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International Journal of Emerging Trends in Engineering and Development
Available online on http://www.rspublication.com/ijeted/ijeted_index.htm
Issue 7, Vol. 2 (March 2017)
ISSN 2249-6149
Mixed convective Heat and Mass transfer flow in a
vertical channel with Soret effect and Radiation
Absorption
J. Deepthi
Research scholar,Department of Mathematics
Rayalaseema University
Kurnool, Andhra Pradesh-INDIA
D.R.V.Prasada Rao
Professor(Rtd) ,Department of Mathematics
S.K. University Anantapur,
Anantapur(Dt), Andhra Pradesh-INDIA
ABSTRACT
In this paper we, investigate the combined influence of Radiation parameter(F),Thermodiffusion(Sr), Radiation absorption(Q1) and Forchheimer number(A) effects on non-Darcy mixed
convective heat and mass transfer flow in a vertical channel with heat generating sources.By
using Galerkin finite element analysis method the governing equations have been solved and the
effect of various parameters on all the flow-characteristics have been investigated.The velocity,
temperature, concentration, and rate of Heat and Mass transfer are evaluated numerically for
different variations of parameter.
INTRODUCTION
Non – Darcy effects on natural convection in porous media have received a great deal of
attention in recent years because of the experiments conducted with several combinations of
solids and fluids covering wide ranges of governing parameters which indicate that the
experimental data for systems other than glass water at low Rayleigh numbers, do not agree with
theoretical predictions based on the Darcy flow model. This divergence in the heat transfer
results has been reviewed in detail in Cheng [5]and Prasad et al. [18] among others. Extensive
effects are thus being made to include the inertia and viscous diffusion terms in the flow
equations and to examine their effects in order to develop a reasonable accurate mathematical
model for convective transport in porous media. The work of Vafai and Tien [23] was one of the
early attempts to account for the boundary and inertia effects in the momentum equation for a
porous medium.They found that the momentum boundary layer thickness is of order of
. Vafai
and Thiyagaraja [24] presented analytical solutions for the velocity and temperature fields for the
interface region using the Brinkman Forchheimer –extended Darcy equation. Detailed accounts
of the recent efforts on non-Darcy convection have been recently reported in Tien and Hong [21],
Cheng [5], and Kalidas and Prasad [8].Poulikakos and Bejan [17] investigated the inertia effects
through the inclusion of Forchheimer’s velocity squared term, and presented the boundary layer
analysis for tall cavities.Prasad and Tuntomo [18] reported an extensive numerical work for a
wide range of parameters, and demonstrated that effects of Prandtal number remain almost
unaltered while the dependence on the modified Grashof number, changes significantly with an
increase in the Forchheimernumber.A numerical study based on the Forchheimer-Brinkman-
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Extended Darcy equation of motion has also been reported recently by Beckerman et al [4].Jha
and Singh [9] studied the free convection and mass transfer flow in an infinite vertical plate
moving impulsively in its own plane taking into account the Soret effect. Kafousias [10] studied
the MHD free convection and mass transfer flow taking into account Soret effect. The analytical
studies of Jha and Singh[9] and Kafousias [10] were based on Laplace transform technique.
Abdul Sattar and Alam [1] have considered an unsteady convection and mass transfer flow of
viscous incompressible and electrically conducting fluid past a moving infinite vertical porous
plate taking into the thermal diffusion effects.Malasetty et al [15] have studied the effect of both
the soret coefficient and Dufour coefficient on the double diffusive convective with compensating
horizontal thermal and solutal gradients. Umadevi et al[22] have studied the chemical reaction
effect on Non-Darcy convective heat and mass transfer flow through a porous medium in a
vertical channel with heat sources. Deepthi et al [7] and Kamalakar et al[11] have discussed the
numerical study of non-Darcy convective heat and mass transfer flow in a vertical channel with
constant heat sources under different conditions.The effects of radiation on MHD flow and heat
transfer problem have become more important industrially.
Keeping these applications in view several authors have studied the effect of radiation on
flow in different configurations under varriedconditions.Bharathi [3] has studied thermo-diffusion
effect on unsteady convective Heat and Mass transfer flow of a viscous fluid through a porous
medium in vertical channel. Radiative flow plays a vital role in many industrial and
environmental process e.g. heating and cooling chambers , fossil fuel combustion energy process,
