Magnetohydrodynamic mixed convection flow and boundary layer control of a nanofluid with heat generation absorption effects

Magnetohydrodynamic Mixed Convection Flow and Boundary Layer Control of A Nanofluid With Heat
Generation/Absorption Effects, M.Chandrasekar, M.S.Kasiviswanathan, Journal Impact Factor (2015): 8.8293
Calculated by GISI (www.jifactor.Com)
Iaeme.com/ijmet.asp 18 editor@iaeme.com
1
Department of Mathematics, Anna University, Chennai-600 025, India,
2
Department of Mathematics, Anna University, Chennai-600 025, India,
ABSTRACT
This research work is focused on the numerical solution of steady MHD mixed convection
boundary layer flow of a nanofluid over a semi-infinite flat plate with heat generation/absorption and
viscous dissipation effects in the presence of suction and injection. Gyarmati’s variational principle
developed on the thermodynamic theory of irreversible processes is employed to solve the problem
numerically. The governing boundary layer equations are approximated as simple polynomial
functions, and the functional of the variational principle is constructed. The Euler-Lagrange
equations are reduced to simple polynomial equations in terms of momentum and thermal boundary
layer thicknesses. The velocity, temperature profiles as well as skin friction and heat transfer rates
are solvable for any given values of Prandtl number Pr, magnetic parameter ξ, heat source/sink
parameter Q, buoyancy parameter Ri, suction/injection parameter H and viscous dissipation
parameter Ec. The obtained results are compared with known numerical solutions and the
comparison is found to be satisfactory.
Keywords: Boundary Layer, Gyarmati’s Variational Principle, Heat Source/Sink, Mixed
Convection, Nanofluid
1. INTRODUCTION
The prime objective of this work is to study the heat transfer enhancement in mixed convection
nanofluid flow over a flat plate with heat source/sink and magneto hydrodynamic effects using a
genuine variational principle developed by Gyarmati. Recently in many industrial applications
nanofluids are used as heat carriers in heat transfer equipment instead of conventional fluids due to
its relatively higher thermal conductivity. The potential benefits of nanofluids are theoretically and
experimentally investigated by many researchers in the past two decades. Buongiorno [1] explained
the seven slip mechanisms as reasons for the heat transfer enhancement observed in nanofluids. Due
to the great potential and characteristics of nanofluid still more research work to be done to study
heat transfer enhancement mechanism.
Khan and Pop [2] solved the numerical solution of a nanofluid flow over a stretching sheet.
The analysis on free convection nanofluid flow over a vertical plate with different boundary
conditions on the nanoparticle volume fraction was investigated by Kuznetsov and Nield [3, 4].
MAGNETOHYDRODYNAMIC MIXED CONVECTION FLOW AND
BOUNDARY LAYER CONTROL OF A NANOFLUID WITH HEAT
GENERATION/ABSORPTION EFFECTS
M.Chandrasekar1
, M.S.Kasiviswanathan2
Volume 6, Issue 6, June (2015), pp. 18-32
Article ID: 30120150606003
International Journal of Mechanical Engineering and Technology
© IAEME: http://www.iaeme.com/IJMET.asp
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
IJMET
© I A E M E

