ORIGINAL ARTICLE

MHD natural convection in a nanofluid filled inclined enclosurewith sinusoidal wall using CVFEM

M. Sheikholeslami • M. Gorji-Bandpy •

D. D. Ganji • Soheil Soleimani

Received: 17 November 2012 / Accepted: 11 December 2012

� Springer-Verlag London 2012

Abstract Magnetohydrodynamic flow in a nanofluid fil-

led inclined enclosure is investigated numerically using the

Control Volume based Finite Element Method. The cold

wall of cavity is assumed to mimic a sinusoidal profile with

different dimensionless amplitude, and the fluid in the

enclosure is a water-based nanofluid containing Cu nano-

particles. The effective thermal conductivity and viscosity

of nanofluid are calculated using the Maxwell–Garnetts

and Brinkman models, respectively. Numerical simulations

were performed for different governing parameters namely

the Hartmann number, Rayleigh number, nanoparticle

volume fraction and inclination angle of enclosure. The

results show that in presence of magnetic field, velocity

field retarded, and hence, convection and Nusselt number

decreases. At Ra = 103, maximum value of enhancement

for low Hartmann number is obtained at c = 0�, but for

higher values of Hartmann number, maximum values of

E occurs at c = 90�. Also, it can be found that for all

values of Hartmann number, at Ra = 104 and 105, maxi-

mum value of E is obtained at c = 60� and c = 0�,

respectively.

Keywords Magnetic field � Nanofluid � CVFEM �Sinusoidal wall � Inclined enclosure

List of symbols

a Dimensionless amplitude of the sinusoidal wall

Cp Specific heat at constant pressure

Gr Grashof number

g~ Gravitational acceleration vector

Ha Hartmann number ð¼ HBx

ffiffiffiffiffiffiffiffiffiffiffiffi

rf =lf

q

ÞH Dimensionless width of the enclosure

k Thermal conductivity

Nu Local Nusselt number

Pr Prandtl number (=tf/af)

T Fluid temperature

u, v Velocity components in the x-direction and

y-direction

U, V Dimensionless velocity components in X-direction

and Y-direction

x, y Space coordinates

X, Y Dimensionless space coordinates

Ra Rayleigh number (=gbfDT(H)3/aftf)

Greek symbols

a Thermal diffusivity

l Dynamic viscosity

t Kinematic viscosity

H Dimensionless temperature

r Electrical conductivity

q Fluid density

/ Volume fraction

c Inclined angle of enclosure

w and W Stream function and dimensionless stream

function

Subscripts

c Cold

h Hot

ave Average

loc Local

nf Nanofluid

f Base fluid

s Solid particles

M. Sheikholeslami (&) � M. Gorji-Bandpy �D. D. Ganji � S. Soleimani

Department of Mechanical Engineering,

Babol University of Technology, Babol, Iran

e-mail: mohsen.sheikholeslami@yahoo.com

123

Neural Comput & Applic

DOI 10.1007/s00521-012-1316-4

1 Introduction

Control Volume based Finite Element Method (CVFEM) is

a scheme that uses the advantages of both finite volume and

finite element methods for simulation of multi-physics

problems in complex geometries [1] and [2]. Soleimani

et al. [3] studied natural convection heat transfer in a semi-

annulus enclosure filled with nanofluid using the Control

Volume based Finite Element Method. They found that the

angle of turn has an important effect on the streamlines,

isotherms and maximum or minimum values of local

Nusselt number.

The geometrical pattern can be useful in improving the

heat transfer performance. Natural convection heat transfer

inside a wavy enclosure is one of the several devices

employed for enhancing the heat and mass transfer effi-

ciency. Flow and heat transfer from irregular surfaces are

often encountered in many engineering applications to

enhance heat transfer such as micro-electronic devices, flat-

plate solar collectors and flat-plate condensers in refriger-

ators, geophysical applications, electric machinery, cooling

system of micro-electronic devices, etc. Saidi et al. [4]

presented numerical and experimental results of flow over

and heat transfer from a sinusoidal cavity. They reported

that the total heat exchange between the wavy wall of the

cavity and the flowing fluid was reduced by the presence of

vortex. Das and Mahmud [5] conducted a numerical

investigation of natural convection in an enclosure con-

sisting of two isothermal horizontal wavy walls and two

adiabatic vertical straight walls. They reported that the

amplitude–wavelength ratio affected local heat transfer

rate, but it had no significant influence on average heat

transfer rate. Adjlout et al. [6] conducted a numerical study

on natural convection in an inclined cavity with hot wavy

wall and cold flat wall. One of their interesting findings was

the decrease in average heat transfer with the surface

waviness when compared with flat wall cavity.

