Magnetic field effect on mixed convection flow in a nanofluid under convective boundary condition

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 6, Issue 4, April (2015), pp. 72-86© IAEME
72
MAGNETIC FIELD EFFECT ON MIXED CONVECTION
FLOW IN A NANOFLUID UNDER CONVECTIVE
BOUNDARY CONDITION
1
M Subhas Abel, 2
Pamita Laxman Rao, 3
Jagadish V.Tawade*
1, 2
Department of Mathematics, Gulbarga University, Gulbarga-585106, Karnataka, India
3*
Department of Mathematics, Bheemanna Khandre Institute of Technology, Bhalki-585328
ABSTRACT
An analysis is carried out to investigate the influence of the prominent magnetic effect on
mixed convection heat and mass transfer in the boundary layer region of a semi-infinite vertical flat
plate in a nanofluid under the convective boundary conditions. The transformed boundary layer,
ordinary differential equations are solved numerically using Runge-Kutta Fourth order method. A
wide range of parameter values is chosen to bring out the effect of Magnetic field parameter on the
mixed convection process with the convective boundary condition. The effect of mixed convection,
Magnetic field and Biot parameters on the flow, heat and mass transfer coefficients is analyzed. The
numerical results obtained for the velocity, temperature and volume fraction profiles are presented
graphically and discussed.
Keywords: Mixed Convection, Nanofluid, Magnetic Effect, Convective Boundary Condition,
Numerical Solution.
1. INTRODUCTION
A Nanofluid is a fluid containing nanometer sized particles, called nanoparticles. These
fluids are engineered colloidal suspensions of nanoparticles in a base fluid. The nanoparticles used in
nanofluids are typically made of metals, oxides, carbides, or carbon nanotubes. Common base fluids
include water, ethylene glycol and oil. Nanofluids have novel properties that make them potentially
useful in many applications in heat transfer, including microelectronics, fuel cells pharmaceutical
processes, and hybrid-powered engines, engine cooling/vehicle thermal management, domestic
refrigerator, chiller, heat exchanger, in grinding, machining and in boiler flue gas temperature
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND
TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 6, Issue 4, April (2015), pp. 72-86
© IAEME: www.iaeme.com/IJMET.asp
Journal Impact Factor (2015): 8.8293 (Calculated by GISI)
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IJMET
© I A E M E

