Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2717-2733
© Research India Publications
http://www.ripublication.com
MHD Boundary Layer Flow of a VISCO-Elastic
Fluid Past a Porous Plate with Varying Suction and
Heat Source/Sink in the Presence of Thermal
Radiation and Diffusion
K. Vijayakumar1 and E. Keshava Reddy1
Department of Mathematics, JNTUA College of Engineering Anantapur,
Anantapuramu -510003, A.P, India.
Abstract
This manuscript consists of the properties of natural convective flow of a viscous
incompressible electrically conducting fluid past a vertical porous plate bounded
by a porous medium in the presence of thermal radiation and variable
permeability. A magnetic field of uniform strength is applied perpendicular to the
plate and the presence of heat source is also considered. The novelty of the study
is to investigate the effects of thermal radiation and Eckert number. The coupled
dimensionless non-linear partial differential equations are solved by finite
difference method. The numerical computations have been studied through
graphs. The presence of thermal radiation decreases the temperature and an
opposite nature is shown in the case of Eckert number. The influence of neat
source leads to enhance the temperature.
Keywords: MHD, thermal radiation, variable suction, variable permeability,
vertical porous plate, heat and mass transfer.
1. INTRODUCTION:
The studies related to MHD visco-elastic fluids with radiation effect past a porous
media in the presence of heat source/sink plays significant role in many scientific,
industrial and engineering applications. These flows were basically utilized in the
fields of petroleum engineering concerned with the oil, gas and water through
reservoir to the hydrologist in the analysis of the migration of underground water. To
recover the water for drinking and irrigation purposes the principles of this flow are
followed. In the recent days many researchers recognized the significance of these
2718 K. Vijayakumar and E. Keshava Reddy
flows and contributed in studying the application of visco-elastic fluid flow of several
types past porous medium in the presence of thermal radiation and heat source/sink.
Chen [1] studied magneto-hydrodynamic mixed convection of a power-law fluid past
a stretching surface in the presence of thermal radiation and internal heat
generation/absorption. Hayat et al. [2] analyzed Soret and Dufour effects for three-
dimensional flow in a viscoelastic fluid over a stretching surface. Ganesan et al. [3]
analyzed and reported radiation and mass transfer effects on flow of an
incompressible viscous fluid past a moving vertical cylinder. Lavanya et al. [4]
discussed radiation and mass transfer effects on unsteady MHD natural convective
flow past a vertical porous plate embedded in a porous medium in a slip flow regime
with heat source/sink and Soret effect. Hayat et al. [5] explained slip and Joule
heating effects in mixed convection peristaltic transport of nanofluid with Soret and
Dufour effects. Abo-Eldahab et al. [6] considered blowing/suction effect on
hydromagnetic heat transfer by mixed convection from an inclined continuously
stretching surface with internal heat generation/absorption. Chamka [7] examined
double-diffusive convection in a porous enclosure with cooperating temperature and
concentration gradients and heat generation or absorption effects. Ahmed et al. [8]
studied boundary layer flow and heat transfer due to permeable stretching tube in the
presence of heat source/sink utilizing nanofluids. Hayat et al. [9] explained effect of
Joule heating and thermal radiation in flow of third-grade fluid over radiative surface.
Motsumi et al. [10] discussed effects of thermal radiation and viscous dissipation on
boundary layer flow of nanofluids over a permeable moving flat plate. Mishra et al.
[11] considered and pointed out the mass and heat transfer effect on MHD flow of a
visco-elastic fluid through porous medium with oscillatory suction and heat source.
Uma et al. [12] discussed Unsteady MHD free convective visco-elastic fluid flow
bounded by an infinite inclined porous plate in the presence of heat source, viscous
dissipation and Ohmic heating. Motivated by the above investigations, in this work a
fully developed free convective flow of a viscous incompressible electrically
conducting fluid past a vertical porous plate bounded by a porous medium in the
presence of thermal radiation, heat source/sink, variable suction and variable
permeability is analyzed. We have extended the work of Chandra Reddy et al. [13]
with the novelty of considering Eckert number, heat generation/absorption and
thermal radiation.