Natural convection under the influence of a magnetic

field is of great importance in many industrial applications

such as crystal growth, metal casting and liquid metal

cooling blankets for fusion reactors. Pirmohammadi et al.

[7] considered the effect of magnetic field on convection

heat transfer inside a tilted square enclosure. Their study

showed that heat transfer mechanism and flow character-

istics inside the enclosure depend strongly upon both

magnetic field and inclination angle. Effect of static radial

magnetic field on natural convection heat transfer in a

horizontal cylindrical annulus enclosure filled with nano-

fluid is investigated numerically using the Lattice Boltz-

mann method by Ashorynejad et al. [8]. They concluded

that the average Nusselt number increases with increase of

nanoparticle volume fraction and Rayleigh number, while

it decreases with increase of Hartmann number. The

problem of laminar viscous flow in a semi-porous channel

in the presence of transverse magnetic field is studied by

Sheikholeslami et al. [9]. They investigated the effects of

some important parameters to evaluate how these param-

eters effect on fluid flow. Rudraiah et al. [10] investigated

numerically the effect of magnetic field on natural con-

vection in a rectangular enclosure. They found that the

magnetic field decreases the rate of heat transfer.

With the growing demand for efficient cooling systems,

particularly in the electronics industry, more effective

coolants are required to keep the temperature of electronic

components below safe limits. Use of nanofluids is a

potential solution to improve heat transfer. Khanafer et al.

[11] conducted a numerical investigation on the heat

transfer enhancement due to adding nanoparticles in a

differentially heated enclosure. They found that the sus-

pended nanoparticles substantially increase the heat trans-

fer rate at any given Grashof number. Bararnia et al. [12]

studied the natural convection in a nanofluid filled portion

cavity with a heated built in plate by lattice Boltzmann

method. Their results have been obtained for different

inclination angles and lengths of the inner plate. Ghasemi

et al. [13] presented the results of a numerical study on

natural convection heat transfer in an inclined enclosure

filled with a water–CuO nanofluid. They found that the heat

transfer rate is maximized at a specific inclination angle

depending on Rayleigh number and solid volume fraction.

Sheikholeslami et al. [14] performed a numerical analysis

for natural convection heat transfer of Cu–water nanofluid

in a cold outer circular enclosure containing a hot inner

sinusoidal circular cylinder in presence of horizontal

magnetic field using the Control Volume based Finite

Element Method. They concluded that in absence of

magnetic field, enhancement ratio decreases as Rayleigh

number increases, while an opposite trend was observed in

the presence of magnetic field. Sheikholeslami et al. [15]

studied the natural convection in a concentric annulus

between a cold outer square and heated inner circular

cylinders in presence of static radial magnetic field. They

reported that average the Nusselt number is an increasing

function of nanoparticle volume fraction as well as Ray-

leigh number, while it is a decreasing function of Hartmann

number. Recently, many researchers used different meth-

ods in order to simulate the effect of adding nanoparticle on

flow and heat transfer [16–19].

The present study represents the results of a numerical

investigation on natural convection of nanofluids in an

enclosure with a cold sinusoidal wall under laminar natural

using the Control Volume based Finite Element Method in

presence of magnetic field. The numerical investigation is

carried out for different governing parameters such as the

Hartmann number, Rayleigh number, nanoparticle volume

fraction and inclination angle of the enclosure.

Neural Comput & Applic

123

2 Geometry definition and boundary conditions

Schematic of the problem and the related boundary conditions

as well as the mesh of enclosure which is used in the present

CVFEM program are shown in Fig. 1. The enclosure has a

width/height aspect ratio of two. The two sidewalls with

length H are thermally insulated, whereas the lower flat and

upper sinusoidal walls are maintained at constant temperatures

Th and Tc, respectively. Under all circumstances, Th [Tc

condition is maintained. The shape of the upper sinusoidal

wall profile is assumed to mimic the following pattern

Y ¼ H � faðH þ sinðpx� p=2ÞÞg ð1Þ

where a is the dimensionless amplitude of the sinusoidal wall.