Chamkha and Aly [5] considered the boundary layer equations for natural convection flow of an
electrically conducting nanofluid past a plate in the presence of heat generation and absorption
effects. The same problem without Brownian motion and thermophoresis effects was analyzed by
Hamad et al. [6]. Rana and Bhargava [7] presented an analysis on mixed convective boundary layer
flow of nanofluid over a vertical flat plate with temperature dependent heat source/sink. The
stagnation point flow of unsteady case in a nanofluid was described by Bachok et al. [8]. The
boundary layer solution for forced convection flow of alumina-water nanofluid over a flat plate in
the presence of magnetic effect was studied by Hatami et al. [9]. Vajravelu et al. [10] observed the
effects of variable viscosity and viscous dissipation on the forced convection flow of water based
nanofluids. The stagnation point flow of nanofluid towards a stretching sheet in the presence of
transverse magnetic field was studied by Ibrahim et al. [11].
By considering all the above facts, in this study non similar mixed convection flow of water
based nanofluid containing one of the nanoparticles Copper (Cu), Silver (Ag), Alumina(Al2O3) with
the volume fraction range 0-4% over a semi-infinite flat plate in the presence of constant magnetic
flux density, heat source/sink, suction/injection and viscous dissipation effects were analyzed.
The Gyarmati’s variational technique has been employed to solve the non-similar boundary
layer equations. The computational results are given for velocity profile temperature profile, the
coefficient of skin friction (shear stress) and local Nusselt number (heat transfer) for various values
of heat generation/absorption parameter Q, magnetic parameter ξ and buoyancy parameter Ri. The
results obtained by the present analysis are compared with the numerical solution of Rana and
Bhargava [7] and the comparison establishes the fact that the accuracy is remarkable. The main
intention of this investigation is to justify that, the Gyarmati’s variational technique is one of the
most general and exact variational techniques in solving flow and heat transfer problems.
Chandrasekar [12, 13], Chandrasekar and Baskaran [14], Chandrasekar and Kasiviswanathan [15]
already applied Gyarmati’s variational technique for steady and unsteady heat transfer and boundary
layer flow problems.
2. THE GOVERNING EQUATIONS OF THE SYSTEM
The system of steady, two dimensional, incompressible and laminar boundary layer flow of
nanofluid over a semi-infinite flat plate with suction and injection is considered. The leading edge of
the plate is at x = 0, the plate is parallel to the x-axis and infinitely long downstream. In this study it
is assumed that the flow is with free stream velocity U∞ and the ambient temperature T∞ which are
parallel to x-axis. And the temperature of the plate is held at a constant temperature T0 which is
greater than the ambient temperature T∞. A uniform magnetic field of strength B0 is applied normal
to the x-axis and assumed that the induced magnetic field, the imposed electric field intensity and the
electric field due to the polarization of charges are negligible. By Boussinesq-boundary layer
approximations and with the assumption that all fluid properties are constants, the governing
boundary layer equations for the present system are as follows, see Aydin and Kaya [16]
2
0
2 2
0 0
0 (1)
1
( ) ( ) ( ) (2)
1
( ) ( ) ( ) (3)
( )
x y
x y nf yy nf
nf
x y nf yy nf y
p nf
u v
uu vu u B U u g T T
uT vT k T u B u U u Q T T
C
µ κ ρβ
ρ
µ κ
ρ
∞ ∞
∞ ∞
+ =
 + = + − + − 
 + = + − − + − 
subject to the boundary conditions

y = 0; u = 0, v = 0v , T = T0,
y → ∞; u=U∞= constant, T = T∞ (4)
Here u, v, 0v , T, κ, 0B , 0Q and g are velocity of the fluid in x-direction, velocity of the fluid
in y-direction, suction/injection velocity, temperature of the fluid, electric conductivity, externally
imposed magnetic field in the y-direction, heat generation/absorption coefficient and acceleration
due to gravity respectively.
The thermophysical properties of nanofluid namely density, dynamic viscosity, thermal
diffusivity, volumetric expansion coefficient, heat capacity and thermal conductivity are denoted by
respectively ρnf, µnf, αnf, (ρβ)nf, (ρCp)nf, knf and have been calculated as functions of thermophysical
properties of nanoparticle (spherical shaped) and base fluid as follows,
2.5
(1 )
(1 )
( )
( ) (1 )( ) ( )
( ) (1 )( ) ( )
2 2 ( )
and
2 ( )
nf f s
f
nf
nf
nf
p nf
nf f s
p nf p f p s
nf s f f s
f s f f s
k
C
C C C
k k k k k
k k k k k
ρ φ ρ φρ
µ
µ
φ
α
ρ
ρβ φ ρβ φ ρβ
ρ φ ρ φ ρ
φ
φ
= − +
=
−
=
= − +
= − +
+ − −
=
+ + −
(5)
Here φ is the particle volume fraction. The thermophysical properties of base fluid and nanoparticle
are distinguished by subscripts f and s respectively.
3. VARIATIONAL FORMULATION OF THE PROBLEM
The purpose of this analysis is to obtain the approximate numerical solution of irreversible
thermodynamics problem by a variational technique. Gyarmati [17, 18] developed a variational
principle known as “Governing Principle of Dissipative Processes” (GPDP) which is given in its
universal form
V
( )dV 0.δ σ − − =∫ ψ Φψ Φψ Φψ Φ (6)
The principle (6) describes the evaluation of linear, quasi linear and some nonlinear
irreversible processes at any instant of time and space under constraints that the balance equations
, ( 1,2,3, )i i ia i fρ σ+ ∇⋅ = =J& L (7)
are satisfied. In Equation (6), δ is the variational symbol, σ is the entropy production, ψψψψ and ΦΦΦΦ are
dissipation potentials and V is the total volume of the thermodynamic system. In Equation (7), ρ is
the mass density and ia& , Ji, σi are respectively substantial variation, flux and source density of the ith
extensive transport quantity ai. The entropy production σ per unit volume and unit time can always
be written in the bilinear form