We also have gone through the literature related to the present work and followed.
Chandra Reddy et al. [14, 15] analyzed magnetohydrodynamic convective double
diffusive laminar boundary layer flow past an accelerated vertical plate as well as
Soret and Dufour effects on MHD free convection flow of Rivlin-Ericksen fluid past a
semi infinite vertical plate. Further Chandra Reddy et al. [16] studied the properties of
free convective magneto-nanofluid flow past a moving vertical plate in the presence
of radiation and thermal diffusion. Mahdy et al. [17] studied thermosolutal marangoni
boundary layer magnetohydrodynamic flow with the Soret and Dufour effects past a
vertical flat plate. Kairi and Murthy [18] discussed the effect of melting and thermo-
diffusion on natural convection heat mass transfer in a non-Newtonian fluid saturated
non-Darcy porous medium. Swati & Murthy [19] analyzed Magnetohydrodynamic
MHD Boundary Layer Flow of a VISCO-Elastic Fluid Past a Porous Plate… 2719
(MHD) mixed convection slip flow and heat transfer over a vertical porous plate.
Mukhopadhyay et al. [20] investigated forced convective flow and heat transfer over a
porous plate in a Darcy-Forchheimer porous medium in presence of radiation. Ahmed
and Kalita [21] discussed magnetohydrodynamic transient flow through a porous
medium bounded by a hot vertical plate in presence of radiation. Kumar and Verma
[22] considered thermal radiation and mass transfer effects on MHD flow past a
vertical oscillating plate with variable temperature effects variable mass diffusion.
Ravi kumar et al. [23] examined combined effects of heat absorption and MHD on
convective Rivlin-Ericksen flow past a semi-infinite vertical porous plate with
variable temperature and suction. Sharma and Ajaib [24] investigated bounds for
complex growth rate in thermosolutal convection in Rivlin–Ericksen viscoelastic fluid
in a porous medium. Raju et al. [25] considered and studied MHD thermal diffusion
natural convection flow between heated inclined plates in porous medium. Raju et al.
[26] attended and discussed Soret effects due to Natural convection between heated
inclined plates with magnetic field. Makinde et al. [27] considered and established
free convection flow with thermal radiation and mass transfer past a moving vertical
porous plate. Samad et al. [28] analyzed thermal radiation interaction with unsteady
MHD flow past a vertical porous plate immersed in a porous medium. Bhattacharyya
et al. [29] analyzed effects of thermal radiation on micropolar fluid flow and heat
transfer over a porous shrinking sheet. Hayat et al. [30] discussed heat and mass
transfer for Soret and Dufour effects on mixed convection boundary layer flow over a
stretching vertical surface in a porous medium filled with a viscoelastic fluid. Khan
and Aziz [31] examined the properties of natural convection flow of a nanofluid over
a vertical plate with uniform surface heat flux.Rohni et al. [32] considered unsteady
mixed convection boundary-layer flow with suction and temperature slip effects near
the stagnation point on a vertical permeable surface embedded in a porous medium.