It is also assumed that the uniform magnetic field

(B~ ¼ Bxe~x þ Bye~y) of constant magnitude B ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B2x þ B2

y

q

is

applied, where e~x and e~y are unit vectors in the Cartesian

coordinate system. The orientation of the magnetic field forms

an angle c with horizontal axis such that c = Bx/By. The

electric current J and the electromagnetic force F are defined

by J ¼ rðV~ � B~Þ and F ¼ rðV~ � B~Þ � B~, respectively.

3 Mathematical modeling and numerical procedure

3.1 Problem formulation

The flow is two dimensional, laminar and incompressible.

The radiation, viscous dissipation, induced electric current

and Joule heating are neglected. The magnetic Reynolds

number is assumed to be small, so that the induced mag-

netic field can be neglected compared to the applied

magnetic field. The flow is considered to be steady, two

dimensional and laminar. Neglecting displacement cur-

rents, induced magnetic field, and using the Boussinesq

approximation, the governing equations of heat transfer

and fluid flow for nanofluid can be obtained as follows:

ou

oxþ ov

oy¼ 0 ð2Þ

uou

oxþ v

ou

oy¼ � 1

qnf

oP

oxþ tnf

o2u

ox2þ o2u

oy2

� �

þ rnfB2

qnf

ðv sin k cos k� u sin2 kÞ ð3Þ

uov

oxþ v

ov

oy¼ � 1

qnf

oP

oyþ tnf

o2v

ox2þ o2v

oy2

� �

þ bnfgðT � TcÞ

þ rnfB2

qnf

ðu sin k cos k� v cos2 kÞ

ð4Þ

uoT

oxþ v

oT

oy¼ anf

o2T

ox2þ o2T

oy2

� �

ð5Þ

where the effective density (qnf) and heat capacitance

ðqCpÞnf of the nanofluid are defined as [14]:

qnf ¼ qf ð1� /Þ þ qs/ ð6Þ

qCp

� �

nf¼ qCp

� �

fð1� /Þ þ qCp

� �

s/ ð7Þ

where / is the solid volume fraction of nanoparticles.

Thermal diffusivity of the nanofluids is

anf ¼knf

ðqCpÞnf

ð8Þ

and the thermal expansion coefficient of the nanofluids can

be determined as (see [3])

bnf ¼ bf ð1� /Þ þ bs/ ð9Þ

The dynamic viscosity of the nanofluids given by

Brinkman [14] is

lnf ¼lf

ð1� /Þ2:5ð10Þ

The effective thermal conductivity of the nanofluid can

be approximated by the Maxwell–Garnetts (MG) model as

[14]:

knf

kf¼ ks þ 2kf � 2/ðkf � ksÞ

ks þ 2kf þ /ðkf � ksÞð11Þ

and the effective electrical conductivity of nanofluid was

presented by Maxwell [14] as below:

rnf

rf¼ 1þ 3ðrs=rf � 1Þ/

ðrs=rf þ 2Þ � ðrs=rf � 1Þ/ ð12Þ

The stream function and vorticity are defined as:

u ¼ owoy; v ¼ � ow

ox; x ¼ ov

ox� ou

oyð13Þ

The stream function satisfies the continuity Eq. (2). The

vorticity equation is obtained by eliminating the pressure

between the two momentum equations, that is, by taking

y-derivative of Eq. (3) and subtracting from it the

x-derivative of Eq. (4). This gives:

owoy

oxox� ow

ox

oxoy¼ tnf

o2xox2þ o2x

oy2

� �

þ bnfgoT

ox

� �

þ rnfB2

qnf

� dv

dysin k cos kþ du

dysin2 k

�

þ du

dxsin k cos k� dv

dxcos2 k

�

ð14Þ

owoy

oT

ox� ow

ox

oT

oy¼ anf

o2T

ox2þ o2T

oy2

� �

ð15Þ

Neural Comput & Applic

123

o2wox2þ o2w

oy2¼ �x ð16Þ

By introducing the following non-dimensional variables:

X ¼ x

L; Y ¼ y

L; X ¼ xL2

af; W ¼ w

af; H ¼ T � Tc

Th � Tc;

U ¼ uL

af; V ¼ vL

afð17Þ

where in Eq. (17) L = rout - rin = rin. Using the

dimensionless parameters, the equations now become:

oWoY

oXoX� oW

oX

oXoY¼ Prf

1� /ð Þ2:5 1� /ð Þ þ / qs

qf

� �

2

4

3

5

o2XoX2þ o2X

oY2

� �

þ Raf Prf 1� /ð Þ þ /bs

bf

" #

oHoX

� �

þHa2Prf 1þ3 rs

rf� 1

� �

/

rs

rfþ 2

� �

� rs

rf� 1

� �

/

0

@

1

A

1

1�/ð Þþ/qs

qf

!