73
reduction. They exhibit enhanced thermal conductivity and the convective heat transfer coefficient
compared to the base fluid. Knowledge of the rheological behavior of nanofluids is found to be very
critical deciding their suitability for convective heat transfer applications.
Merkin [1] investigated the mixed convection boundary layer flow on a semi-infinite vertical
flat plate when the buoyancy forces aid, and oppose the development of the boundary layer. Ding et
al. [2] Nanofluid are also important for the production of nanostructured materials, for the
engineering of complex fluids, as well as for cleaning oil from surfaces due to their excellent wetting
and spreading behavior. After the poineering work by Sakiadis [3], a large amount of literature is
available on boundary layer flow of Newtonian and non-Newtonian fluids over linear and nonlinear
stretching surface. A detailed review of mixed convective heat and mass transfer can be found in the
book by Bejan [4]. Recently, Subhashini et al. [5] discussed the simultaneous effects of thermal and
concentration diffusion on a mixed convection boundary layer flow over a permeable surface under
convective surface boundary condition. The general topic of heat transfer in nanofluids has been
surveyed in a review article by Das and Choi [6] and a book by Das et al. [7]. Buongiorno [8] noted
that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a
relative velocity. He considered in turn seven slip mechanism: inertia, Brownian diffusion,
thermophoresis, diffusiophoresis, Magnus effect, fluid drainage and gravity setting. He concluded
that in the absence of turbulent effects, it is the Brownian diffusion and the thermophoresis that will
be important. Buongiorno proceeded to write down conservation equations based on these two
effects.
The problem of natural convection in a regular fluid past a vertical plate is a classical
problem first studied theoretically by E. Pohlhausen in contribution to an experimental study by
Schmidt and Beckmann[9]. unfortunately the boundary layer scaling used by early researchers and
text book authors did not properly incorporate the papers by Kuiken [10, 11]. An extension to the
case of heat and mass transfer was made by Khair and Bejan [12]. Makinde, and Aziz [13]
considered to study the effect of a convective boundary condition on boundary layer flow, heat and
mass transfer and nanoparticle fraction over a stretching surface in a nanofluid. Kuznetsov and Nield
[14] studied, the natural convection boundary layer flow, heat and mass transfer of nanofluid past a
vertical plate. The transformed non-linear ordinary differential equations governing the flow are
solved numerically by the Runge-Kutta Fourth order method.
These authors discussed about the convective heat transport in nanofluids. They studied
natural convective flow of nanofluids over a vertical plate and their similarity analysis is identical
with four parameters governing the transport process, namely a Lewis number Le , a Buoyancy- ratio
Number Gr , a Brownian motion number Nb and a thermophoresis number Nt , A.V. Kuznetsov, D.
A. Nield [2010].
Ali et al. [2011] included the idea of induced magnetic field to the problem of Ashraf [2011]
and analyzed MHD stagnation-point flow and heat transfer towards stretching sheet with induced
magnetic field. The application of Magnetic Field to convection process will play as a control factor
in the convection by damping both the flow and temperature oscillation material manufacturing
fields.
Motivated by the above referenced work, and the vast possible industrial applications, it is
paramount interest to consider the effect of magnetic parameter on mixed convective flow along a
vertical plate in a nanofluid under the convective boundary condition. A similarity solution is
presented. This solution is depends on Prandtl number Pr , a Lewis number Le, a Brownian motion
number Nb , a nanoparticle buoyancy ratio Nr , thermophoresis number Nt , Biot number Bi ,
mixed convection number λ and a magnetic number M .The dependency of the Skin friction
coefficient, Nusselt number, nanoparticle Sherwood number and regular Sherwood number on these
six parameters is numerically investigated. Consideration of the nanofluid and the convective
boundary conditions enhanced the number of non-dimensional parameters considerably. The effect

74
of mixed convection, Magnetic and Biot parameters on the physical quantities of the flow, heat and
mass transfer coefficients are analyzed. To examine the convergence of the numerical code written to
solve the present problem, we compare the present result for the clear fluid mixed convection results
with previously published works with convective boundary conditions and the comparison shows
that the results are in very good agreement.
2. MATHEMATICAL FORMULATION
Choose the coordinate system such that the x-axis is along the vertical plate and y-axis
normal to the plate. The physical model and coordinate system are shown in fig. 1. Consider the
steady laminar two-dimensional mixed convection heat and mass transfer along a flat vertical surface
embedded in a nanofluid having T
∞
and φ
∞
as the temperature and nanoparticle volume fraction
respectively in the ambient medium. Also assume that a free stream with uniform velocity u
∞
goes
past the flat plate. The plate is either heated or cooled from left by convection from a fluid of
temperature T
f
with T T
f
>
∞
corresponding to a heated surface (assisting flow) and T T
f
>
∞
corresponding to a cooled surface (opposing flow) respectively. On the wall the nanoparticle volume
fraction is taken to be constant and is given by
w
φ respectively.
By employing Oberbeck-Boussinesq approximation, making use of the standard boundary layer
approximation and eliminating pressure, the, the governing equations for the nanofluid are given by
Fig: 1
0
u v
x y
∂ ∂
+ =
∂ ∂ (1)
2
(1 )[ ( )]
2
u u u
u v g T T
f f Tx y y
ρ µ ρ φ β
 ∂ ∂ ∂
+ = + − − ∞ ∞ ∞ ∞∂ ∂  ∂