Nomenclature:
Cl
D
Gr
Species concentration
Molecular diffusivity
Grashof number of heat transfer
C Non-dimensional Species
concentration
σ Electrical conductivity ω Non-dimensional frequency of
oscillation
K1 Permeability of the medium Gc Grashof number for mass transfer
k Thermal diffusivity G Acceleration due to gravity
Rc
Ec
Elastic parameter
Eckert number
Kp
F
Porosity parameter
Thermal radiation
Nu Nusselt number M Magnetic parameter
S Heat source parameter B0 Magnetic field of uniform strength
Sh Sherwood number Pr Prandtl number
T Non-dimensional temperature Sc Schmidt number
T Non-dimensional time T1 Temperature of the field
2720 K. Vijayakumar and E. Keshava Reddy
u Non-dimensional velocity tl Time
vo Constant suction velocity ul Velocity component along x-axis
y Non-dimensional distance along y-axis V Suction velocity
Ԑ A small positive constant yl Distance along y-axis
β Volumetric coefficient of expansion for
heat transfer
ρ
τ
Density of the fluid
Skin friction
β1 Volumetric coefficient of expansion
with species concentration
ω1
So
Frequency of oscillation
Soret number
υ Kinematic coefficient of viscosity W Condition on porous plate
2. FORMULATION OF THE PROBLEM:
The unsteady free convection heat and mass transfer flow of a well-known non-
Newtonian fluid, namely Walters B visco-elastic fluid past an infinite vertical porous
plate, embedded in a porous medium in the presence of thermal radiation, oscillatory
suction as well as variable permeability is considered. In addition to this the existence
of heat generation / absorption is also considered. A uniform magnetic field of
strength B0 is applied perpendicular to the plate. Let x1 axis be taken along with the
plate in the direction of the flow and y1 axis is normal to it. Let us consider the
magnetic Reynolds number is much less than unity so that the induced magnetic field
is neglected in comparison with the applied transverse magnetic field. The basic flow
in the medium is, therefore, entirely due to the buoyancy force caused by the
temperature difference between the wall and the medium. It is assumed that initially,
at tl ≤ 0, the plate as fluids are at the same temperature and concentration. When tl >
0, the temperature of the plate is instantaneously raised to 1
wT and the concentration of
the species is set to 1
wC . Under the above assumption with usual Boussinesq’s
approximation, the governing equations and boundary conditions are given by
1
13
11
13
0
11
1
12
0111
1
12
1
1
1
1
2
2
)(
)()(
y
uvytuk
tKu
uBCCgTTgyu
yuv
tu
(1)
2
2*1 1 2 1 *
1 1
1 1 * *1( ) rqT T T uv S T T
t y y yy
(2)
22 1
12
11
12
1
1
1
1
yTD
yCD
yCv
tC
(3)
MHD Boundary Layer Flow of a VISCO-Elastic Fluid Past a Porous Plate… 2721
with the boundary conditions
,)(,0111 tn
ww eTTTTu 0)(111 yateCCCC tn
ww (4)
yasCCTTu 11 ,,0
Let the permeability of the porous medium and the suction velocity be of the form
)1()(11111 tn
p eKtK (5)
)1()(11
0
1 tnevtv (6)
where v0 ˃ 0 and ԑ « 1 are positive constants.
Introducing the non-dimensional quantities
(7)
The equations (1), (2), (3) reduce to the following non-dimensional form: 2
2
2
3
2
1(1 )
4
(1 ) 4
nt
nt
u u ue GrT GcC M ut y y
u Rc uKp e t y
(8)
(9)
2 2
2 2
1 1(1 )
4
ntC C C Te Sot y Sc y y
(10)
11 2 1 11 1
0 0
2
0 0
2 1 2 210 2 0 0 0
2 2 2 2
0 0
11
1
2 3 3
0 0 0
*
4, , , , , ,
4
, , Pr , , , ,
( ) ( ) ( )4, , , ,
( )
4(
w w
p
w w w
w
r
v y v t T T C Cw uy t w u T Cv v T T C C
v K B k vSS Kp Sc M Rcv D v
g C C g T T D T Tnn Gc Gr Sov v v C C
q T Ty
2
0
2 1 10
4) , , .
p p w
UII F EC U C T T
22
2
1 1(1 )
4 Pr
ntT T T ue E ST FTt y y y
2722 K. Vijayakumar and E. Keshava Reddy
with the boundary conditions
01,1,0 yateCeTu ntnt
yasCTu 0,0,0 (11)
3. SOLUTION OF THE PROBLEM:
Equations (8)-(10) are coupled non-linear partial differential equations and are to be
solved by using the initial and boundary conditions (11). However exact solution is
not possible for this set of equations and hence we solve these equations by finite-
difference method. For this, a rectangular region of the flow field is chosen, and the
region is divided into a grid of lines parallel to X and Y-axes, where the X-axis is
taken along the plate and the Y-axis is normal to the plate as shown in Fig.1.