evaporation form larger open water reservoirs, astrophysical flows, solar power technology and
space vehicle re-entry. Taneja et al [20] studied the effects of magnetic field on free convective
flow through porous medium with radiation and variable permeability in the slip flow regime.
Kumar et al [13] studied the effect of MHD free convection flow of viscous fluid past a porous
vertical plate through non homogeneous porous medium with radiations and temperature gradient
dependent heat source in slip flow regime. The effect of free convection flow with thermal
radiation and mass transfer past a moving vertical porous plate was studied by Makinde [14].
Ayani et al [2] studied the effect of radiation on the laminar natural convection induced by a line
source. Raptis [19] have discussed the effect of radiation and free convection flow through
porous medium. MHD oscillating flow on free convection radiation through porous medium with
constant suction velocity was investigated by El.Hakiem[8]. Muralidhar[16] has analysed the
thermo-diffusion effect of non-Darcy convective heat and mass transfer flow in a vertical
channel. Recently, Das et al [6] have studied the mixed convective magnetohydrodynamic flow in
a vertical channel filled with nanofluid. Recently,Kristaiah [12] investigated the combined
influence of thermal radiation ,chemical reaction and thermo-dffusion effects on convective heat
and mass transfer flow of an electrically conducting fluid in vertical channel.
In this paper, to investigate the combined influence of Radiation parameter(F),Thermodiffusion(Sr), Radiation absorption(Q1) and Forchheimer number(A) effects on non-Darcy mixed
convective heat and mass transfer flow in a vertical channel with heat generating sources.By
using Galerkin finite element analysis method the governing equations have been solved and the
effect of various parameters on all the flow-characteristics have been investigated.The velocity,
temperature, concentration, and rate of Heat and Mass transfer are evaluated numerically for
different variations of parameter.
FORMULATION OF THE PROBLEM
We consider a fully developed laminar convective heat and mass
transfer flow of a viscous, electrically conducting fluid through a porous medium confined in a
vertical channel bounded by flat walls. We choose a Cartesian co-ordinate system O(x,y,z) with
x- axis in the vertical direction and y-axis normal to the walls and the walls are taken at y=  L.
The walls are maintained at constant temperature and concentration. The temperature gradient in
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ISSN 2249-6149
the flow field is sufficient to cause natural convection in the flow field .A constant axial pressure
gradient is also imposed so that this resultant flow is a mixed convection flow.The porous
medium is assumed to be isotropic and homogeneous with constant porosity and effective thermal
diffusivity. The thermo physical properties of porous matrix are also assumed to be constant and
Boussinesq- approximation is invoked by confining the density variation to the buoyancy term. In
the absence of any extraneous force flow is unidirectional along the x-axis which is assumed to
be infinite.
x
Tw=T0+Ax
Tw=T0+Ax
C w= C0+Bx
C w= C0+Bx
y
y=-L
g
y=+L
Fig.1 : Configuration of the problem
The equations governing the non-darcy flow heat and mass transfer taking thermal
radiation into account are
p   2u  2 H 2
 F 2
  ( ) 2  ( e o )u 
u  g  0
(1)
x  y
o
k
16 2Te3  2T
T
 2T
 k f 2  Q(T  To )  Q1' (C  Co ) 
x
y
3 r y 2
C
 2C
 2T
u
 D1 2  k1C  k11 2
x
y
y
0C p u
The relevant boundary conditions are
T=Tw , C=Cw at
y=L
u  0,
(2)
(3)
(4)
Following Tao and Das et.al [29],we assume that the temperature and concentration of the both
walls is Tw  T0  Ax , Cw  C0  Bx where A and B are the vertical temperature and
concentration gradients which are positive for buoyancy –aided flow and negative for buoyancy –
opposed flow, respectively, T0 and C 0 are the upstream reference wall temperature and
concentration respectively. For the fully developed laminar flow in the presences of transverse
magnetic field, the velocity depend only on the normal coordinate and all the other physical
variables except temperature, concentration and pressure are functions of y and x, x being the
vertical co-ordinate.
The temperature and concentration inside the fluid can be written as
T  T  ( y )  Ax
,
C  C  ( y )  Bx
We define the following non-dimensional variables as
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u 
Issue 7, Vol. 2 (March 2017)
ISSN 2249-6149
u
p
, ( x, y)  ( x, y ) / L , p 
( / L)
(  2 / L2 )
(5)
T  -T0
C -C0
 (y) 
,
, C,
ALP1
BLP1
Introducing these non-dimensional variables the governing equations in the dimensionless form
reduce to (on dropping the dashes)
d 2u
 1   ( M12 )u   G (  NC )   2 u 2
(6)
2
dy
d 2
4 F d 2
    (Pr)u  Q1C 
dy 2
3 dy 2
d 2C
ScSo d 2