�dV

dYtankþ dU

dYtan2 kþ dU

dXtank� dV

dX

� �

ð18Þ

oWoY

oHoX�oW

oX

oHoY¼

knf

kf

1�/ð Þþ/ qCpð ÞsqCpð Þf

� �

2

4

3

5

o2HoX2þo2H

oY2

� �

ð19Þ

o2WoX2þ o2W

oY2¼ �X ð20Þ

where Raf = gbfL3(Th - Tc)/(aftf) is the Rayleigh number

for the base fluid, Ha ¼ LBx

ffiffiffiffiffiffiffiffiffiffiffiffi

rf =lf

q

is the Hartmann

Fig. 1 a Geometry and the

boundary conditions with

(b) the mesh of enclosure

considered in this work;

c Comparison of the

temperature on axial midline

between the present results and

numerical results obtained by

Khanafer et al. [11] for

Gr = 104, / = 0.1 and

Pr = 6.2 (Cu - Water)

Neural Comput & Applic

123

number, and Prf = tf/af is the Prandtl number for the base

fluid. The thermo physical properties of the nanofluid are

given in Table 1 [14]. The boundary conditions as shown

in Fig. 1 are:

H ¼ 1:0 on the hot wall

H ¼ 0:0 on the cold wall

oH=on ¼ 0:0 on the two other insulation boundaries

W ¼ 0:0 on all solid boundaries

ð21ÞThe values of vorticity on the boundary of the enclosure

can be obtained using the stream function formulation and

the known velocity conditions during the iterative solution

procedure.

The local Nusselt number of the nanofluid along the hot

wall can be expressed as:

Nuloc ¼knf

kf

oHon

hot wall

ð22Þ

where n is normal to surface. The average Nusselt number

on the hot wall is evaluated as:

Nuave ¼1

2

Z

2

0

NulocdS: ð23Þ

3.2 Numerical procedure

A control volume finite element method is used in this work.

The building block of the discretization is the triangular

element, and the values of variables are approximated with

linear interpolation within the elements. The control vol-

umes are created by joining the center of each element in the

support to the mid points of the element sides that pass

through the central node i which creates a close polygonal

control volume (see Fig. 1b). The set of governing equations

is integrated over the control volume with the use of linear

interpolation inside the finite element and the obtained al-

gebretic equations are solved by the Gauss–Seidel Method.

A FORTRAN code is developed to solve the present prob-

lem using a structured mesh of linear triangular.

4 Grid testing and code validation

To allow grid-independent examination, the numerical pro-

cedure has been conducted for different grid resolutions.

Table 2 demonstrates the influence of number of grid points

for the case of Ra = 105, Ha = 100, a = 0.3, c = 0�,

/ = 0.06 and Pr = 6.2. The present code is tested for grid

independence by calculating the average Nusselt number on

the hot wall. In harmony with this, it was found that a grid

size of 81 9 161 ensures a grid-independent solution. The

convergence criterion for the termination of all computa-

tions is:

maxgrid

Cnþ1 � Cn

Cnþ1

� 10�7 ð24Þ

where n is the iteration number, and C stands for the

independent variables (U, V, T). The results have been

validated for the natural convection flow in an enclosed

cavity filled by a pure fluid, as reported by Khanafer et al.

Table 1 Thermo physical properties of water and nanoparticles [14]

q (kg m-3) Cp (J kg-1 k-1) k (W m-1 k-1) b (K-1) r (X m-1)

Pure water 997.1 4,179 0.613 21 9 10-5 0.05

Copper (Cu) 8,933 385 401 1.67 9 10-5 5.96 9 107

Table 2 Comparison of the average Nusselt number Nuave for different grid resolution at Ra = 105, Ha = 100, a = 0.3, c = 0�, / = 0.06 and

Pr = 6.2

Mesh size

41 9 81 51 9 101 61 9 121 71 9 141 81 9 161 91 9 181 101 9 201

2.206345 2.162699 2.15851 2.155584 2.153422 2.15176 2.150442

Table 3 Comparison of the present results with previous works for

different Rayleigh numbers when Pr = 0.7

Ra Present Khanafer et al. [11] De Vahl Davis [20]