75
2
0
( ) ( )p f
B u
g
σ
ρ ρ φ φ
ρ∞ ∞− − − − (2)
22
2
DT T T T TTu v D
m Bx y y y T yy
φ
α τ
  ∂ ∂ ∂ ∂ ∂ ∂ + = + +  
 ∂ ∂ ∂ ∂ ∂ ∂ ∞ 
(3)
2 2
2 2
D TTu v D
Bx y Ty y
φ φ φ∂ ∂ ∂ ∂
+ = +
∂ ∂ ∂ ∂∞ (4)
where u and v are the velocity components along the x and y axes, respectively, T is the
temperature, φ is the nanoparticle volume fraction, g is the gravitational acceleration,
/( )k c
m f
α ρ= is the thermal diffusivity of the fluid, / fν µ ρ ∞= is the kinematic viscosity
coefficient and ( ) /( )p fc cτ ρ ρ= . Further, fρ ∞ is the density of the base fluid and , , , Tkρ µ β and
cβ are the density , viscosity, thermal conductivity, volumetric thermal expansion coefficient and
volumetric solutal expansion coefficient of the nanofluid, while is pρ the density of the
nanoparticles, ( )fcρ is the heat capacity of the fluid and ( )pcρ is the effective heat capacity of the
nanoparticle material. The coefficients that appear in Eqs. (3) and (4) are the Brownian diffusion
coefficient BD , the thermophoretic diffusion coefficient TD . For, details of the derivation of
equations (1) - (4), one can refer the papers by Buonggiorno [8] and Nield and Kuznetsov ([14, 15]).
The boundary conditions are
0, 0, ( ), w
T
u v k h T T
f fy
φ φ
∂
= = − = − =
∂
at y=0 (5a)
u u∞→ , T T∞→ , φ φ∞→ as y → ∞ (5b)
here, fh is the convective heat transfer coefficient and the subscripts w and ∞ indicate the
conditions at the surface and at the outer edge of the boundary layer respectively.
In view of the continuity equation (1), we introduce the stream function ψ by
u
y
ψ∂
=
∂
,v
x
ψ∂
= −
∂ (6)
Substituting (6) in eqs. (2)-(4) and then using the following non-dimensional transformation
/u yη ν∞= , ( )u xfψ ν η∞= , (7a)

76
( ) ,
f
T T
T T
θ η ∞
∞
−
=
−
( )
w
φ φ
γ η
φ φ
∞
∞
−
=
−
(7b)
with the local Reynolds's number Rex
u x
υ
∞
= , we get the transformed system of ordinary
differential equations as
2
''' '' ' '
[ ] 0f ff f Nry Mfλ θ+ − + − − = (8)
2
'' ' ' ' '1
0f Nb y Nt
Pr
θ θ θ θ+ + + =
(9)
'' ' ''
0
Nt
y Lefy
Nb
θ+ + =
(10)
where the primes indicate differentiation with respect toη . In usual notations,
m
Pr
ν
α
= is the
Prandtl number and
B
Le
D
υ
= is the Lewis number.
( )( )
( )(1 )
p f w
f T f
Nr
T T
ρ ρ φ φ
ρ β φ
∞ ∞
∞ ∞ ∞
− −
=
− −
is the nanofluid
buoyancy ratio. Further,
( ) ( )
( )
p B w
f
c D
Nb
c
ρ φ φ
ρ ν
∞−
= is the Brownian motion parameter,
( ) ( )
( )
p T f
f
c D T T
Nt
c T
ρ
ρ υ
∞
∞
−
= is the thermophoresis parameter, (1 ) ( )x T fGr g T Tφ β∞ ∞= − − is the
local Grashof number and 2
xGr
u x
λ
∞
= the mixed convection parameter. Finally =
∞ ∞
is the
magnetic number
Boundary conditions (5) in terms of , ,f θ γ become
' '
0: (0) 0, (0) 0, (0) [1 (0)], (0) 1f f Biη θ θ γ= = = = − − = (11a)
'
: ( ) 1, ( ) 0, ( ) 0fη θ γ→ ∞ ∞ → ∞ → ∞ → (11b)
where Bi=
2c
k u
ν
∞
is the Biot number. This boundary conditions will be free from the local variable
x when we choose
1/ 2
fh cx−
= . The Biot number Bi is a ratio of the internal thermal resistance of
the plate to the boundary layer thermal resistance of the hot fluid at the bottom of the surface.
It is important to note that this boundary value problem reduces to the classical problem of flow
and heat and mass transfer due to the Blasius problem of flow when Nb and Nt are zero. Most
nanofluids examined to date, have large values for the Lewis number Le>1.For water nanofluids at
room temperature with nanoparticles of 1-100nm diameters, the Brownian diffusion coefficient BD

