microchannel wall. The slip velocity value is estimated by the following equation:

β

= ± ∂

∂ = ( )

⎛

⎝⎜ ⎞

⎠⎟

u u

y 11

s

y 0,h

where,βrepresents the slip coefﬁcient. The dimensionless form of the Eq.(11)on the wall is:

= ± *β ∂

∂ = ( )

⎛

⎝⎜ ⎞

⎠⎟

U U

Y 12

s

y 0,1

where, β* =_h^β is the dimensionless slip coefﬁcient. Other dimensionless boundary conditions are as follows:

θ

θ θ

= = = = ≤ ≤

= ∂

∂ = = ≤ ≤

= = ∂

∂ = = ≤ ≤

= = ∂

∂ = ∂

∂ =

= ≤ ≤ ( )

U V X Y

V U

X Y

V Y X

V k

k Y

X

1 , 0 at 0 , 0 1

0 ,

X 0 at 30 , 0 1

0 , U U ,

Y 0 at 0 , 0 30

0 , U U ,

Y 0 or

Y at

1 , 0 30 13

f nf s

s

Eqs.(5)–(8)were solved numerically under the aforementioned boundary conditions using the SIMPLE algorithm and a FORTRAN code. The local Nusselt number is:

Fig. 1. The schematic of the studied microchannel.

Table 1

Empirical thermo-physical properties of the FMWNT/water nanoﬂuid at 33°C for different values of nanoparticles volume fraction[35]^*.

wt% FMWNT/water ρ(Kg/m³) K (W/mK) μ(Pa s)

Pure water 996 0.62 7.6510⁴

Pure waterþ0.1% of FMWNT 1003 0.66 7.8110⁴

Pure waterþ0.2% of FMWNT 1006 0.71 7.9010⁴

*The speciﬁc heat capacity of nanoﬂuid almost equals to that of water as cp¼4182(J/Kg K).

Table 2

The values of averaged Nusselt number in different grids forβ*¼0.04, Re¼10, Ha¼0,ϕ¼0.1%, 0.2%.

90030 120040 150050

ϕ¼0.1% 2.359 2.355 2.353

ϕ¼0.2% 2.469 2.466 2.464

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( )

= ″

( ) − ( )

Nu = q h

T x T k 14

x y h

s c nf

0,

0

where, T x_s( ) is the microchannel wall surface temperature. The dimensionless form of the Eq.(14)is:

( ) =θ

( ) ( )

Nu X x

1

x 15

s

The averaged Nusselt number is calculated by the integration along the constantﬂux wall. It can be expressed as follows:

∫

= ( )

( ) Nu 1L Nu X x

0.7 d

m 16

L L 0.3 x

4. Numerical procedure, grid study and validations

A FORTRAN computer code is developed to solve the governing equations and related boundary conditions numerically. However the uniform staggered grids inﬁnite volume approach and power law scheme are applied to discretize the diffusion and convective terms of the governing equations. The SIMPLE algorithm based on the implicit scheme is employed to make couple the velocity and modiﬁed-pressure equations. Moreover the system of equations is solved by using TDMA method in this way[40].

Table 2 shows the averaged Nusselt number on the microchannel wall in three different grids (i.e., 90030, 120040 and 150050) with

ϕ

¼0.1, 0.2% and Re¼10. The Hartmann number and the slip coefﬁcient was assumed 0 and 0.04, respectively.

Negligible differences existed between the results of the 120040 Fig. 2.(a) Horizontal velocity proﬁles compared to those of Aminossadati et al.[34]

at different sections of the microchannel at Ha¼20, Re¼100, ϕ¼0.02.

(b) Comparison of the variation of Nusselt number with those of Aminossadati et al.

[34]along the microchannel wall at different Hartmann numbers for Re¼100 and ϕ¼0.02.

Fig. 3. (a) Centerline horizontal velocity variation in the microchannel compared to those of Raisi et al.[24]for nanoﬂuid and pureﬂuid for different values of Re andϕ.

(b) Local Nusselt number variation compared to those of Raisi et al.[24]for na- noﬂuid and pureﬂuid for different values of Re andϕ.

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and 150050 grids; therefore, the 120040 grid was selected for further calculations.

To validate the computer code, the results of this study were compared to those of Aminossadati et al.[34](Fig. 2a). They investigated theflow and heat transfer of a nanofluid in a microchannel under heatflux and a constant magneticfield. The initial andfinal parts of the microchannel were insulated, and the slip velocity was neglected.Fig. 2a shows the horizontal velocity profile at various sections along the microchannel. Obviously, the results were in good agreement with those of Aminossadati et al.

[34]. The local Nusselt number proﬁles were also compared to the results of reference [34] for different values of the Hartmann number (Fig. 2b).