1
0
f
i
i iσ
=
= ⋅ ≥∑J X (8)
where Ji and Xi are fluxes and forces respectively. According to Onsager’s linear theory [19, 20]
the fluxes are linear functions of forces, that is
1
, ( 1,2,3, )
f
i ik k
k
L i f
=
= =∑J X L (9)
or alternatively
1
, ( 1,2,3, )
f
i ik k
k
R i f
=
= =∑X J L (10)
The constants Lik and Rik are conductivities and resistances respectively satisfying the reciprocal
relations [19, 20]
Lik = Lki and Rik = Rki, (i, k = 1,2,3,…f ) (11)
The matrices of Lik and Rik are mutually reciprocals and they are symmetric, that is
1 1
, ( , 1,2,3, )
f f
im mk mk im ik
m m
L R L R i k fδ
= =
= = =∑ ∑ L (12)
where δik is the Kronecker delta. The local dissipation potentials ψψψψ and ΦΦΦΦ are defined [19, 20] as,
, 1
( , ) (1/ 2) 0
f
ik i k
i k
L
=
= ⋅ ≥∑X X X Xψψψψ (13)
, 1
( , ) (1/ 2) 0
f
ik i k
i k
R
=
= ⋅ ≥∑J J J JΦΦΦΦ (14)
In the case of transport processes, the forces Xi can be generated as gradients of certain “Γ” variables
and can be written as
Xi =∇Γi (15)
The principle (6) with the help of Equations (8), (13), (14) and (15), takes the form
1 , 1 , 1V
(1/ 2) (1/ 2) V 0
f f f
i i ik i k ik i k
i i k i k
L R dδ
= = =
 
⋅∇Γ − ∇Γ ⋅∇Γ − ⋅ = 
 
∑ ∑ ∑∫ J J J (16)
This variational principle has been already applied for various dissipative systems and was
established as the most general and exact variational principle of macroscopic continuum physics.
Many other variational principles have already been shown as partial forms of Gyarmati’s principle.
The balance equations of the system play a central role in the formulation of Gyarmati’s
variational principle and hence the governing boundary layer Equations (1-3) are written in the
balance form as
0, ( )u v∇⋅ = = +V V i j (17)

2
0( ) P = ( B )[ ( )] ( ) ( )nf nfU g T Tρ κ ρβ∞ ∞⋅∇ + ∇⋅ − ⋅ + −V V i V i (18)
2 2
0 0( ) ( ) = ( ) ( B )( )[ ( )] ( )p nf q nf yC T u U Q T Tρ µ κ ∞ ∞⋅∇ +∇⋅ − ⋅ − ⋅ + −V J i V i V (19)
These equations represent the mass, momentum and energy balances respectively. Here i and
j being unit vectors in the directions of x and y axes respectively. In Equation (18) P denotes the
pressure tensor which can be decomposed [17] as
o
P Pvs
pδ= + (20)
where p is the hydrostatic pressure, δ is the unit tensor and
o
Pvs
is the symmetrical part of the viscous
pressure tensor, whose trace is zero.
In the study of heat transfer and fluid flow problems, the energy picture of Gyarmati’s
principle is always advantageous over entropy picture. Therefore, the energy dissipation Tσ is used
instead of entropy production σ. The energy dissipation for the present system is given [17] by,
12( / ) ( / )qT J lnT y P u yσ = − ∂ ∂ − ∂ ∂ (21)
where Jq is the heat flux and P12 is the only component of momentum flux
o
Pvs
, satisfy the
constitutive relations connecting the independent fluxes and forces as
12( / ) and ( / )q sJ L lnT y P L u yλ= − ∂ ∂ = − ∂ ∂ (22)
Here Lλ = λT and Ls = µ, where λ and µ are the thermal conductivity and viscosity
respectively. With the help of Equations (22) the dissipation potentials in energy picture are found as
follows
2 2
(1/ 2) ( ( / )sT L lnT / y) L u yλ
 = ∂ ∂ + ∂ ∂ ψψψψ (23)
2 2
12(1/ 2) q sT R J R Pλ
 = + ΦΦΦΦ (24)
where 1 1
and s sL R L Rλ λ
− −
= = .
Using Equations (21-24), Gyarmati’s variational principle (6) is formulated in the following form
2
12
2 2 2
0 0 12
( ) ( ) ( 2)( )
0
( 2)( ) ( 2) ( 2)
l
q
s q s
J lnT y P u y L lnT y
dydx
L u y R J R P
λ
λ
δ
∞ − ∂ ∂ − ∂ ∂ − ∂ ∂
 =
− ∂ ∂ − −  
∫∫ , (25)
in which l is the representative length of the surface.
4. METHOD OF SOLUTION
It is assumed that the trial functions for velocity and temperature fields inside the respective
boundary layers are as follows