C

(
Sc
)
u

N dy 2
dy 2
(7)
(8)
where
 gAL3
  FD
( Inertia or Fochhemeir parameter), G 
(Grashof Number)
2
 e2 H o2 L2

M2 
(Hartmann Number), Sc 
(Schmidt number)
2
D1

 Cp
 B
N
(Buoyancy ratio), Pr 
(Prandtl Number)
kf
A
1 / 2
k1 L2
QL2
(Heat source parameter),  
(Chemical reaction parameter)

D1
kf
F
k T
4 Te3
(Radiation parameter), S 0  11
(Soret parameter)
k f R
T
3F
3F
Q1' BL2
, 1  
(Radiation absorption parameter), P1  Pr
3  4F
3  4F
kf A
dp

(Constant pressure gradient)
dx
Q1 
The corresponding boundary conditions are
u  0 ,   0 , C  0 on y  1
THE METHOD OF SOLUTION
(9)
The Galerkinfinite element method has been implemented to obtain numerical solutions
of coupled non-linear equations [6] to [8] of third-order inf and second order in h,,  under
boundary conditions [9]. This technique is extremely efficient and allows robust solutions of
complex coupled, nonlinear multiple degree differential equation systems. The fundamental steps
comprising the method are
1]Discretization of the domain into elements
2] Derivation of element equations
3] Assembly of Element Equations
4] Imposition of boundary conditions
5] Solution of assembled equations
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COMPARISON
In the absence of radiation absorption parameter (Q1=0) and Forchheimernumber(A=0) the results
are in good agreement with Kristahiah (12).
Sr
0.5
1.0
1.5
0.5
0.5
Table 1 :
Comparison of Nu and Sh at η=±1 with Kristaiah(12) with A=0, Q1=0
Kristaiah[12]Results
Present Results(A=0, Q1=0)
Pr
Nu(+1) Nu(-1) Sh(+1) Sh(-1) Nu(+1) Nu(-1) Sh(+1) Sh(-1)
0.71
0.71
0.71
1.71
7.0
-0.2607
-0.2598
-0.2599
-0.2617
-0.4595
-4.0053
-3.9895
-3.9885
-4.1086
-5.9582
0.2534
0.2516
0.2508
0.2686
0.4595
14.7935
14.7929
14.7915
14.8592
15.1084
-0.2606
-0.2596
-0.2598
-0.2616
-0.4593
-4.0054
-3.9896
-3.9884
-4.1084
-5.9599
0.2536
0.2518
0.2507
0.2685
0.4591
14.8014
14.7932
14.7916
14.8591
15.1079
DISCUSSION OF THE NUMERICAL RESULTS
In order to get physical insight into the problem we have carried out numerical
calculations for non-dimensional velocity, temperature and concentration, Nusselt number
andSherwood number by assigning some specific values to the parameters entering into the
problem.We have investigated the influence of F, Sr,Q1andAby fixing the other parameters as
G=2,M=2,D-1=0.2,Sc=1.3,N=1,Pr=0.71,=0.5 unless otherwise stated.
Figs.1a-1c represents the velocity components,temperature and concentration
withRadiation parameter(F). From Fig.1a it can be seen from the profiles that an increase in F
increases the magnitude of the velocity thus higher the radiative heat flux larger the magnitude of
u in the flow region.Fig.1b represents  with radiation parameter F.It can be seen from the
profiles that increase in F leads to an enhancement in the actual temperature in the flow region,
this may be attributed to the fact that increasing F results in the enhancement of thickness of the
thermal boundary layer.Fig.1c represents the concentration C with radiation parameter(F).It can
be seen from the profiles that higher the radiative heat flux larger the actual concentration in the
flow field.
Figs. 2a-2c shows the variation of the velocity components,temperature and concentration
with Soret parameter(Sr).Fig.2a shows the variation of u with Soret parameter Sr , and it can be