Fig. 3a and b shows the velocity variations in the microchannel centerline and the local Nusselt number variations along the microchannel wall for different volume fractions and Reynolds numbers in comparison with Raisi et al.[24]. They compared the flow and heat transfer of a nanofluid in two different conditions of slip flow regime and the lack of slippage. There is also good

agreement between these results.

5. Results and discussions

Forced convective heat transfer of nanofluids composed of water and FMWNT carbon nanotube was numerically investigated in a two-dimensional microchannel. As shown inFig. 1, the microchannel was simultaneously exposed to a uniform magnetic field with strength ofB0and a constant heatflux ofq″₀.Here, the effects of pertinent parameters are explained.

5.1. Effect of dimensionless slip velocity

Fig. 4 shows the velocity and temperature proﬁles of the FMWNT nanoﬂuids on the microchannel vertical centerline (x¼L/

2) for

ϕ

¼0.2%, Ha¼0 and Re¼10 for different values of slip coefficient (β*). As can be seen inFig. 4, an increase in slip coefficient significantly affects the velocity profiles, reducing the maximum velocity, and increases the slip velocity in the vicinity of Fig. 4.Velocity and temperature profiles of the FMWNT nanofluid along the mi-

crochannel vertical centerline (x¼L/2) forϕ¼0.2%, Ha¼0 and Re¼10 for different values ofβ*.

Fig. 5.Dimensionless temperature proﬁles at different sections of the microchannel at Re¼10, Ha¼0, andϕ¼0.2% for different values ofβ*.

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the walls. In other words, theﬂow is accelerated near the wall while it decreases at the central region, as β*increases. In addition, the difference between velocities from β*_¼_{0.0 to} β*_¼_0.04 was greater compared to the difference fromβ*_¼_{0.04 to}β*_¼_0.08.

The dimensionless temperature profile evidently shows the in- sulation in the bottom walls. Increasing the fluid temperature caused by the heat flux from the upper wall is evident in this figure. Obviously, the slip coefficient did not significantly affect the temperature profiles.Fig. 5shows the dimensionless temperature profiles in developing region at different sections of the microchannel at Re¼10, Ha¼0, and

ϕ

¼0.2% for different values of β*_. The nanofluid temperature raised along the microchannel so that atX¼0.8 L,fluid temperature was almost entirely affected by the inletflux and was changed. It can be seen that, increasing the slip coefficient slightly changed the nanofluid temperature.

5.2. Effect of Hartmann number

Fig. 6shows the dimensionless horizontal velocity proﬁle in the vertical centerline of the microchannel at Ha¼0, Ha¼20 and Ha¼40, and

ϕ

¼0.2% and Re¼10 for the slip coefficients of β* =0.04and β* =0.08. The magneticfieldB0acting on the microchannel created Lorentz force is in the opposite direction of the X-axis. Therefore, an increased Hartmann number reduced the maximum dimensionless velocity and increased the velocity near the walls. The effect of the Hartmann number on the velocity profile is an interesting result of thisfigure, especially on the slip velocity adjacent to the wall. As can be seen in the diagram corresponding to β* =0.08, an increased Ha number increased the slip velocity on the (Y¼0) wall. In fact, due to the Lorentz force, all the incoming flow shift toward the microchannel wall. Fig. 7 shows the temperature profiles in the microchannel vertical centerline to illustrate the effect of the Hartmann number on theflow thermal domain for different Hartmann numbers for

ϕ

¼0.2% and Re¼10 and slip coefﬁcients of β* =0.0and β* =0.08. InFig. 7(in Fig. 6.Dimensionless horizontal velocity proﬁles along the vertical centerline of

the microchannel at different Hartmann numbers andϕ¼0.2% and Re¼10 for β*¼0.04 andβ*¼0.08.

Fig. 7.Dimensionless temperature proﬁles along the microchannel vertical centerline at different Hartmann numbers andβ* forϕ¼0.2% and Re¼10.

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Fig. 8. Streamlines atϕ¼0.2% andβ*¼0.08 for Ha¼0 (top) and Ha¼40 (bottom).

Fig. 9.Slip velocity,Us, along the upper wall of the microchannel at different values ofβ* forϕ¼0.2% and Re¼10 at Ha¼0, 40.

Fig. 10.Variations of local Nusselt number along the microchannel wall at different values of Reynolds number and Hartmann number forϕ¼0.2% andβ*¼0.08.