103 1.1432 1.118 1.118

104 2.2749 2.245 2.243

105 4.5199 4.522 4.519

Table 4 Average Nusselt number versus at different Grashof number

under various strengths of the magnetic field at Pr = 0.733

Ha Gr = 2 9 104 Gr = 2 9 105

Present Rudraiah et al. [10] Rudraiah Rudraiah et al. [10]

0 2.5665 2.5188 5.093205 4.9198

10 2.26626 2.2234 4.9047 4.8053

50 1.09954 1.0856 2.67911 2.8442

100 1.02218 1.011 1.46048 1.4317

Neural Comput & Applic

123

310Ra = 410Ra = 510Ra =

γ= 0

ο

Ha=

0

max 0.190Ψ = max 4.523Ψ = max 21.35Ψ =

Ha

= 1

00

max 0.006548Ψ = max 0.06763Ψ = max 1.012Ψ =

γ= 3

0ο

Ha=

0

max 0.5215Ψ = max 4.73Ψ = max 24.31Ψ =

Ha=

100

max 0.0522Ψ = max 0.5372Ψ = max 5.882Ψ =

Fig. 2 Isotherms (up) and

streamlines (down) contours for

different values of Rayleigh

number, Hartmann number and

inclination angle at a = 0.3,

/ = 0.06 and Pr = 6.2

Neural Comput & Applic

123

[11] and De Vahl Davis [20] to observe a good agreement;

see Table 3. Another test for validation of the current code

was performed for the case of natural convection in a

square enclosure in the presence of magnetic field. In this

test case, the average Nusselt number using different Gr

and Ha number has been compared with those obtained by

Rudraiah et al. [10] as shown in Table 4. In Fig. 1c, the

present computation also is validated against the results of

Khanafer et al. [11] carried for natural convection in an

enclosure filled with Cu–water nanofluid for different

Grashof numbers. All of the previous comparisons indicate

the accuracy of the present code.

5 Results and discussion

Numerical simulations of natural convection nanofluid flow

in an enclosure with one sinusoidal wall in presence of

magnetic flied were performed. Calculations are made for

various values of Hartmann number (Ha = 0, 20, 60 and

100), Rayleigh number (Ra = 103, 104 and 105), volume

fraction of nanoparticles (/ = 0, 2, 4 and 6 %) and incli-

nation angle (c = 0�, 30�, 60� and 90�) at constant

dimensionless amplitude of the sinusoidal wall (a = 0.3)

and Prandtl number (Pr = 6.2).

Figures 2 and 3 show isotherms (up) and streamlines

(down) contours for different values of Rayleigh number,

Hartmann number and inclination angle. The figures show

that the absolute value of stream function increases as

Rayleigh number enhances, and it decreases as Hartmann

number increases. Also, it can be seen that maximum

values of Wmaxj j are observed at c = 90� for Ra = 103, 104

while it is obtained at c = 60� for Ra = 105. At Ra = 103,

for all inclination angles, the isotherms are nearly smooth

curves and nearly parallel to each other which follow the

geometry of the sinusoidal surfaces; this pattern is the

characteristic of conduction dominant mechanism of heat

transfer at low Rayleigh numbers. At c = 0�, two counter

rotating vortices cores are observed. This bi-cellular flow

pattern divides the cavity into two symmetric parts respect

to vertical centerline of the enclosure. By increasing

inclination angle, at first, the upper vortex becomes stron-

ger, and then, at c = 90�, streamlines become symmetric

with respect to horizontal centerline of the enclosure. In

general, as Rayleigh number increases, the buoyancy-dri-

ven circulations inside the enclosure become stronger as it

is clear from greater magnitudes of stream function, and

more distortion appears in the isotherms. When the mag-

netic field is imposed on the enclosure, the velocity field

suppressed owing to the retarding effect of the Lorenz

force. So, intensity of convection weakens significantly.