3 3 4 4
1 1 1 1
1
0
3 3 4 4
2 2 2 2
2
2 2 ( )
( )
( ) ( )
1 2 2 ( )
( )
u U y d y d y d y d
u U y d
T T T T
y d y d y d y d
T T y d
θ
∞
∞
∞ ∞
∞
= − + <
= ≥
− − =
= − + − <
= ≥
(26)
where d1, d2 are the velocity and temperature boundary layer thicknesses which are to be determined
from the variational procedure. The trial functions (26) satisfy the following compatibility
conditions,
y = 0 ; u = 0, v = 0v , T = T0, ∂ 2
T/∂ y2
= 0
y = d1 ; u = U∞ = constant, ∂ u /∂ y = 0 (smooth fit), ∂ 2
u /∂ y2
= 0 (27)
y = d2 ;T = T∞, ∂ T/∂ y = 0 (smooth fit), ∂ 2
T/∂ y2
= 0
The smooth fit conditions ∂ u /∂ y = 0 and ∂ T/∂ y = 0 correspond to P12 = 0 and Jq = 0 at
their respective edges of the boundary layer. Using the boundary conditions (27), the transverse
velocity component v is obtained from the mass balance equation (17) as
5 5 4 4 2 2 '
1 1 1 1 0(4 / 5 3 / 2 / )y dv y d y d d vU∞ − + += , (28)
where 0v is the suction/injection velocity.
The velocity and temperature functions (26) and the boundary conditions (27) are used in the
governing boundary layer Equations (17-19) and on direct integration with respect to y with the help
of their corresponding smooth fit conditions uy = 0 and Ty = 0, the momentum flux P12 and energy
flux Jq are obtained. The momentum flux P12 remains the same for any Prandtl number Pr but the
energy flux Jq has different expressions for Pr ≤ 1 and Pr ≥ 1. When Pr ≤ 1 the expression for Jq in
the range d1 ≤ y ≤ d2 is obtained first and the expression for Jq in the range 0 ≤ y ≤ d1 is determined
subsequently by matching the expressions of the two regions at the interface. The expressions for
momentum and the energy fluxes P12 and Jq are obtained respectively as follows,
2 9 9 8 8 7 7 6 6 5 5
12 1 1 1 1 1 1
3 3 4 4 3 3
1 0 1 1 1
2 5 4 4 3 2
0 1 1 1 1 1
5 4 4 3 2
0 2 2 2
/ ( / )( 4 45 2 5 3 7 11 15 7 5
2 3 101/1800) ( )( 2 2 7/10)
( )( 5 2 7 30)
( )/ ( 5 2
s nf
nf
nf
nf nf
P L U d y d y d y d y d y d
y d vU y d y d y d
BU y d y d y d y d U d
gT T y d y d y d y
υ
υ
κ µ
β υ
∞
∞
∞ ∞
∞
′− = − + − − +
− + + − + −
+ − + − + +
− − − + − + + 5 4 4 3
1 2 1 2
2
1 2 1 1
30 10
3 2) (0 )
d d d d
d d d y d
−
+ − ≤ ≤
(29)