Fig. 1 Finite difference space grid
The equivalent finite difference schemes of equations for (8)-(10) are as follows:
, 1 , 1, , 1, , 1,
, , 2
1, 1 , 1 1, 1 1, , 1, 2,2
,
21(1 )
4
2 2
4
1
(1 )
i j i j i j i j i j i j i jnti j i j
i j i j i j i j i j i ji j
i jntp
u u u u u u ue GrT GcC
t y yu u u u u uRc M u
t y
uK e
(12)
MHD Boundary Layer Flow of a VISCO-Elastic Fluid Past a Porous Plate… 2723
2
, 1 , , 1 , 1, , 1, 1, ,
2
, ,
21 1(1 )
4 Pr
i j i j i j i j i j i j i j i j i jnt
i j i j
T T T T T T T u ue E
t y yyST FT
(13)
, 1 , , 1 , 1, , 1,
2
1, , 1,
2
21 1(1 )
4
2
i j i j i j i j i j i j i jnt
i j i j i j
C C C C C C Ce
t y Sc y
T T TSo
y
(14)
Here, the index i refer to y and j to time. The mesh system is divided by taking
∆y = 0.1. From equation (11), we have the following equivalent initial condition
( ,0) 0, ( ,0) 0, ( ,0) 0u i T i C i for all i (15)
The boundary conditions from (11) are expressed in finite-difference form as follows
1, 1,
max max max
(0, ) 1, (0, ) 1, 2.
( , ) 0, ( , ) 0, ( , ) 0
i j i ju j T j C C y for all ju i j T i j C i j for all j
(16)
(Here imax was taken as 20)
First the velocity at the end of time step viz, u(i,j+1)(i=1,20) is calculated from (12) in
terms of velocity, temperature and concentration at points on the earlier time-step.
Then T(i, j +1) is computed from (13) and C(i, j +1) is computed from(14). The
procedure is repeated until t = 0.5 (i.e. j = 500). During computation ∆t was chosen as
0.001.
4. RESULTS AND DISCUSSION:
The influence of different physical parameters like Grashof number modified Grashof
number, magnetic parameter, thermal radiation, Prandtl number, Eckert number, Soret
number and Schmidt number on velocity, temperature and concentration is discussed
by using graphical representations. The general nature of the velocity profile is
parabolic with picks near the plate. Figure 2 show that the velocity enhances for rising
values of magnetic parameter. The effect of Prandtl number on velocity is displayed
in figure 3. The Prandtl number is a dimensionless number approximating the ratio of
momentum diffusivity (kinematic viscosity) and thermal diffusivity. The fluid
velocity decreases for increasing values of Prandtl number. This is due to the effect of
transverse magnetic field, which has the nature of reducing the velocity. These results
are similar to that of Mishra et al. [11]. Figures 4, 5 depict the velocity variations
2724 K. Vijayakumar and E. Keshava Reddy
under the effect of Grashof number and modified Grashof number respectively. The
velocity of the flow grows when the values of these parameters increases. Figure 6
represents the impact of porous medium on velocity. It is evident that the velocity
enhances for increasing values of porosity parameter. The changes in velocity under
the existence of heat source / sink are depicted in figure 7. It is noticed that the
velocity enhances in the presence of heat source where as it falls down in the case of
heat sink. The variation in velocity under the influence of Schmidt number is shown
in figure 8. It is evident that velocity comes down when the values of Schmidt number
are increased. The existence of thermal diffusion results in improving the flow
velocity which is clear from figure 9.These results coincide with that of Chandra
Reddy et al. [13]. The temperature decreases in the presence of thermal radiation
which is shown in figure 10. The effect of Prandtl number on temperature is presented
in figure 11. The temperature decreases for increasing values of Prandtl number.
Figure 12 reveals the same nature as that of velocity under the influence of heat
source/sink. Figure 13 depicts the influence of Eckert number on temperature. It is
observed that the temperature rises with increasing values of Eckert number. The
effect of Schmidt number on concentration is shown in figure 14. The concentration
reduces with an increase in Schmidt number. Increasing values of Soret number leads
to enhance the concentration which is clear from figure 15.