77
ranges from
4
4 10−
× to
12
4 10−
× 2
/m s−
. Furthermore, the ratio of the Brownian diffusivity
coefficient to the thermophoresis coefficient for particles with diameters of 1-100nm can be varied in
the ranges of 2-0.02 for alumina, and from 2 to 20 for copper nanoparticles, Hence, the variation of
the non-dimensional parameters of nanofluids in the present study is considered in the mentioned
range.
3. SKIN FRICTION, HEAT AND MASS TRANSFER COEFFICIENTS
The primary objective of this study is to estimate the skin friction coefficientC
f , Nusselt
number Nu and the nanoparticle Sherwood number NSh . These parameters characterize the surface
drag, the wall heat and nanoparticles mass transfer respectively.
The shearing stress, local heat and local nanoparticle mass from the vertical plate can be obtained
from
,
0 0
u T
q k
w wy yy y
τ µ
   ∂ ∂
= = −   ∂ ∂   = =
and
0
q D
n B y y
φ ∂
= −  ∂  =
(12)
The non -dimensional shear stress 2
wC
f
u
f
τ
ρ
=
∞ ∞
, the Nusselt number
( )
q x
wNu
x k T T
f
=
−
∞
and the nanoparticle Sherwood number
( )
q x
nNSh
x D
B w
φ φ
=
−
∞
, are given by
1/ 2 ''(2Re ) (0),C f
f x
=
1/ 2 '(Re / 2) (0),Nu
x x
θ− = −
1 / 2 '
(R e / 2 ) (0 )x xN S h γ−
= −
Effect of the various parameters involved in the investigation on these coefficients is
discussed in the following section.
4. RESULT AND DISCUSSION
The resulting transport Eqns. 8-11 are non-linear, coupled, ordinary differential equations,
which possess no closed-form solution. Therefore, these are solved numerically subject to the
boundary conditions given by Eqn. 12. The computational domain in the η -direction was made up
of 196 non-uniform grid points. It is expected that most changes in the dependent variable occur in
the region close to the plate where viscous effect dominate. However, small changes in the
dependent variables are expected far away from the plate surface. For these reasons, variable step
sizes in the η -direction are employed. The initial step size 1η∆ and growth factor K* employed
such that 1i iKη η∗
+∆ = ∆ (where the subscript i indicates the grid location) were
3
10−
and 1.0375
respectively. These values were found (by performing many numerical experimentations) to give
accurate and grid-independent solutions. The solution convergence criterion employed in the present
work was based on the difference between the values of the dependent variables at the current and

78
the previous iteration. When this difference reached 5
10−
, the solution was assumed converged and
the iteration process was terminated.
To have a better understanding of the flow characteristics, numerical results for the velocity,
temperature, volume fraction are calculated for different sets of values of the parameters
, , , , , ,M Le Nb Nt Nr Biλ also, the effect of these parameters on skin fraction, non-dimensional heat
and nanoparticle mass coefficients is discussed.
Fig. 2(a)-(c) show the non-dimensional velocity, temperature and volume fraction profiles
for various values of magnetic along with varying values of the Lewis number Le for a given
0.2, 0.2, 0.2,Pr 1.0, 0.4, 0.01Nt Nb Nr Biλ= = = = = = . The magnetic number M accounts for the
additional mass diffusion due to the temperature gradients. It is noticed from Fig. 2 that an increase
in the magnetic number resulted in an increase in the velocity while decrease in temperature and
nanopartical volume fraction is noted within the boundary layer. The present analysis shows that the
flow field is appreciably influenced by magnetic parameter. It is clear from these figures that an
increase in Lewis number Le increased the momentum boundary layer thickness, while reduction in
the thermal and nanopartical volume fraction boundary layer thickness is noted.
The non-dimensional velocity for different values of Biot number Bi with fixed values of
the other parameters is plotted in Fig. 3(a). Increased convective heating associated with an increase
in is seen to thicken the momentum boundary layer. A reverse trend is seen in the case of
nanofluid buoyancy ratio . Given that convective heating increases with Biot number simulates
the isothermal surface, which is clearly seen from Fig.3 (b), where, In fact, a high Biot number
indicates higher internal thermal resistance of the plate than the boundary layer thermal resistance.
As a result, an increase in the Biot number leads to increase of fluid temperature efficiently, these
figures confirm this fact also. As the parameter values and increases, the volume fraction
increased for the fixed values of the other parameters seen from Fig.3(c).
Fig. 4. presents the effect of the Brownian motion Nb and thermophoresis Nt parameters on
the velocity, temperature and volume fraction. It is observed that the momentum boundary layer
thickness increases with the increase in values of Nb but it decreases with increasing values of Nt .
The nanoparticle volume fraction decreased with increase in Nb and it increased with increasing
values of Nt . It is also noticed that the nanoparticle volume fraction increased with an increase in
Nb in the case of forced convection flow. > 0 indicates a cold surface while negative < 0
corresponds a hot surface, in case of hot surface, thermophoresis tends to blow the nanoparticle
volume fraction away from the surface since a hot surface repels the sub-micron sized particles from
it, thereby forming a relative particle-free layer near the surface.
Variation of non-dimensional velocity and temperature against the similarity variable η , is
shown respectively in Fig.5, for a few set of values of λ and Pr with fixed values of other
parameters. As the parameter λ and Pr increase, the velocity increased whereas temperature of
nanofluid decreased.

