The effect of magnetic field on the boundary layer flow over a stretching sheet in a nanofluid with 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. 87-100© IAEME
87
THE EFFECT OF MAGNETIC FIELD ON THE
BOUNDARY LAYER FLOW OVER A STRETCHING
SHEET IN A NANOFLUID WITH 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
The boundary layer flow created due to a linearly stretching sheet in a nanofluid is studied
numerically. The boundary value problem consisting of nonlinear partial differential equations are
converted into nonlinear ordinary differential equations, using similarity transformation and are
solved numerically using Runge-Kutta Fourth order method, with shooting technique. The transport
equations include the effects of Brownian motion and thermophoresis. Unlike the commonly
employed thermal conditions of constant temperature or constant heat flux, the present study uses a
convective heating boundary conditions. The solutions for the temperature and nanoparticle
concentration distribution depend on six parameters, Prandtl number, Lewis number, Brownian
motion parameter, thermophoresis parameter, convective Biot number and magnetic field parameter.
Numerical results are presented both in tabular and graphical forms illustrating the effects of these
parameters on thermal and concentration boundary layers. The thermal boundary layer thickness
increases, with a rise in the local temperature as the Brownian motion, thermophoresis and
convective heating, each intensify. The effect of Lewis number on the temperature distribution is
insignificant. With the other parameters unchanging, the local concentration of nanoparticle
increases as the convective Biot number increases but decreases as the Lewis number increases. As
both reduced Nusselt number, and the reduced Sherwood number increases, when, Brownian motion
and thermophoresis effects become stronger.
Keywords: Nanofluid; Boundary Layer Flow; Stretching Sheet; Convective Biot Number;
Thermophorosis; and Brownian Motion.
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND
TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 6, Issue 4, April (2015), pp. 87-100
© 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

88
1. INTRODUCTION
A Nanofluid is a fluid containing nanometer sized particles, called Nanoparticles. These
fluids are engineered colloidal suspension 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 engine, engine cooling/vehicle thermal management, domestic
refrigerator, chiller, heat exchanger, in grinding, machining and in boiler gas temperature reduction.
They demonstrate 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
vital deciding their suitability for convective heat transfer applications.
The fluid flow over a stretching surface is important in applications such as extrusion, wire
drawing, metal spinning, hot rolling, etc[1-3]. A wide variety of problems dealing with heat and fluid
flow over a stretching sheet have been studied with both Newtonian and non-Newtonian fluids and
with the inclusion of imposed electric and magnetic fields, different thermal boundary conditions,
and power law variation of the stretching velocity. A representative sample of the recent literature
on the topic is provided by references [4-12]. After the poineering work by Sakiadis [13], a large
amount of literature is available on boundary layer flow of Newtonian and non-Newtonian fluids
over linear and nonlinear stretching surface. 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 [14].
In the past few years, convective heat transfer in nanofluids has become a topic of major
current interest. Recently Khan and Pop [15] used the model of Kuznetsov and Nield [16] to study
the boundary layer flow of a nanofluid past a stretching sheet with a constant surface temperature.
Makinde, and Aziz [17] 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. The transformed non-linear ordinary differential equations governing the flow are solved
numerically by the Runge-Kutta Fourth order method.
The solution of boundary layer equation for a power law fluid in MHD was obtained by
Helmy[20]. Chiam[21] investigated hydromagnetic flow over a surface stretching with power law
velocity using shooting method. Ishak etal[22] investigated MHD flow and heat transfer adjacent to a
stretching vertical sheet. Nourazar etal[23] investigated MHD forced convective flow of nanofluid
over a horizontal stretching sheet with variable magnetic field with the effect of viscous dissipation.
Zeeshan etal[24]. Chamkaand aly etal[25] have considered MHD free convective boundary layer
flow of a nanofluid along a permeable isothermal vertical plate in the prescence heat source/sink.
In this paper, our main objective is to investigate the effect of a convective boundary condition
and magnetic field effect on boundary layer flow, heat transfer and nanoparticle fraction over a
stretching surface in nanofluid. The governing boundary layer equations have been transformed to a
two-point boundary value problem in similarity variables, and these have been solved numerically.
The effects of embedded parameters on fluid velocity, temperature and particle concentration have
been shown graphically. It is hoped that the results obtained will not only provide useful information
for applications, but also serve as a complement to the previous studies.