seen that higher the thermo-diffusion effect larger u in the flow region and for higher Sr1.5 the
velocity enhances u in the entire flow region.Fig.2b shows the variation of  with respect to
Soret parameter(Sr) we find that the higher the thermo-diffusion effect(Sr1.0) larger the actual
temperature and for higher thermo-diffusion effects(Sr1.5) lesser the actual temperature in the
entire flow region except in a narrow region adjacent to the left wall y=-1.From fig.2c we find
that higher the thermo-diffusion(Sr)effects, smaller the actual concentration in the flow-region.
Due to the fact that increase in Sr reduces the thickness of the solutary boundary layer,also higher
the thermo-diffusion effects larger the actual concentration in the flow.
Figs.3a-3c represents the velocity components,temperature and concentration with
Radiation absorption parameter(Q1). Fig.3a represents the variation of u with radiation absorption
parameter(Q1),it can be observed from the figure that |u| reduces with increase in Q 1.This can be
attributed to the fact that the thickness of the momentum boundary layer decreases with increase
in Q1.From fig.3b we find that the actual temperature decreases with increase in radiation
absorption parameter(Q1≤1) and enhances with higher Q1≥1.5.Due to the fact that the thickness of
the boundary layer increases with increase inQ1. Fig.3c represents the concentration with radiation
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Issue 7, Vol. 2 (March 2017)
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absorption parameter(Q1). An increase inQ1 reduces the thickness of the solutary boundary layer,
which in turns leads to a depreciation in the actual concentration.

u
y
0
-1
-0.5
0
0.5
1
0.0010
F=0.5,1.5,3.5,5
-0.1
0.0008
-0.2
0.0006
-0.3
F=0.5,1.5,3.5,5
0.0004
-0.4
0.0002
-0.5
y
1.0
Fig. 1a: Variation of u with F
G=2,M=2,D-1=0.2,Sc=1.3,N=1, Pr=0.71,
A=0.01,Q=0.5,Q1=0.5, =0.5,Sr=0.5
0.5
0.5
1.0
Fig. 1b: Variation of  with F
G=2,M=2,D-1=0.2,Sc=1.3,N=1, ,Pr=0.71,
A=0.01,Q=0.5,Q1=0.5, =0.5,Sr=0.5
Figs.4a-4c represents the velocity components, temperature and concentration with
Forchheimer number(A).Fig.4a represents the velocity u with Forchheimer parameter A. From
the figures we find that an increase in ‘A’ reduces |u| in the flow region ,the inclusion of nondarcy effect results in a reduction(or)depreciation in the velocity.Fig.4b depeicts a temperature 
with Forchheimer number(A).It can be seen from the profiles that an increase in A leads to a
depreciation in the actual temperature in the flow region.Thus the inclusion of intertia and
boundary effects reduces the actual temperature.Fig12c depicts the concentration with
Forchheimernumber(A). From the fig we notice an enhancement in the actual concentration with
increase in A. Thus the inclusion of the inertia and boundary effects results in an enhancement in
the actual concentration in the flow region.
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C
0.045
uf
0.04
1.0
0.035
0.5
0.5
1.0
y
0.03
0.1
Sr=1.0,1.5,0.5,2.0
0.025
0.02
0.2
0.015
0.01
F=0.5,1.5,3.5,5
0.3
0.005
0
-1
-0.5
y
0
0.5
0.4
1
Fig. 2a : Variation of u with Sr
G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5,
Pr=0.71,A=0.01,Q=0.5,Q1=0.5, =0.5
Fig. 1c : Variation of C with F
G=2,M=2,D-1=0.2,Sc=1.3,N=1, Pr=0.71,
A=0.01,Q=0.5,Q1=0.5, =0.5,Sr=0.5