Fig.2: Effect of magnetic parameter on velocity
0 5 10 15 20 250
0.5
1
1.5
2
2.5
3
3.5
4
y
u
M=1,1.2,1.4,1.6
Sc=0.22; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; Gr=5;F=0.2; S=0.1;S0=2; Pr=0.71;
=0.01; n=1;
MHD Boundary Layer Flow of a VISCO-Elastic Fluid Past a Porous Plate… 2725
Fig.3: Effect of Prandtl number on velocity
Fig.4: Effect of Grashof number on velocity
Fig.5: Effect of modified Grashof number on velocity
0 5 10 15 20 250
1
2
3
4
y
u
S=0.1;F=5;n=1;Sc=0.22;S0=0.1;M=1;Kp=5;Gr=5;Gc=5;Rc=0.001;Ec=0.1;t=0.1;
Pr=0.71,1,7.1
M=1; S=0.1;F=5; n=2; Sc=0.22; So=0.1;Kp=5; Gr=5;Gc=5; Rc=0.001;Ec=0.1; t=0.1;
=0.01
0 2 4 6 8 10 12 14 16 18 20 22
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
y
u
Sc=0.22; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;S0=2; Pr=0.71;
=0.01; n=1;
Gr=5,10,15,20
0 2 4 6 8 10 12 14 16 18 20 22
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
y
u
Sc=0.22; Gr =5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;S0=2; Pr=0.71;
=0.01; n=1;
Gc=5,10,15,20
2726 K. Vijayakumar and E. Keshava Reddy
Fig.6: Effect of porosity parameter on velocity
Fig.7: Effect of heat source/sink on velocity
Fig.8: Effect of Schmidt number on velocity
0 1 2 3 4 5 6 7 80
0.5
1
1.5
2
y
u
K=0.2,0.4,0.6,0.8
Sc=0.22; Gc=5;Gr=5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;So=2; Pr=0.71;
=0.01; n=1;
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
y
u
_____ S = 0.2, 0.4, 0.6........... S = -0.2, -0.4, -0.6
Sc=0.22; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; Gr=0.1;So=2; Pr=0.71;
=0.01; n=1;
0 2 4 6 8 10 12
0
0.05
0.1
0.15
0.2
0.25
0.3
y
u
Gr=5; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;S0=2; Pr=0.71;
=0.01; n=1;
Sc=0.60,0.78,0.96
MHD Boundary Layer Flow of a VISCO-Elastic Fluid Past a Porous Plate… 2727
Fig.9: Effect of Soret number on velocity
Fig.10: Effect of radiation parameter on temperature
Fig.11: Effect of Prandtl number on temperature
0 2 4 6 8 10 12 14 16 18 20 220
0.5
1
1.5
2
y
u
Gr=5; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;Sc=0.22; Pr=0.71;
=0.01; n=1;
So=0.5,1,1.5,2
2 4 6 8 10 12 140
0.2
0.4
0.6
0.8
1
F=1,3,5,7
y
T
Gr=5; Gc=5;
Kp=0.5; Rc=0.1;
Ec=0.01; M=1;
Pr=0.71; S=0.1;
Sc=0.22; So=1;
=0.01; n=1;
2 4 6 8 10 12 140
0.2
0.4
0.6
0.8
1
y
T
Pr=0.71,1,3,5,7.1
Gr=5; Gc=5;
Kp=0.5; Rc=0.1;
Ec=0.01; M=1;
F=0.2; S=0.1;
Sc=0.22; So=1;
=0.01; n=1;
2728 K. Vijayakumar and E. Keshava Reddy
Fig.12: Effect of heat source/sink on temperature
Fig.13: Effect of Eckert number on temperature
Fig.14: Effect of Schmidt number on concentration
0 1 2 3 4 50
0.5
1
1.5
2
2.5
y
_____ S = 0.2, 0.4, 0.6, 0.8........... S = -0.2, -0.4, -0.6, -0.8
Sc=0.22; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; Gr=0.1;So=2; Pr=0.71;
=0.01; n=1;
5 10 15 20 250
0.2
0.4
0.6
0.8
1
y
T
Gr=5; Gc=5;
Kp=0.5; Rc=0.1;
Pr=0.71; M=1;
F=0.2; S=0.1;
Sc=0.22; So=1;
=0.01; n=1;
Ec=0.1,0.3,0.5,0.7
0 2 4 6 8 10 120
0.5
1
1.5
2
y
C
Pr=0.71; So=2;F=5; n=2;S=0.1; M=2;Kp=20; Gr=5;Gc=5; Rc=0.001;Ec=0.1; t=2;
=0.001
Sc=0.60,0.78,0.96
MHD Boundary Layer Flow of a VISCO-Elastic Fluid Past a Porous Plate… 2729
Fig.15: Effect of Soret number on concentration
The present work is extended to observe the changes in skin friction, Nusselt number
and Sherwood number under the influence of thermal radiation, Eckert number,
Schmidt number and magnetic parameter. This is done with the help of tabular values.