89
2. CONVECTIVE TRANSPORT EQUATIONS
Consider steady two-dimensional ( ),x y boundary layer flow of a nanofluid past a stretching
sheet with a linear velocity variation with the distance x i.e. wu cx= where c is a real positive
number, is stretching rate, and x is the coordinate measured from the location, where the sheet
velocity is zero
Fig 1. Nano boundary layer flow over a stretching sheet.
The sheet surface temperature wT , to be determined later, is the result of a convective
heating process which is characterized by temperature fT and a heat transfer coefficient h . The
nanoparticle volume fraction C at the wall is wC , while at large values of y ,the value is C∞ . The
Boungiorno model may be modified for this problem to give the following continuity, momentum,
energy and volume fraction equations.
(1)
22
0
2
,
B uu u u
u v
x y y
σ
ν
ρ
 ∂ ∂ ∂
+ = − 
∂ ∂ ∂ 
(2)
22
2
,T
B
DT T T C T T
u v D
x y y y y T y
α τ
∞
       ∂ ∂ ∂ ∂ ∂ ∂ 
+ = + +        
∂ ∂ ∂ ∂ ∂ ∂         
(3)
2 2
2 2
,T
B
DC C C T
u v D
x y y T y∞
    ∂ ∂ ∂ ∂
+ = +     
∂ ∂ ∂ ∂    
( )4
where u and v are the velocity components along the x and y directions, respectively, p is the
fluid pressure, fρ is the density of base fluid, ν is the kinematic viscosity of the base fluid, α is the
thermal diffusivity of the base fluid, ( ) ( )p f
c cτ ρ ρ= is the ratio of nanoparticle heat capacity and
the base fluid heat capacity, BD is the Brownian diffusion coefficient, TD is the thermophoretic
diffusion coefficient and T is the local temperature. The subscript ∞ denotes the values of at large
values at large values of y where the fluid is quiescent. The boundary condition may b written as
0,
u v
x y
∂ ∂
+ =
∂ ∂

90
( ) ,0, , 0, ,f w
T
y u cx v k h T T C C
y
∂
= = = − = − =
∂
(5)
, 0, 0, , ,y u v T T C C∞ ∞→ ∞ = = = = (6)
We introduce the following dimensionless quantities
( ) ( ) ( )
1 2 1 2
, , ,
,
f
w
T T
c y c xf
T T
C C
C C
η υ ψ υ η θ
φ
∞
∞
∞
∞
−
= = =
−
−
=
−
(7)
Whereψ is the stream function with ,u y v xψ ψ= ∂ ∂ = −∂ ∂
2
''' '' ' ' 0,f ff f Mf+ − − = (8)
2
'' ' ' ' '
0,Prf PrNb PrNtθ θ φθ θ+ + + = (9)
'' ' ''
0,
Nt
Lef
Nb
φ φ θ+ + = (10)
subject to the following boundary conditions.
( ) ( ) ( ) ( ) ( )0 0, ' 0 1, ' 0 1 0 , 0 1,f f Biθ θ φ= = = − − =   (11)
( ) ( ) ( )'
0, 0, 0,f θ φ∞ = ∞ = ∞ = (12)
where primes denote differentiation with respect to η and the five parameters appearing in Eqs. (9-
12) are defined as follows.
(13)
With 0Nb = there is no thermal transport due to buoyancy effects created as a result of
nanoparticle concentration gradients.
Here, we note that Eq. (8) with the corresponding together with the corresponding boundary
conditions on f provided by Eq. (11) has a closed form solution which is given by
( ) 1 .f e η
η −
= − (14)
In Eq. (14), , , ,Pr Le Nb Nt and Bi denote the Prandtl number, the Lewis number, the
Brownian motion parameter, the thermophoresis parameter and the Biot number respectively. The
reduced Nusselt number Nur and the reduced Sherwood number Shr may found in terms of the
dimensionless temperature at the surface, ( )' 0θ and the dimensionless concentration at the sheet
surface, ( )' 0φ , respectively i.e.
1 2
RexNur −
= ( )' 0 ,Nu θ= − (15)
1 2
RexShr −
= ( )' 0 ,Nu φ= − (16)
( ) ( )
( )
( ) ( )
( )
( )
1 2
, , ,
,
B wp
B f
T fp
f
c D C C
Pr Le Nb
D c
c D T T h a
Nt Bi
c T k
ν
ν
ρν ν
α ρ
ρ ν
ρ
∞
∞
∞
−
= = =
−
= =