C0.
0.008
0.035
0.030
0.006
0.025
0.020
0.004
0.015
Sr=1.0,1.5,0.5,2.0
0.010
0.002
0.005
Sr=0.5,1,1.5,2.0
y
1.0
0.5
0.5
1.0
1.0
Fig. 2b : Variation of  with Sr
G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5,
Pr=0.71,A=0.01,Q=0.5,Q1=0.5, =0.5
0.5
0.5

u
-0.5
y
Fig. 2c : Variation of C with Sr
G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5,
Pr=0.71,A=0.01,Q=0.5,Q1=0.5, =0.5
0
-1
1.0
0.16
y
0
0.5
1
-0.05
0.14
Q=-0.5,-1.5
-0.1
0.12
-0.15
0.1
-0.2
0.08
-0.25
0.06
-0.3
0.04
-0.35
0.02
-0.4
Q1=0.5,1,1.5,2.0
Q1=0.5,1,1.5,2.0
0
-0.45
-1
Fig. 3a : Variation of u with Q1
G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5,
Pr=0.71,A=0.01,Q=0.5, =0.5,Sr=0.5
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-0.5
y
0
0.5
1
Fig. 3b : Variation of  with Q1
G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5,
Pr=0.71,A=0.01,Q=0.5, =0.5,Sr=0.5
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C
0.25
u
y
0
-1
-0.5
0
0.5
1
-0.05
0.2
-0.1
0.15
A= 0.05,0.07
-0.15
-0.2
0.1
-0.25
-0.3
0.05
Q1=0.5,1,1.5,2.0
0
-1
-0.5
-0.4
y
0
0.5
A= 0.01,0.03
-0.35
1
-0.45
Fig. 3c : Variation of C with Q1
G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5,
Pr=0.71,A=0.01,Q=0.5, =0.5,Sr=0.5
Fig. 4a : Variation of u with A
G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5,
Pr=0.71,Q=0.5,Q1=0.5, =0.5,Sr=0.5