9 4 5 8 3 5 8 4 4 7 3 4
0 2 1 2 1 2 1 2 1 2
6 5 6 4 2 5 4 5 3 2 3 2 5 5
1 2 1 2 1 2 1 2 1 2 1 2
4 4 2 2 9 5 4 8 5 3
1 2 1 2 1 1 2 1 2
8 4 4
1 2
( ) (4 9 3 4 12 7
4 3 3 12 5 4 5 4 3 45
9 140 2 15 3/10) ( 16 45 3 5
3 4 9
q nfJ L U T T d y d d y d d y d d y d d
y d d y d d y d d y d d y d d d d
d d d d d y d d y d d
y d d
λ α∞ ∞
 ′ − = − − − + 
+ + − − + +
′− + − + − +
+ − 7 4 3 6 2 4 6 5 5 4
1 2 1 2 1 2 1 2
5 2 3 3 2 4 4 3 3
1 2 1 2 1 2 1 2 1 2
4 4 3 3 4 4 3 3
0 2 2 2 1 2 1 2 1 2
2 7 8 6 7 5 6 4 5
1 1 1 1
3
1
7 2 3 4 15 3 5
6 5 2 3 49 180 18 35 3 )
( )( 2 2 2 2 )
( ( ) )( 16 7 8 36 5 4
8
nf nf p nf
y d d y d d y d d y d d
y d d y d d d d d d d d
v U y d y d y d d d d d d d
U C y d y d y d y d
y d
µ α ρ
∞
∞
− − +
+ − + − +
+ − + − + − + 
+ − + − −
+ 4 2 2 2 9 8
1 1 0 1
8 7 7 6 6 5 5 4 4 3 3 2
1 1 1 1 1 1
2 5 4 4 3 2
1 1 0 0 2 2 2
2 1
4 52 35 ) ( ( ) )( 9
2 4 7 2 3 9 5 2 4 3
37 315) ( ( ) ( ) )( 5 2
3 10) (0 ); ( 1)
nf p nf
nf p nf
y d d B U C y d
y d y d y d y d y d y d
y d d Q T T C y d y d y d
y d y d Pr
κ α ρ
α ρ
∞
∞
− + + −
+ − − + − −
+ − + − − +
− + ≤ ≤ ≤
(30)
5 5 4 4 2 2
0 2 2 2 2
5 4 4 3 2
0 0 2 2 2 2
1 2
( ) (4 5 3 2 3/10)
( ( ) ( ) )( 5 2 3 10)
( ); ( 1)
q nf
nf p nf
J L U T T d y d y d y d
Q T T C y d y d y d y d
d y d Pr
λ α
α ρ
∞ ∞
∞
 ′− = − − + −
 
+ − − + − +
≤ ≤ ≤
(31)
9 4 5 8 3 5 8 4 4 7 3 4
0 2 1 2 1 2 1 2 1 2
6 5 6 4 2 5 4 5 3 2 3 2
1 2 1 2 1 2 1 2 1 2
4 4 3 3 9 5 4 8 5 3
2 1 2 1 2 1 1 1 2 1 2
8 4 4 7 4
1 2 1 2
( ) (4 9 3 4 12 7
4 3 3 12 5 4 5 4 3
36 3 35 4 15 ) ( 16 45 3 5
3 4 9 7
q nfJ L U T T d y d d y d d y d d y d d
y d d y d d y d d y d d y d d
d d d d d d d y d d y d d
y d d y d d
λ α∞ ∞
 ′ − = − − − + 
+ + − − +
′− + − + − +
+ − 3 6 2 4 6 5 5 4
1 2 1 2 1 2
5 2 3 3 2 5 5 4 4 2 2
1 2 1 2 2 1 2 1 2 1
4 4 3 3 2
0 2 2 2
7 8 6 7 5 6 4 5 3 4 2
1 1 1 1 1 1
7 8 6
2 1 2
2 3 4 15 3 5
6 5 2 3 45 9 140 2 15 )
( )( 2 2 1) ( ( ) )
(16 7 8 36 5 4 8 4 (32)
16 7 8
nf nf p nf
y d d y d d y d d
y d d y d d d d d d d d
v U y d y d y d U C
y d y d y d y d y d y d
d d d d
µ α ρ∞ ∞
− − +
+ − + − +
+ − + − + −
− + + − +
− + 7 5 6 4 5 3 4
1 2 1 2 1 2 1
2 2 2 9 8 8 7 7 6
2 1 0 1 1 1
6 5 5 4 4 3 3 2 2 9 8
1 1 1 1 1 2 1
8 7 7 6 6 5 5 4 4 3 3 2
2 1 2 1 2 1 2 1 2 1 2 1
2
2 1 0 0
36 5 4 8
4 ) ( ( ) )( 9 2 4 7
2 3 9 5 2 4 3 9
2 4 7 2 3 9 5 2 4 3
) ( ( ) (
nf p nf
nf p
d d d d d d
d d B U C y d y d y d
y d y d y d y d y d d d
d d d d d d d d d d d d
d d Q T T C
κ α ρ
α ρ
∞
∞
− − +
− + − + −
− + − − + +
− + + − + +
− + − 5 4 4 3 2
2 2 2 2
2
) )( 5 2 3 10)
(0 ); ( 1)
nf
y d y d y d y d
y d Pr
− + − +
≤ ≤ ≥
The prime indicates differentiation with respect to x. Using the expressions of P12 and Jq
together with velocity and temperature functions (26), the variational principle (25) is formulated
independently for Pr ≤ 1 and Pr ≥ 1 cases. After performing the integration with respect to y, one can
obtain the variational principle in the following forms,
1 1 2 1 2
0
[ , , , ] 0, ( 1)
l
L d d d d dx Prδ ′ ′ = ≤∫ (33)