79
Fig. 2(a): Effect of M and Le on velocity '( )f η
Fig. 2(b): Effect of M and Le on temperature ( )θ η
0 1 2 3 4 5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
Le=0.2,0.3,0.5
Nt=0.2
Nb=0.2
Nr=0.2
Pr=1.0
Lam=0.4
Bi=0.01
____M=1.0
-------M=1.1
........M=1.2
f'(ηηηη)
ηηηη
C
E
F
G
H
I
J
K
L
0 1 2 3 4
0.000
0.005
0.010
0.015
0.020
Nt=0.2
Nb=0.2
Nr=0.2
Pr=1.0
Lam=0.4
Bi=0.01
Le=0.2,0.3,0.5
___M=1.0
-----M=1.1
......M=1.2
θ(θ(θ(θ(ηηηη))))
ηηηη
E
G
H
I
J
K
L
M
N

80
Fig. 2(c): Effect of M and Le on volume fraction ( )γ η
Fig. 3(a): Effect of Bi and Nr on the velocity
'( )f η
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Le=0.2,0.3,0.5
Nt=0.2
Nb=0.2
Nr=0.2
Pr=1.0
Lam=0.4
Bi=0.01
____M=1.0
-------M=1.1
........M=1.2
γ(γ(γ(γ(ηηηη))))
ηηηη
G
I
J
K
L
M
N
O
P
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Nt=0.2
Nb=0.2
Pr=1.0
M=1.0
Lam=1.0
Le=0.2
______Nr=0.1
----------Nr=0.3
............Nr=0.5
Bi=0.1,0.3,0.5
f'(ηηηη)
ηηηη
C
E
F
G
H
I
J
K
L

81
Fig. 3(b): Effect of Bi and Nr on the temperature ( )θ η
Fig.3(c): Effect of Bi and Nr on the volume fraction ( )γ η
0 1 2 3
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
Bi=0.1,0.3,0.5
Nt=0.2
Nb=0.2
Pr=1.0
M=1.0
Lam=1.0
Le=0.2
Nr=0.1
Nr=0.3
Nr=0.5
θθθθ(η)(η)(η)(η)
ηηηη
E
G
H
I
J
K
L
M
N
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Nt=0.2
Nb=0.2
Pr=1.0
M=1.0
Lam=1.0
Le=0.2
Bi=0.1,0.3,0.5
____Nr=0.1
-------Nr=0.3
........Nr=0.5
ηηηη
G
I
J
K
L
M
N
O
P