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contrast toFig. 6), in the diagrams corresponding to β* =0.0and β* =0.08, the effects of slip coefficient was negligible while the effects of Hartmann number was significant. This means that the higher Hartmann numbers increase the nanofluid temperature;

however, this increase was not considerable. The effect of Ha number at

ϕ

¼0.2% and β* =0.08for Ha¼0 and Ha¼40 on the streamlines can be clearly seen inFig. 8. At Ha¼0, the streamlines are perfectly parallel and uniform, while at Ha¼40, changes in streamlines is evident immediately after theflow enters the area under subjected to magneticfield. It can be clearly seen that, the higher Ha numbers increased the slip velocity at the walls and transferred theflow towards the walls. Moreover,Fig. 9illustrates the slip velocityUsalong the microchannel walls at different values of β*_for

ϕ

¼0.2% and Re¼10 at Ha¼0, 40. The slip velocity had its maximum value at the inlet. It decreased by increase inX, and ultimately, approached a constant value along the wall. In addition, it can be observed that the slip coefﬁcient increases the slip velocity. The diagram corresponding to Ha¼40 had interesting

The variations of the local Nusselt number was examined to further study the heat transfer mechanisms in the microchannel.

Fig. 10depicts the variations of NuXalong the microchannel wall for different values of Reynolds number and Hartmann number for

ϕ

¼0.2% and β* =0.08. Evidently, the Nusselt number is zero on the insulated walls. Along the constantflux upper wall, Nusselt number began from its maximum atX¼9, then, it experienced a steep decline, and finally, it approached a constant value at the outlet of the microchannel. It can be observed that an increased Reynolds number significantly increased the heat transfer rate through the walls. However, given that the value of the Reynolds number is small in nanofluids flowing in microchannels, this method is costly for increasing the heat transfer rate and is not practical. In the present research, the main part of the results corresponds to Re¼10, and the results corresponding to Re¼100 is reported in this section only. In addition, an increase in the Hartmann number strongly increases the local Nusselt profiles at developing region, although this increase continue to dwindle down along the microchannel. Thus, magneticfield effects on heat transfer rate at the fully developed region are lower than that at the developing region. This positive effect of the Ha number, however, was more significant at higher Re numbers. Interestingly, the diagrams corresponding to Ha¼20 and Ha¼40 were almost coincident and had a greater difference with the diagram corresponding to Ha¼0. This means that increasing the magneticfield strength in order to increase the heat transfer rate is applicable only in a limited range and is not economical beyond that range.

To further illustrate the heat transfer rate through the microchannel wall, the mean Nusselt number, Num, on the upper wall of the microchannel at different Ha,

ϕ

^,

β

^{* for Re}¼10 and Re¼100 is shown inFig. 11. The effects of various parameters, such as Re, Ha, β*and

ϕ

, is evident on Nu_m. Comparing the two separate diagrams inFig. 11shows that the values of Numwere higher at higher Re numbers. The positive and increasing effect of

ϕ

in increasing the heat transfer rate in each of these diagrams is also evident;

however, the increase is more significant at higher Reynolds numbers. Num increased by increasing the slip coefficient. The most interesting matter was the significant and increasing effect of the Hartmann number on increasing the heat transfer rate from the wall. It can be argued that the maximum heat transfer rate at the microchannel walls can be achieved at higher values of volume fraction and Reynolds number (as high as possible but not inter- fering with the physical conditions of the problem) and in a stronger magneticfield. An increased slip velocity coefficient further increased the heat transfer rate.

6. Conclusion

The forced convective heat transfer of a nanofluid composed of water and FMWNT particles in a developing two-dimensional microchannel was numerically investigated using a FORTRAN code. Microchannel walls were insulated in the initial section. In the next section, a uniform heatflux was imposed to thefluid from the upper wall. The microchannel had the slipflow regime. In the constantflux section, thefluid was exposed to a uniform magnetic field.

Fig. 11.Numon the upper wall of the microchannel at different Ha,ϕandβ* for Re¼10 (top) and Re¼100 (bottom).

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Magneticfield produced a Lorentz force in the opposite direction of theXaxis. Increasing the Hartmann number reduced the maximum velocity in the centerline of the microchannel and increased the velocity near the walls. Therefore, the fully developed velocity profile varied with the Hartmann number. The slip velocity had its maximum value at the inlet. It decreased by increase in X, and ultimately, approached a constant value along the wall. An increased slip coefficient increased the slip velocity of the fluid near the walls and reduced the maximum velocity at the microchannel center. An increased magnetic field strength (increased Ha) resulted in a significant increase in the slip velocity. This means that, immediately after theflow entered the region exposed to the magnetic field, the slip velocity reached Us¼0.72 from Us¼0.33 at

β

^*¼0.08.

Local Nusselt diagrams for Ha¼20 and Ha¼40 were almost coincident and had a more visible difference with the diagram of Ha¼0. This means that increasing the magneticfield strength in order to increase the heat transfer rate is applicable only in a limited range, and it is not effective beyond that range. It can be argued that, the maximum heat transfer rate from the microchannel walls occurred at higher volume fractions and Reynolds numbers an in a stronger magneticfield. An increased slip velocity coefficient further increased the heat transfer rate. In addition, it is observed that imposing the magneticfield is very effective at the thermally developing region. In contrast, the magneticfield effect at fully developed region is insignificant, especially at low Rey- nolds number.

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