2 1 2 1 2
0
[ , , , ] 0, ( 1)
l
L d d d d dx Prδ ′ ′ = ≥∫ (34)
where L1, L2 are the Lagrangian densities of the principle. The variation is carried out with respect to
the independent parameters d1 and d2. These variational principles (33), (34) are found identical
when d1=d2.
The Euler-Lagrange equations corresponding to these variational parameters are
1,2 1 1,2 1( ) ( )( ) 0L d d dx L d′∂ ∂ − ∂ ∂ = (35)
1,2 2 1,2 2( ) ( )( ) 0, ( 1, 1)L d d dx L d Pr Pr′∂ ∂ − ∂ ∂ = ≤ ≥ (36)
where L1,2 represents the Lagrangian densities L1 and L2 respectively. These Equations (35) and (36)
are second order ordinary differential equations in terms of d1 and d2. The procedure for solving
Equations (35) and (36) can be considerably simplified by introducing the non-dimensional
boundary layer thicknesses *
1d , *
2d and are given by
* *
1 1 2 2/ and /f fd d x U d d x Uυ υ∞ ∞= = (37)
These variational principles (33) and (34) are subject to the transformations (37). The
resulting Euler-Lagrange equations are obtained as simple polynomial equations,
*
1,2 1 0L d∂ ∂ = (38)
*
1,2 2 0,( 1, 1)L d Pr Pr∂ ∂ = ≤ ≥ (39)
The coefficients of these Equations (38) and (39) dependent on the independent parameters
Pr, ξ, Q, Ri, H and Ec where f fPr υ α= (Prandtl number), 2
0B x Uξ κ ρ ∞= (magnetic parameter),
0 / ( )p fQ Q x U Cρ∞= (heat generation/ absorption parameter) Ri=Gr/Re2
(Richardson number),
3 2
0( )f fGr g T T xβ υ∞= − (Grashof number), fRe U x υ∞= (Reynolds number), 0 fH v x Uυ ∞=
(suction/injection parameter) and 2
0( )pEc U C T T∞ ∞= − (Eckert number).
In the present analysis heat generation and absorption are presented by Q > 0 and Q < 0
respectively and the suction and injection are represented by H < 0 and H > 0 respectively.
Equations (38) and (39) are simple coupled polynomial equations and it can be solved for any values
of Pr, ξ, Q, Ri, H and Ec and it is found that the obtained simultaneous solutions *
1d and *
2d are as
the only one set of positive real roots. After obtaining the values of *
1d and *
2d for given Pr, ξ, Q, Ri,
H and Ec the values of velocity, temperature profiles, skin friction (shear stress) and heat transfer
(local Nusselt number) are calculated with the help of the following expressions,
fy U xη υ∞= (40)
3
12 0( )w f s yx U P Lτ υ ∞ == − (41)