82
Fig. 4(a): Effect of Nb and Nt on the volume velocity '( )f η
Fig. 4(b): Effect of Nb and Nt on the temperature ( )θ η
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Lam=0.4,
Nr=0.2,
Pr=1.0,
M=1.0,
Le=0.2
Bi=0.1
_____ Nb=0.1
--------- Nb=0.5
........... Nb=0.9
Nt=0.1,0.5,0.9
f'(ηηηη)
ηηηη
C
E
F
G
H
I
J
K
L
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
0.0
0.1
0.2
0.3
Lam=0.4,
Pr=1.0,
M=1.0,
Le=0.2,
Bi=0.1,
Nr=0.2
Nt=0.1,0.5,0.9
______Nb=0.1
-----------Nb=0.5
.............Nb=0.9
θθθθ(ηηηη)
ηηηη
E
G
H
I
J
K
L
M
N

83
Fig. 4(c): Effect of Nb and Nt on the volume fraction ( )γ η
Fig. 5(a): Effect of λ and Pr on Velocity '( )f η
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Lam=0.4 Nr=0.2 Pr=1.0
M=1.0 Le=0.2 Bi=0.1
____Nb=0.1
-------Nb=0.5
........Nb=0.9
Nt=0.1,0.5,0.9
ηηηη
G
I
J
K
L
M
N
O
P
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Nt=0.2
Nb=0.2
Nr=0.2
M=1.0
Le=0.4
Bi=0.01
____Pr=1.0
-------Pr=2.0
........Pr=3.0
lam=0.4,0.8,1.2
f'(ηηηη)
ηηηη
C
E
F
G
H
I
J
K
L

84
Fig. 5(b): Effect of λ and Pr on Temperature ( )θ η
5. CONCLUSION
In this paper, the effect of Magnetic parameter on mixed convection flow along a vertical
plate in a nanofluid is analyzed under the convective boundary conditions. Using the similarity
variables, the governing equations are transformed into a set of non-dimensional parabolic equations.
These equations are solved numerically using the Runge-Kutta Fourth order method. The numerical
results are obtained for a wide range of values of the physical parameters. To ascertain the
convergence of the numerical method adopted. The nanoparticle is considered in the analysis. The
skin friction, heat and nanopartical mass coefficients are obtained for a physically realistic values of
governing parameters. The results are analyzed thoroughly for different values of M , Bi , and λ
on the flow, thermal and solutal field. The major conclusion is that the magnetic effect enhanced the
skin friction, heat and nanoparticle mass in the medium.
NOMENCLATURE
Bi Biot number
c constant
BD Brownian diffusion coefficient
TD Thermophoretic diffusion coefficient
f Dimensionless steam function
g Gravitational acceleration
xGr Local Grashof number
0 1 2 3 4
0.000
0.005
0.010
0.015
0.020
0.025
Nt=0.2
Nb=0.2
Nr=0.2
M=0.2
Le=0.4
Bi=0.01
___Pr=1.0
-----Pr=2.0
......Pr=3.0
Lam=0.4,0.8,1.2
θ(θ(θ(θ(ηηηη))))
ηηηη
E
G
H
I
J
K
L
M
N

85
fh Convective heat transfer coefficient
τ Ratio between the effective heat capacity of the nanoparticle material and heat capacity of
the fluid
k Thermal conductivity of the nanofluid
Le Lewis number
Nb Brownian motion parameter
Nr Nanopartical buoyancy ratio
Nt Thermophoresis parameter
xNu Local Nusselt number
Pr Prandtl number
nq Nanoparticle mass flux at the wall
wq Heat flux at the wall
Rex Local Reynolds number
xNSh Local nanoparticle Sherwood number
M Magnetic number
T Temperature
fT Temperature of the hot fluid
T∞ Ambient temperature
u∞ Characteristic velocity
,u v Velocity components in x and y direction
,x y coordinates along and normal to the plate
mα Thermal diffusivity
η Similarity variable
γ Dimensionless volume fraction
λ Mixed convection parameter
θ Dimensionless temperature
φ Nanoparticle volume fraction
wφ Nanoparticle volume fraction at the wall
φ∞ Nanoparticle volume fraction at large values of y(ambient)
µ Dynamic viscosity of the base fluid
υ Kinematic viscosity
ρ Density of the fluid
fρ ∞ Density of the base fluid
pρ Nanoparticle mass density
( )f
cρ Heat capacity of the fluid
( )p
cρ Effective heat capacity of the nanoparticle material
wτ Wall shear stress
ψ Stream function