91
where
( ) ( )
( ), ,Re ,ww m
x
w B w
u x xq x q x
Nu Sh
k T T D φ φ ν∞ ∞
= = =
− −
(17)
where wq is the surface (wall) heat flux and mq is the surface (wall) mass flux.
3. RESULT AND DISCUSSION
Eqs. (8-10) subject to the boundary conditions, Eqs. (11) and (12), were solved numerically
using Runge- kutta -Fehlberg fourth order method. As a further check on the accuracy of our
numerical computations, Table1 contains a comparison of our results for the reduced Nusselt number
and the reduced Sherwood number with those reported by Khan and Pop [15] for
10, 10, , 10.Le Pr Bi M= = = ∞ = The infinitely large Biot number simulates the isothermal stretching
model used in[15] as noted earlier. The results for all combination values of Brownian motion
parameter Nb and the thermophoresis parameter Nt used in our computations, showed an exact
match between our results and the results reported in [15]. The first five entries show that for a fixed
thermophoresis parameter, 0.1Nt = , the reduced Nusselt number decreases sharply with the
increasing in Brownian motion, that as Nb is increased from 0.1 to 0.5. However, the reduced
Sherwood number increases substantially as Nb is increased from 0.1 to 0.2 but tends to plateau
beyond 0.2Nb = . These observations are consistent with the initial slopes of the temperature and
concentration profiles to be discussed later. As the Brownian motion intensifies, it impacts a larger
extent of the fluid, causing the thermal boundary layer to thicken, which in turn decreases the
reduced Nusselt number. The thickening of the boundary layer due to stronger Brownian motion will
be high-lighted again when the temperature profiles are discussed. It will be seen from the
concentration profiles appearing later in the discussion that the initial slope of the curve and the
extend of the concentration boundary layer are not affected significantly beyond 0.2Nb = and
hence the plateau in the sherwood number behavior. The last four entries in Table2 show that the
reduced Nusselt number decreases as the thermophoresis diffusion penetrates deeper into the fluid
and causes the thermal boundary layer to thicken. However, the increase in the thermophoresis
parameter enhance the Sherwood number, a conclusion that is consistent with the results of Khan
and Pop [15].
We now turn our attention to the discussion of graphical results that provide additional insights
into the problem under investigation.
Temperature profiles
Fig. 2 shows the temperature distribution in the thermal boundary layer for different values of
Brownian motion and the thermophoresis parameters. As both Nb and Nt increase, the boundary
layer thickens, as noted earlier in discussing the tabular data, the surface temperature increases , and
the curves become less step indicating a diminution of the reduced Nusselt number. As seen in Fig.
3, the effect of Lewis number on the temperature profiles is noticeable only in a region close to the
sheet as the curves tend to merge at larger distances from the sheet.
The Lewis number expresses the relative contribution of thermal diffusion rate to species
diffusion rate in the boundary layer regime. An increase of Lewis number will reduce thermal
boundary layer thickness and will be accompanied with a decrease in temperature. Larger Le will
suppress concentration values. I.e inhibit nanoparticle species diffusion. There will be much greater
reduction in concentration bopundary layer thickness than thermal boundary layer thickness over an
increment in Lewis Number.
Fig. 4 illustrates the effect of Biot number on the thermal boundary layer. As expected, the
stronger convection results in higher surface temperatures, causing the thermal effect to penetrate
deeper into the quiescent fluid. The temperature profiles depicted in Fig. 5 show that as the Prandtl