and
2
0 0( ) ( )l f q yNu x U T T J Lλυ ∞ ∞ == − . (42)
5. RESULTS AND DISCUSSION
The main and important characteristics of the problem analyzed are skin friction and heat
transfer values. The energy equation has been solved for two cases * *
1 2 ( 1)d d Pr≤ ≤ and
* *
1 2 ( 1)d d Pr≥ ≥ . These two independent analyses yield solutions and it is matching at Pr =1. It is
found that both the analyses lead to satisfactory results in the respective ranges of Pr.
The thermophysical properties of water and nanoparticles given in Table 1 are used to
compute each case of nanofluid.
Table 1: Thermophysical properties of water and nanoparticles.
ρ (kgm-3
) Cp (Jkg-1
K-1
) k (Wm-1
K-1
) β×10-5
(K-1
)
H2O 997.1 4179 0.613 21
Al2O3 3970 765 40 0.85
Cu 8933 385 401 1.67
Ag 10500 235 429 1.89
It is customary that when a new mathematical method is applied to a problem, the obtained
results are compared with the available solution in order to determine the accuracy of the results
involved in the present technique.
In Table 2, the heat transfer values of regular fluid for various values of Pr (Pr ≤ 1 and Pr ≥
1) when ξ = Q = Ri = H = Ec = 0 are obtained by the present variational technique. From this table
it is evidently clear that the present results are in good agreement with Chamkha et al. [21], Aydin
and Kaya [16], Rana and Bhargava [7]. It is also observed that the heat transfer increases with the
values of Prandtl number. Since the higher Prandtl number has very low thermal conductivity, the
local Nusselt number increases rapidly. This means that the variation of the heat transfer rate is more
sensitive to the larger Prandtl number than the smaller one.
Table 2: Local Nusselt number for various values of Pr when ξ = Q = Ri = H = Ec = φ =0.
Pr
Present Results
Nul
Chamkha et al. [21]
Nul
Aydin & Kaya [16]
Nul
Rana & Bhargava [7]
Nul
0.01 0.054742313 0.051830 0.051437 0.0596
0.1 0.147754551 0.142003 0.148123 0.1579
1 0.334277544 0.332173 0.332000 0.3319
10 0.738452128 0.728310 0.727801 0.7278
100 1.599967934 1.572180 1.573141 1.5721
Figs. 1-4, represent the effects of buoyancy parameter Ri on the velocity profile, temperature
profile, local Nusselt number and skin friction respectively. These results are obtained for Pr = 6.2,
Q = 0.05 corresponding to pure water and copper-water nanofluid with volume fraction φ = 0.04.

Fig. 1: Velocity profile for different values of Fig. 2: Temperature profile for different
Ri with Pr = 6.2 and Q = 0.05 when ξ = H= values of Ri with Pr = 6.2 and Q = 0.05
Ec = 0. when ξ = H = Ec = 0.
From Figs. 1 and 2, it can be easily observed that as buoyancy force increases accordingly,
the non-dimensional velocity also increases and the temperature profile decreases. In addition, the
effect of Cu/H2O nanofluid on velocity and temperature profiles is depicted that nanofluid makes an
increase in temperature profile also it causes decrease in velocity profile as compared to pure water.
Figs. 3 and 4 represent respectively the local Nusselt number and skin friction values as a
function of magnetic parameter ξ, for different values of Ri. From these two figures it is observed
that both local Nusselt number and skin friction increases with buoyancy parameter Ri and due to the
higher thermal conductivity of nanofluid, heat transfer as well as skin friction increase when
compared to the pure water.
Fig. 3: Variation of local Nusselt number as a Fig. 4: Skin friction values as a function of
function of ξ for different values of Ri with ξ for different values of Ri with Pr = 6.2
Pr = 6.2 and Q = 0.05 when H = Ec = 0. and Q = 0.05 when H = Ec = 0.