Table 1 shows that the skin friction coefficient reduces for increasing values of Eckert
number and Schmidt number. A reverse trend is shown in the case of radiation
parameter and magnetic parameter. The rate of heat transfer increases under the
influence of thermal radiation whereas it decreases in the case of Eckert number.
Increasing values of Schmidt number leads to enhance the Sherwood number.
Table1. Effect of various physical parameters on skin friction, Nusselt number and
Sherwood number
Ec F Sc M Nu Sh
0.2 1 0.22 5 2.0359 0.6742 0.2116
0.4 1 0.22 5 1.9848 0.6645 0.2116
0.6 1 0.22 5 1.5212 0.5932 0.2116
0.2 1 0.22 5 2.0361 0.6742 0.2116
0.2 3 0.22 5 2.0856 0.7356 0.2116
0.2 5 0.22 5 2.1589 0.7931 0.2116
0.2 1 0.60 5 1.9611 0.6742 0.5755
0.2 1 0.78 5 1.9292 0.6742 0.7472
0.2 1 0.96 5 1.9002 0.6742 0.9194
0.2 1 0.22 1 6.9669 0.6742 0.2116
0.2 1 0.22 1.2 7.9647 0.6742 0.2116
0.2 1 0.22 1.4 9.3238 0.6742 0.2116
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
1.2
y
C
So=1,2,3,4
Gr=5; Gc=5;Kp=0.5; Rc=0.1;Ec=0.01; M=1;F=0.2; S=0.1;Sc=0.22; So=1;
=0.01; n=1;
2730 K. Vijayakumar and E. Keshava Reddy
4. CONCLUSION:
In the present study the effect of thermal radiation due to natural convection on MHD
flow of a visco-elastic fluid past a porous plate with variable suction and heat
source/sink is analyzed. The governing equations for the velocity field, temperature
and concentration by finite difference method. The main findings of this study are as
follows.
Velocity of the fluid reduces for increasing values of Prandtl number and
magnetic parameter.
Temperature of the fluid grows for rising values of Eckert number, but a
reverse effect is noticed in the case of Prandtl number and radiation
absorption parameter.
The concentration reduces with an increase in Schmidt number.
The existence of heat source leads to enhance the temperature and a
reverse trend is observed in the presence of heat sink.
Skin friction decreases with an increase of Eckert number and Schmidt
number but a reverse effect is noticed in the case of radiation absorption
parameter and magnetic parameter.
Nusselt number increases as radiation absorption parameter increases but
in the case of Eckert number it decreases.
Sherwood number increases with an increase in Schmidt number.
REFERENCES:
[1] C.H. Chen, Magneto-hydrodynamic mixed convection of a power-law fluid
past a stretching surface in the presence of thermal radiation and internal heat
generation/absorption, Int. J. Non-Linear Mech. 44 (2009) 596–603.
[2] T. Hayat, Ambreen Safdar, M. Awais, S. Mesloub, Soret and Dufour effects
for three-dimensional flow in a viscoelastic fluid over a stretching surface,
Int. J. Heat Mass Transf. 55 (2012) 2129–2136.
[3] P. Ganesan, P. Loganathan, Radiation and mass transfer effects on flow of an
incompressible viscous fluid past a moving vertical cylinder, Int. J. Heat
Mass Trans. 45 (21) (2002) 4281–4288.