0.09
C0.
0.08
A= 0.01,0.03,0.05,0.07
0.20
A= 0.01,0.03,0.05,0.07
0.07
0.06
0.15
0.05
0.04
0.10
0.03
0.02
0.01
0.05
y
0
-1
-0.5
0
0.5
Fig. 4b : Variation of  with A
G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5,
Pr=0.71,Q=0.5,Q1=0.5, =0.5,Sr=0.5
1
1.0
0.5
0.5
1.0
y
Fig. 4c : Variation of C with A
G=2,M=2,D-1=0.2,Sc=1.3,N=1,F=0.5,
Pr=0.71,Q=0.5,Q1=0.5, =0.5,Sr=0.5
Table.1
Values of Nusselt number, Sherwood number at ƞ=1
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Parameter
Nu(1)
Nu(-1)
Sh(1)
Sh(-1)
0.5
0.0182438
-0.0182438
0.0585079
-0.0585079
F
International Journal of Emerging Trends in Engineering and Development
Issue 7, Vol. 2 (March 2017)
1.5
0.0181263 -0.0181263 0.0570322 -0.0570322
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3.5
0.018034
-0.018034
0.0570511 -0.0570511
5.0
0.0179654 -0.0179654 0.0570652 -0.0570652
0.5
0.0182442 -0.0182442 0.0603368 -0.0603368
Sr
1.0
0.0181724 -0.0181724 0.0551637 -0.0551637
1.5
0.018172
-0.018172
0.0533047 -0.0533047
2.0
0.0181715 -0.0181715 0.0514458 -0.0514458
0.5
0.0182438 -0.0182438 0.0585079 -0.0585079
Q1
1.0
0.018208
-0.018208
0.0570156 -0.0570156
1.5
0.0182431 -0.0182431 0.0570085 -0.0570085
2.0
0.0182783 -0.0182783 0.0570014 -0.0570014
0.01
0.0182438 -0.0182438 0.0585079 -0.0585079
A
0.03
0.0181729 -0.0181729 0.0570227 -0.0570227
0.05
0.0181728 -0.0181728 0.0570221 -0.0570221
0.07
0.0181726 -0.0181726 0.0570212 -0.0570212
The rate of heat transfer (Nusselt number) Nu enhances on y=1, with increase in the strength of
the heat generating source while an increase in the strength of the heat absorption source
enhances Nu on y=+1 and reduces on y=-1. An increase in Forchheimer number A reduces |Nu|
on y=+1,thus the inclusion of inertia and boundary effect reduces the rate of heat transfer on
y=±1. The rate of heat transfer reduces with increase in Q1 ≤1.0 and enhances with higher Q1≥1.5
on y=±1. An increase in F≤3.5 enhances |Nu| and reduces with higher F≥5. Higher the thermo
diffusion (Sr) lesser the rate of heat transfer on the walls y=1.
The rate of mass transfer (Sherwood number) enhances with increase in F≤3.5 and
reduces with higher F≥5 on y=±1. An increase in the radiation absorption Q1 (or) Forchheimer
number A results in depreciation |Sh| on both the walls. Higher the thermo diffusion (Sr) lesser
the rate of mass transfer on the walls y=1.
CONCLUSIONS
The coupled equations governing the flow,heat and mass transfer have been solved by
using Galerkin finite element technique.The important conclusions of this analysis are
1) Higher the thermo-diffusion(Sr1.5) larger the velocity, temperature and concentration
and for higher Sr1.5 ,the velocity and temperature reduces and the concentration
enhances in the flow region. The rate of heat and mass transfer reduces on the walls with
increase in Sr.It is found that an increase in soret parameter S r≤1.0 increases the
velocity,temperature and reduces the concentration for higher Sr ≥1.5,we notice a
depreciation in them.
2) Higher the thermo-diffusion(Sr1.5) larger the velocity, temperature and concentration
and for higher Sr1.5 ,the velocity and temperature reduces and the concentration
enhances in the flow region. The rate of heat and mass transfer reduces on the walls with
increase in Sr.It is found that an increase in soret parameter S r≤1.0 increases the
velocity,temperature and reduces the concentration for higher S r ≥1.5,we notice a
depreciation in them.
3) The velocity and temperature decreases while the concentration increases with increase
inForchheimer number A. The rate of heat and mass transfer reduces with increase in A.
4) The velocity, temperature and concentration reduces with increase in radiation absorption
Q1>1 and for higherQ1≥1.5 the velocity and concentration reduces while the temperature
enhances in the flow region .The Sherwood number reduces and Nusselt number
increases with increase in Q1.
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Issue 7, Vol. 2 (March 2017)
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Issue 7, Vol. 2 (March 2017)
ISSN 2249-6149
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