92
number increases, the thickness of the thermal boundary layer decreases as the curve become
increasingly steeper. As a consequence, the reduced Nusselt number, being proportional to the initial
slope, increases. This pattern is reminiscent of the convective of the free convective boundary layer
flow in a regular fluid[20].Fig 6 shows that the effect of magnetic number on the temperature
profiles is noticeable only in region close to the sheet as the curves tend to merge at larger distances
from the sheet.
Concentration profiles
The effect of nanoparticle concentration profiles, is shown in Fig. 7. Unlike the temperature
profiles, the concentration profiles are only slightly affected by the strength of the Brownian motion
and thermophoresis. A comparison of Fig. 3 and Fig. 8 shows that the Lewis number significantly
affected the concentration distribution (Fig. 8), but has little influence on the temperature distribution
(Fig. 3). For a base fluid of certain kinematic viscosityν ,a higher Lewis number implies a lower
Brownian diffusion coefficient BD (see Eq.(13)) which must result in a shorter penetration depth for
the concentration boundary layer. This is exactly what we see in Fig. 8 it was observed in Fig. 4 that
as the convective heating of the sheet is enhanced i.e. Bi increases, the thermal penetration depth
increases. Because the concentration distribution is driven by the temperature field, one anticipates
that a higher Biot number would promote a deeper penetration of the concentration. This anticipation
is indeed realized in Fig. 9 which predict higher concentration at higher values of the Biot number. A
comparison of Fig6 and fig. 10 shows that the Magnetic number significantly affected the
concentration distribution (Fig.10), but has little influence on the temperature distribution (Fig.6).
Fig. 2. Effect of Nt and Nb on temperature profiles when 5, 5, 0.1Le Pr Bi= = = .
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00
0.05
0.10
0.15
0.20
0.25
____Nb=Nt=0.1
____Nb=Nt=0.2
____Nb=Nt=0.3
____Nb=Nt=0.5
θ(θ(θ(θ(ηηηη))))
ηηηη
E
G
H
I

93
Fig. 3. Effect of Le on temperature profiles when 0.1, 5, 0.1.Nt Nb Pr Bi= = = =
Fig. 4. Effect of Bi on temperature profiles when 0.1, 5.Nt Nb Pr Le= = = =
0.0 0.5 1.0 1.5 2.0 2.5
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
____Le=5
____Le=10
____Le=15
____Le=20
ηηηη
E
G
H
I
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
____Bi=0.1
____Bi=1.0
____Bi=5.0
____Bi=10.0
ηηηη
E
G
H
I

94
Fig. 5. Effect of Pr on temperature profiles when 0.1, 5Nt Nb Bi Le= = = = .
Fig. 6. Effect of M on temperature profiles when 0.1, 5.Nt Nb Bi Le Pr= = = = =
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
____M=1.0
____M=2.0
____M=5.0
____M=6.0
ηηηη
E
G
H
I
0 1 2 3 4 5 6 7 8
0.00
0.05
0.10
0.15
_____Pr=1
_____Pr=2
_____Pr=5
_____Pr=10
ηηηη
E
G
H
I

95
Fig. 7. Effect of Nt and Nb on concentration profiles when 5, 5, 0.1Le Pr Bi= = = .
Fig. 8. Effect of Le on concentration profiles when 0.1, 5, 0.1.Nt Nb Pr Bi= = = =
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
____Nb=Nt=0.1
____Nb=Nt=0.2
____Nb=Nt=0.3
____Nb=Nt=0.5
φ(φ(φ(φ(ηηηη))))
ηηηη
G
I
J
K
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
____Le=5
____Le=10
____Le=15
____Le=20
ηηηη
G
I
J
K

96
Fig.9.Effect of Bi on concentration profiles when 0.1, 5.Nt Nb Pr Le= = = =
Fig. 10. Effect of M on concentration profiles when 0.1, 5.Nt Nb Bi Le Pr= = = = =
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
_____Bi=0.1
_____Bi=1.0
_____Bi=5.0
_____Bi=10.0
ηηηη
G
I
J
K
0 1 2 3 4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
_____M=1.0
_____M=2.0
_____M=5.0
_____M=6.0
ηηηη
G
I
J
K