[4] B. Lavanya, A.L. Ratnam, Radiation and mass transfer effects on unsteady
MHD natural convective flow past a vertical porous plate embedded in a
porous medium in a slip flow regime with heat source/sink and Soret effect,
Int. J. Eng. Tech. Res. 2(2014).
[5] T. Hayat, F.M. Abbasi, M. Al-Yami, S. Monaquel, Slip and Joule heating
effects in mixed convection peristaltic transport of nanofluid with Soret and
Dufour effects, J. Mol. Liq. 194 (2014) 93–99.
[6] E.M. Abo-Eldahab, M. Abd El-Aziz, Blowing/suction effect on
hydromagnetic heat transfer by mixed convection from an inclined
MHD Boundary Layer Flow of a VISCO-Elastic Fluid Past a Porous Plate… 2731
continuously stretching surface with internal heat generation/absorption, Int.
J. Therm. Sci. 43 (2004) 709–719.
[7] A.J. Chamkha, Double-diffusive convection in a porous enclosure with
cooperating temperature and concentration gradients and heat generation or
absorption effects, Numer. Heat Transfer, Part A: Appl. 41 (1) (2002) 65–87.
[8] S. Ahmed, A. Hussein, H. Mohammed, S. Sivasankaran, Boundary layer flow
and heat transfer due to permeable stretching tube in the presence of heat
source/sink utilizing nanofluids, Appl. Math. Comput. 238 (2014) 149–162.
[9] T. Hayat, A. Shafiq, A. Alsaedi, Effect of Joule heating and thermal radiation
in flow of third-grade fluid over radiative surface, Plos One 9 (1) (2014)
e83153.
[10] T.G. Motsumi, O.D. Makinde, Effects of thermal radiation and viscous
dissipation on boundary layer flow of nanofluids over a permeable moving
flat plate, Phys. Scr. 86 (2012) 045003.
[11] S.R. Mishra, G. C. Dash, M. Acharya: Mass and heat transfer effect on MHD
flow of a visco-elastic fluid through porous medium with oscillatory suction
and heat source, International Journal of Heat and Mass Transfer, 57 (2013)
433-438.
[12] M. Umamaheswar, S.V.K.Varma and M. C. Raju, Unsteady MHD free
convective visco-elastic fluid flow bounded by an infinite inclined porous
plate in the presence of heat source, viscous dissipation and Ohmic heating,
International journal of advanced science and technology, 61 (2013) 39-52.
[13] P. Chandra Reddy, M.C. Raju, G.S.S. Raju, Thermal and solutal buoyancy
effect on MHD boundary layer flow of a visco-elastic fluid past a porous
plate with varying suction and heat source in the presence of thermal
diffusion, Journal of Applied & Computational Mathematics, 2015, 4(5), 1-7.
doi:10.4172/2168-9679.1000249.
[14] P. Chandra Reddy, M.C. Raju, G.S.S. Raju, “Magnetohydrodynamic
convective double diffusive laminar boundary layer flow past an accelerated
vertical plate” International Journal of Engineering Research in Africa, 20
(2016) 80-92.
doi:10.4028/www.scientific.net/JERA.20.80.
[15] P. Chandra Reddy, M.C. Raju, G.S.S. Raju, “Soret and Dufour effects on
MHD free convection flow of Rivlin-Ericksen fluid past a semi infinite
vertical plate” Advances and Applications in Fluid Mechanics, 19 (2016)
401-414. doi:10.17654/FM019020401.
[16] P. Chandra Reddy, M.C. Raju, G.S.S. Raju, S.V.K. Varma “Free convective
magneto-nanofluid flow past a moving vertical plate in the presence of
radiation and thermal diffusion”, Frontiers in Heat and Mass Transfer, Int.
Journal, 7, 28 (2016) DOI: 10.5098/hmt.7.28.
2732 K. Vijayakumar and E. Keshava Reddy
[17] S. Mahdy, E.A. Sameh, Thermosolutal Marangoni boundary layer MHD flow
with the Soret and Dufour effects past a vertical flat plate, Engineering
Science and Technology, an International Journal, 18 (2015) 24-31.
