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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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