310Ra = 410Ra = 510Ra =

γ= 6

0ο

Ha=

0

max 0.6786Ψ = max 5.484Ψ = max 23.43Ψ =

Ha

= 1

00

max 0.08917Ψ = max 0.9019Ψ = max 7.834Ψ =

γ= 9

0ο

Ha=

0

max 0.6694Ψ = max 4.979Ψ = max 19.11Ψ =

Ha=

100

max 0.1026Ψ = max 1.019Ψ = max 7.858Ψ =

Fig. 3 Isotherms (up) and streamlines (down) contours for different

values of Rayleigh number, inclination angle and inclination angle at

a = 0.3, / = 0.06 and Pr = 6.2

Neural Comput & Applic

123

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 4 Effects of the

nanoparticle volume fraction,

Hartmann number, inclination

angle and Rayleigh number on

Local Nusselt number when a,

b Ra = 105, c = 90�; c,

d Ra = 105, / = 0.06; e,

f Ra = 105, / = 0.06; g, h/ = 0.06, c = 90�

Neural Comput & Applic

123

The braking effect of the magnetic field is observed from

the maximum stream function value. Increase of Hartmann

number merges two vortexes into one except for c = 0�.

Also, magnetic field disappears the thermal plume over the

hot wall and makes the isotherms parallel to each other due

to domination of conduction mode of heat transfer.

Figure 4 depicts the effects of the nanoparticle volume

fraction, Hartmann number, inclination angle and Rayleigh

number on Local Nusselt number. Generally, increasing

the nanoparticles volume fraction and Rayleigh number

leads to an increase in local Nusselt number. In absence of

magnetic field, at c = 0�, the local Nusselt profile is

symmetric respect to the vertical center line of the enclo-

sure. But, in presence of magnetic field, because of dom-

ination of conduction mechanism, maximum value of local

Nusselt number occurs at vertical center line. At c = 90�,

the local Nusselt decreases with increase of S, and

increasing Hartmann number leads to decrease in Nusselt

number. When Ha = 0, the number of extremum in the

local Nusselt number profile is corresponding to exist of

thermal plume.

Effects of the Hartmann number, Rayleigh number and

inclination angle on the average Nusselt number are shown

in Fig. 5a, b. Generally, the average Nusselt number

increases with increase of Rayleigh number, while it

decreases as Hartmann number increases. At Ra = 105, in

absence of magnetic field, maximum value of average

Nusselt number is obtained at c = 0�, but for higher values

of Hartmann number, maximum value of Nuave occurs at

c = 90�.

To estimate the enhancement of heat transfer between

the case of / = 0.06 and the pure fluid (base fluid) case,

the enhancement is defined as:

E ¼ Nuð/ ¼ 0:06Þ � NuðbasefluidÞNuðbasefluidÞ � 100 ð25Þ

The effects of Hartmann number, Rayleigh number

and inclination angle on heat transfer enhancement ratio

are shown in Fig. 5c. At Ra = 103, maximum value of

enhancement for low Hartmann number is observed at

c = 0�, but for Ha [ 20, maximum values of it occur for

c = 90�. Also, it can be seen that for Ra = 104 and 105,

Fig. 5 Effects of the Hartmann number, Rayleigh number and

inclination angle on the average Nusselt number when a Ra = 105;

b c = 90� at a = 0.3 and / = 0.06; c Effects of Hartmann number,

Rayleigh number and inclination angle on the ratio of heat transfer

enhancement due to addition of nanoparticles a = 0.3

Neural Comput & Applic

123

maximum value of E is obtained for c = 60� and c = 0�,

respectively. It is an interesting observation that at

Ra = 105 the enhancement in heat transfer for case of

c = 0� increases with increase of Hartmann number

when Ha \ 60, while opposite trend is observed for

Ha [ 60. For other value of inclination angles,

enhancement in heat transfer is an increasing function

of Hartmann number.

6 Conclusions

In this study, Control Volume based Finite Element

Method is used to solve the problem of heat and fluid flow

of a nanofluid in an enclosure with a cold sinusoidal wall

under laminar natural. The effects of Hartmann number,

Rayleigh number, volume fraction of nanoparticles and

inclination angle on the flow and heat transfer character-

istics have been investigated. The results indicate that

Hartmann number and the inclination angle of the enclo-

sure can be control parameters at different Rayleigh num-

ber. In presence of magnetic field, velocity field retarded,

and hence, convection and Nusselt number decreases. The

average Nusselt number increases with increase of Ray-

leigh number and nanoparticle volume fraction, while it

decreases as Hartmann number increases. For high value of

Rayleigh number, the enhancement in heat transfer

increase with increase of Hartmann number expect for the

case of c = 0� in which Ha = 60 role as critical Hartmann

number.

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