97
Table 1. Comparison of results for the reduced Nusselt number (0)θ′− and the reduced Sherwood
number (0)ϕ′ with Khan and Pop [15]
Nb Nt Nur Shr Nur present Shr present
0.1 0.1 0.9524 2.1294 0.5230 2.0507
0.2 0.1 0.5056 2.3819 0.3561 2.2346
0.3 0.1 0.2522 2.4100 0.2082 2.2797
0.4 0.1 0.1194 2.3997 0.1077 2.2846
0.5 0.1 0.0543 2.3836 0.0513 2.2767
0.1 0.2 0.6932 2.2740 0.4761 1.9851
0.1 0.3 0.5201 2.5286 0.44235 2.0231
4. CONCLUSION
A numerical study of the boundary layer flow in a nanofluid induced as as a result of motion
of a linearly stretching sheet has been performed. The use of a convective hating boundary condition
instead of a constant temperature or a constant heat flux makes this study more general novel. The
following conclusions are derived
1. The transport of momentum, energy and concentration of nanoparticles in the respective
boundary layers depends on six parameters: Brownian motion parameter Nb , thermophoresis
parameter Nt , Prandtl number Pr , Lewis number Le, convection Biot number Bi and
Magnetic parameter M .
2. For infinitely large Biot number characterizing the convective heating (which corresponds to
the constant temperature boundary condition), the present results and those reported by Khan
and Pop [27] match up to four places of decimal.
3. For a fixed Pr , Le and Bi , the thermal boundary thickens and the local temperature rises as
the Brownian motion and thermophoresis effects intensify. A similar effect on the thermal
boundary is observed when , ,Nb Nt Leand Bi are kept fixed and the Prandtl number Pr is
increased or when , ,Pr Nb Nt and Le are kept fixed and the Biot number is increased.
However, when , ,Pr Nb Nt and Bi are kept fixed, and the Lewis number is increased, the
temperature distribution is affected only minimally.
4. With the increase in Bi , the concentration layer thickens but the concentration layer becomes
thinner as Le increases.
5. For fixed ,Pr Le and Bi , the reduced Nusselt number decreases but the reduced Sherwood
number increases as Brownian motion and thermophoresis effects intenify.

98
NOMENCLATURE
iB Biot number
a A positive constant associated with linear stretching
BD Brownian diffusion coefficient
TD Thermophoretic diffusion coefficient
( )f η Dimensionless steam function
g Gravitational acceleration
h Convective heat transfer coefficient
k Thermal conductivity of the nanofluid
Le Lewis number
Nb Brownian motion parameter
Nt Thermophoresis parameter
Nu Nusselt number
Nur Reduced Nusselt number
Pr Prandtl number
p Pressure
''
mq Wall mass flux
''
wq Wall heat flux
Rex Local Reynolds number
Sh Sherwood number
Shr Reduced Sherwood number
M Magnetic number
T Local fluid Temperature
fT Temperature of the hot fluid
T
w
Sheet surface (wall) temperature
T∞ Ambient temperature
,u v Velocity components in x and y directions
C Nanoparticle volume fraction
C
w
Nanoparticle volume fraction at the wall
C
∞
Nanoparticle volume fraction at large values of y (ambient)
Greek symbol
α Thermal diffusivity of the base fluid
η Similarity variable
θ Dimensionless temperature
ϕ Dimensionless volume fraction
µ Absolute viscosity of the base fluid
υ Kinematic viscosity of the base fluid
f
ρ Density of the base fluid
pρ Nanoparticle mass density

99
( )f
cρ Heat capacity of the base fluid
( )p
cρ Heat capacity of the nanoparticle material
( ) ( )c c
p f
τ ρ ρ=
ψ Stream function
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Engineering & Technology (IJMET), Volume 3, Issue 2, 2012, pp. 559 - 572, ISSN Print:
0976 – 6340, ISSN Online: 0976 – 6359.
26. Anand H. Agadi, M. Subhas Abel, Jagadish V. Tawade And Ishwar Maharudrappa, “Mhd
Boundary Layer Flow of A Maxwell Fluid Past A Porous Stretching Sheet In Presence of
Viscous Dissipation” International Journal of Mechanical Engineering & Technology
(IJMET), Volume 4, Issue 5, 2013, pp. 156 - 163, ISSN Print: 0976 – 6340, ISSN Online:
0976 – 6359.
ACKNOWLEDGEMENT
Authors are thankful to the reviewers for their useful comments and suggestions.nOne 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).

The effect of magnetic field on the boundary layer flow over a stretching sheet in a nanofluid with convective boundary condition

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The effect of magnetic field on the boundary layer flow over a stretching sheet in a nanofluid with convective boundary condition