86
Subscripts
w Wall condition
∞ Ambient condition
c Concentration
T Temperature
REFERENCES
1. J. H. Merkin, The effects of buoyancy forces on the boundary layer flow over a semi-infinite vertical
flat plate in a uniform stream, J. Fluid Mech.35 (1996) 439-450.
2. Y. Ding, H. Chen, L. Wang, C.-Y. Yang, y. He, W. Yang, W. P. Lee, L. Zhang, R. Huo, Heat transfer
intensification using nanofluids, Kona 25 (2007) 23-38.
3. B. C. Sakiadas, Boundary layer behavior on continuous solid surfaces: I Boundary layer equations for
two dimensional and flow, AIChE J.7 (1961) 26-28.
4. A. Bejan, Convection Heat Transfer, John Wiley, New York, 2004.
5. S. V. Subhashini, Samuel Nancy, I. Pop, Double-diffusive convection from a permeable vertical
surface under convective boundary condition, Int. Commun. Heat Mass Transfer 38 (2011)1183-
1188.
6. S. K. Das, S.U.S. Choi, A review of heat transfer in nanofluids, Adv. Heat Transfer 41(2009) 81-197.
7. S. K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: Science and Technology. Wiley, Hoboken,
NY, 2008.
8. J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240-250.
9. E. Schmidt, W. Beckmann, Das Temperature-und Geschwindikeitsfeld voneiner warme abgebenden
senkrechten platte bei naturlicher konvection, II. Die Versuche und ihre Ergibnisse, Forcsh,
Ingenieurwes 1 (1930) 391-406.
10. H.K. Kuiken, An asymptotic solution for large Prandtl number free convection, J. Engng. Math. 2
(1968) 355-371.
11. H.K. Kuiken, Free convection at low Prandtl numbers, J. Fluid Mech 39 (1969) 785-798.
12. K.R. Khair, A. Bejan, Mass transfer to natural convection boundary-layer flow driven by heat transfer,
ASME J. Heat Transfer 107 (1985) 979-981.
13. O.D. Mankinde, A. Aziz, Boundary layer flow of a nanofluid past a stretching sheet with a convective
boundary condition ,Int. J. Therm. Sci. 50(2011) 1326-1332.
14. A.V. Kuznetsov, D.A. Nield, natural convective boundary-layer flow of a nanofluid past a vertical
plate, Int. J. Therm. Sci. 49 (2010) 243-247.
15. A.V. Kuznetsov, D.A. Nield, double-diffusive natural convective boundary-layer flow of a nanofluid
past a vertical plate, Int. J. Therm. Sci. 50 (2011) 712-717.
16. Dr.N.G.Narve and Dr.N.K.Sane, “Experimental Investigation of Laminar Mixed Convection Heat
Transfer In The Entrance Region of Rectangular Duct” International Journal of Mechanical
Engineering & Technology (IJMET), Volume 4, Issue 1, 2013, pp. 127 - 133, ISSN Print: 0976 –
6340, ISSN Online: 0976 – 6359.
17. Dr. B.Tulasi Lakshmi Devi, Dr. B.Srinivasa Reddy, G.V.P.N.Srikanth and Dr. G.Srinivas,
“Hydromagnetic Mixed Convection Micro Polar Flow Driven by A Porous Stretching Sheet – A
Finite Element Study” International Journal of Mechanical Engineering & Technology (IJMET),
Volume 5, Issue 2, 2014, pp. 52 - 63, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.
ACKNOWLEDGEMENT
Authors are thankful to the reviewers for their useful comments and suggestions. One of the author
Dr. Jagadish V. Tawade wishes to thank Bharat Ratna Prof. C.N.R.Rao, Hon’ble Chairman, Dr. S. Anant Raj
Consultant and Prof. Roddam Narasimha, Hon’ble member VGST, Department of IT, BT S & T, GoK, India,
for supporting this work under Seed Money to Young Scientists for Research (F.No.VGST/P-3/
SMYSR/GRD-286/2013-14).

Magnetic field effect on mixed convection flow in a nanofluid under convective boundary condition

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