Figs. 5-8 present the effects of three different types of nanofluids containing the
nanoparticles, namely Copper (Cu), Alumina (Al2O3) and Silver (Ag) on velocity, temperature,
Nusselt number and skin friction respectively. These results are obtained by considering Pr = 6.2, Ri
= 1, Q = 0.05 and the volume fraction as φ = 4%. The velocity profile increases from Al2O3 to Ag and
the trend reverses in thermal boundary layers as shown in Figs. 5 and 6.
Fig. 5: Velocity profile for different nano- Fig. 6: Temperature profile for different
particles with φ = 0.04, Pr = 6.2, Ri = 1 and nanoparticles with φ = 0.04, Pr = 6.2,
Q = 0.05 when ξ = H = Ec = 0. Ri = 1 and Q = 0.05 when ξ = H = Ec = 0.
From Figs. 7 and 8, it is found that the nanofluid has higher values in heat transfer rates and
skin friction when it is compared with pure water. In addition, the heat transfer rate in Cu/H2O
nanofluid is higher than Ag/H2O nanofluid even though Ag has higher thermal conductivity than that
of Cu and also the skin friction increases from Al2O3/H2O nanofluid to Ag/H2O nanofluid.
Fig. 7: Variation of local Nusselt number as a Fig. 8: Skin friction values as a function of
a function of ξ for different nanoparticles with ξ for different nanoparticles with φ = 0.04,
φ = 0.04, Pr = 6.2, Ri = 1 and Q = 0.05 when Pr = 6.2, Ri = 1 and Q = 0.05 when
H = Ec = 0. H = Ec = 0.

Fig. 9: Velocity profile for different volume Fig. 10: Temperature profile for different
fraction of Cu-Water nanofluid Pr = 6.2, volume fraction of Cu-Water nanofluid
Ri = 1 and Q = 0.05 when ξ = H = Ec = 0. Pr = 6.2, Ri = 1 and Q = 0.05 when ξ = H = Ec = 0.
In Figs. 9-12, the effects of the volume fraction (φ) on velocity, temperature, Nusselt number
and skin friction are presented respectively. The numerical results are obtained by considering Pr =
6.2, Q = 0.05 and Ri = 1. For increasing volume fraction, the velocity profile decreases but the
increase is not in significant level. The thermal boundary layer increases with volume fraction. These
studies show that thermal conductivity of the fluid-particle system increases when volume fraction
increases. Hence heat transfer increases with volume fraction as shown in Fig. 11 and the skin
friction also follows the same trend as in Fig. 12.
Fig. 11: Variation of local Nusselt number as a Fig. 12: Skin friction values as a function
function of ξ for different volume fraction of of ξ for different volume fraction of
Cu-Water nanofluid with Pr = 6.2, Ri = 1 and Cu-Water nanofluid with Pr = 6.2, Ri = 1
Q = 0.05 when H = Ec = 0. and Q = 0.05 when H = Ec = 0.

Fig. 13: Velocity profile for different values of Fig. 14: Temperature profile for different
heat source/sink parameter Q with Pr = 6.2 and values of heat source/sink parameter Q
Ri = 1 when ξ = H = Ec = 0. with Pr = 6.2 and Ri =1 when ξ=H=Ec =0.
Figs. 13-16 describe the effects of heat generation or absorption (Q) on velocity, temperature,
Nusselt number and skin friction respectively. The results obtained for Pr = 6.2, Ri = 1
corresponding to pure water and copper-water nanofluid with volume fraction φ = 0.04. It is
observed that increasing of heat generation or adsorption (Q) increases both velocity and temperature
profiles.
Fig. 15: Variation of the local Nusselt number Fig. 16: Skin friction values as a function of
as a function of ξ for different values of heat ξ for different values of heat source/sink
source/sink parameter Q with Pr = 6.2 and parameter Q with Pr = 6.2 and Ri = 1
Ri = 1 when H = Ec = 0. when H = Ec = 0.
From Figs. 15 and 16, it is observed that the local Nusselt number decreases as Q increases.
Contrarily skin friction increases with the increasing values of Q. From this analysis it is evidently
clear that heat transfer rate with in the boundary layer is enhanced by a nanofluid when we compared
to the conventional fluid.

6. CONCLUSION
This work deals with the effects of heat generation/absorption, buoyancy parameter, volume
fraction, different types of nanofluids, skin friction and surface heat transfer over a semi infinite flat
plate. The governing partial differential equations are reduced to simple polynomial equations whose
coefficients are of independent parameters Pr, ξ, Q, Ri, H and Ec. These equations offer a practicing
engineer a rapid way of obtaining shear stress and heat transfer for any combinations of Pr, ξ, Q, Ri,
H and Ec. The great advantage involved in the present technique is that the results are obtained with
high order of accuracy and the amount of calculation is certainly less when compared with more
conventional methods. Hence the practicing engineers and scientists can employ this unique
approximate technique as a powerful tool for solving boundary layer flow and heat transfer
problems. Further, the work can be extended by considering Brownian motion and thermophoresis
effects in the nanofluid flow model.
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