[18] R.R. Kairi, P.V.S.N. Murthy, The effect of melting and thermo-diffusion on
natural convection heat mass transfer in a non-Newtonian fluid saturated non-
Darcy porous medium, Open Transp. Phenom. J., 1 (2009) 7-14.
[19] Swati Mukhopadhyay, I.C. Mandal, Magnetohydrodynamic (MHD) mixed
convection slip flow and heat transfer over a vertical porous plate,
Engineering Science and Technology, an International Journal, 18 (2015) 98-
105.
[20] S. Mukhopadhyay, P.R. De, K. Bhattacharyya, G.C. Layek, Forced
convective flow and heat transfer over a porous plate in a Darcy-Forchheimer
porous medium in presence of radiation, Meccanica, 47 (1) ( 2012) 153-161.
[21] S. Ahmed, K. Kalita, Magnetohydrodynamic transient flow through a porous
medium bounded by a hot vertical plate in presence of radiation: a theoretical
analysis, J. Eng. Phys. Thermophys, 86 (1) (2012) 31–39.
[22] A.G.V. Kumar, S.V.K. Varma, Thermal radiation and mass transfer effects
on MHD flow past a vertical oscillating plate with variable temperature
effects variable mass diffusion, Int. J. Eng., 3 (2011) 493–499.
[23] V. Ravi kumar, M.C. Raju, G.S.S. Raju, Combined effects of Heat absorption
and MHD on Convective Rivlin-Ericksen flow past a semi-infinite vertical
porous plate with variabletemperature and suction, Ain Shams Engineering
Journal, 5 (3) (2014) 867–875.
[24] K. D. Sharma, S. Ajaib, Bounds for complex growth rate in thermo solutal
convection in Rivlin–Ericksen viscoelastic fluid in a porous medium.
International J Eng Sci Advan Technol, 2(6) (2012)1564–71.
[25] M. C. Raju, S.V.K.Varma, N. AnandaReddy: MHD Thermal diffusion
Natural convection flow between heated inclined plates in porous medium,
Journal on Future Engineering and Technology, 6, No.2, 2011, 45-48.
[26] M. C. Raju, S.V.K Varma, P. V. Reddy and SumonSaha: Soret effects due to
Natural convection between heated inclined plates with magnetic field,
Journal of Mechanical Engineering, ME39, No.Dec 2008, 43-48.
[27] O.D. Makinde, Free convection flow with thermal radiation and mass transfer
past a moving vertical porous plate, Int. Commun. Heat Mass Trans. 32 (10)
(2005) 1411–1419.
[28] M.A. Samad, M.M. Rahman, Thermal radiation interaction with unsteady
MHD flow past a vertical porous plate immersed in a porous medium, J.
Naval Arch. Marine Eng. 3 (1) (2006) 7–14.
MHD Boundary Layer Flow of a VISCO-Elastic Fluid Past a Porous Plate… 2733
[29] K. Bhattacharyya, S. Mukhopadhyay, G.C. Layek and I.Pop, Effects of
thermal radiation on micropolar fluid flow and heat transfer over a porous
shrinking sheet, Int J Heat Mass Transf. 55(1) (2012) 2945–2952.
[30] T. Hayat, M. Mustafa, and Pop, I., 2010, Heat and mass transfer for Soret
and Dufour effects on mixed convection boundary layer flow over a
stretching vertical surface in a porous medium filled with a viscoelastic fluid,
Comm. Nonlinear Sci. Num. Sim. 15 (2010) 1183-1196.
[31] W. A. Khan and A. Aziz, Natural convection flow of a nanofluid over a
vertical plate with uniform surface heat flux, International Journal ofThermal
Sciences. 50(7) (2011) 1207–1214.
[32] A.M. Rohni, S. Ahmadm, I. Pop, J.H. Merkin, Unsteady mixed convection
boundary-layer flow with suction and temperature slip effects near the
stagnation point on a vertical permeable surface embedded in a porous
medium, Transp. Porous Med. 92 (2012) 1–14.
2734 K. Vijayakumar and E. Keshava Reddy