-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
447 Copyright © 2016. Vandana Publications. All Rights
Reserved.
Volume-6, Issue-3, May-June 2016
International Journal of Engineering and Management Research
Page Number: 447-459
Chemical Reaction Effects on Heat and Mass Transfer in MHD
Boundary Layer Flow past an Inclined Plate with Viscous
Dissipation
and Radiation in Porous Medium
V. Subhakanthi1, N.Bhaskar Reddy2 1,2
Department of Mathematics, Sri Venkateswara University,
Tirupati, A.P, INDIA
ABSTRACT This paper analyzes the chemical reaction effects on
heat and mass transfer in magnetohydrodynamic boundary layer flow
past an inclined plate with viscous dissipation and radiation in
porous medium. A suitable similarity transformation is used to
transform the non linear system of partial differential equations
into a system of ordinary differential equations. To solve the
resultant system an efficient numerical technique Runge-Kutta
fourth order is used along with shooting technique. The behavior of
the velocity, temperature, concentration for variations in the
thermo physical parameters are presented in graphs. The values of
skin friction coefficient, Nusselt number and Sherewood number are
also computed and are reported in tables. Keywords--- heat and mass
transfer-MHD- radiation – chemical reaction-similarity
parameter
I. INTRODUCTION The MHD boundary layer theory has a powerful
place in the development of the magnetohydrodynamics. It has many
applications in engineering problems such as geophysics,
astrophysics, boundary layer control in the field of aerodynamics.
So that the study of mixed convection flow and heat transfer for
electrically conducting fluids over a surface has attracted much
interest of researchers. Pioneer work is done by Ostrach [1] on
convection flow and obtained a similarity solution of transient
free convection flow over a semi infinite vertical plate using the
integral method. Yih [2] presented the free convection effect on
MHD combined heat and mass transfer of a moving permeable vertical
surface. Non similar analytic solution for MHD flow and heat
transfer in a third-order fluid past a stretching sheet was
examined by Sajid et al. [3]. More recently the problem of
magnetohydrodynamic flow over infinite surfaces has become more
important due its applications in areas like nuclear fusion,
chemical engineering, medicine, and high-speed, noiseless printing.
MHD flow in the vicinity of infinite plate has
been investigated intensively by many investigators [4-9].
Investigation (theoretical and experimental) of natural convection
flow under the influence of gravitational force over a solid body
with different geometries embedded in a fluid-saturated porous
medium is of considerable importance due to frequent occurrence of
such fluid flow in nature as well as in science and technology viz.
geothermal reservoirs, drying of porous solids, thermal insulators,
heat exchanger devices, enhanced recovery of petroleum resources,
underground energy transport etc. Nakayama and Koyama [10] carried
out an analysis to study the free and forced convection flow in
Darcian and non- Darcian porous medium. Similarity solution for
free convection flow from a vertical plate fixed in a
fluid-saturated porous medium was studied by Cheng and Minkowycz
[11]. Makinde [12] examined MHD mixed convection flow and mass
transfer over a vertical porous plate with constant heat flux
embedded in a porous medium. Recently, Zeeshan and Ellahi [13]
obtained series solutions for nonlinear partial differential
equations with slip boundary conditions for non-Newtonian
magnetohydrodynamic fluid in porous space. Viscous dissipation
changes the temperature distributions by playing a role like energy
source, which leads to influence the heat transfer rates. The
effect of viscous dissipation depends on whether the plate is being
cooled or heated. Alam et al. [14, 15] studied the combined effects
of viscous dissipation and Joule heating on steady magneto
hydrodynamic free convective heat and mass transfer flow of a
viscous incompressible fluid past a semi-infinite inclined radiate
isothermal permeable moving surface in the presence of
thermophoresis. Singh [16] reported the effect of viscous
dissipation on heat and mass transfer in MHD boundary layer flow
over an inclined plate in porous medium. The radiative effects have
dominent applications in physics and engineering. In space
technology and high temperature processes the radiative heat
transfer effects on different flows are very
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
448 Copyright © 2016. Vandana Publications. All Rights
Reserved.
important and very little is known about the effects of
radiation on the boundary layer of a radiative-MHD fluid past a
body. Viskanta and Grosh [17] were one of the initial investigators
to study the effects of thermal radiation on temperature
distribution and heat transfer in an absorbing and emitting media
flowing over a wedge. To simplify the energy equation they used
Rosseland approximation for the radiative flux vector. The
radiative effects on heat transfer in nonporous and porous medium
utilizing the Rosseland or other radiative flux model was
investigated by number of researchers such as Raptis [18], Hall et
al. [19], Bakier [20], Raptis and Perdikis [21] and Rao [22] etc.
Recently Mishra et al. [23] done their work on the effect of
radiation on free convection heat and mass transfer flow through
porous medium in a vertical channel with heat absorption/
generation. Heat and mass transfer problems with chemical reactions
are of importance in many processes, and therefore have received a
considerable attention in recent years. In many chemical
engineering processes, a chemical reaction between a foreign mass
and the fluid does occur. These processes take place in numerous
industrial applications, such as the polymer production, the
manufacturing of ceramics or glassware, food processing. . Sharma
and Nabajyoti Dutta[24] analyzed the effect of chemical reaction
and thermal radiation effects on MHD boundary layer flow past a
moving vertical porous plate. magnetohydrodynamic boundary layer
flow of heat and mass transfer over a moving vertical plate with
suction and chemical reaction is investigated by Ibrahim, and
Makinde[25].
However, the interaction of chemical reaction and thermal
radiation with MHD boundary layer flow of heat and mass transfer in
porous medium in the presence of viscous dissipation received
little attention. Hence, the object of the present paper is to
study the combined effect of thermal radiation and a first-order
chemical reaction on heat and mass transfer effects on MHD boundary
layer flow past an inclined plate with viscous dissipation in
porous medium in the presence of a uniform transverse magnetic
field. The dimensionless governing equations of the flow, heat and
mass transfer are solved numerically using Runge-Kutta fourth order
method along with shooting technique. Numerical results are
reported in figures for various values of the physical parameters
of interest.
II. MATHEMATICAL ANALYSIS
A steady two dimensional hydromagnetic
convective flow of a viscous, incompressible electrically
conducting fluid past an inclined plate with an acute angle to the
vertical is considered in porous medium. x-axis is taken along the
leading edge of the inclined plate and y -axis normal to it, i.e.,
the plate starts at x = 0 and extends parallel to the x axis and is
of semi infinite length. A magnetic field of uniform strength is
introduced normal to the direction of the flow. The uniform plate
temperature and concentration are maintained at wT and wC . The
plate temperature is
higher than the temperature of the fluid far away from the plate
and concentration at the surface of the plate is greater than the
free stream concentration. A steady flow parallel to the plate with
free stream velocity is assumed. The convective flow starts under
the simultaneous action of the buoyancy forces used by the
variations in density due to temperature and species concentration
differences. The magnetic Reynold number is assumed to be small, so
that the induced magnetic field is neglected. The Hall effects term
is neglected. The effects of viscous dissipation, radiation and
chemical reaction have been taken into the account. Then, under the
Boussinesq and the usual boundary layer approximations, the
governing equations for the Darcy type flow, following Schlichting
[16] and Nield and Bejan[17], are given by
Fig. 1 Physical configuration and coordinate system
0=∂∂
+∂∂
yv
xu (1)
( ) ( ) ( )∞∞∞ −′−−−+−+∂∂
=∂∂
+∂∂ Uu
ku
BCCgTTg
yu
yuv
xuu CT
υρ
σγβγβυ
20
2
2
coscos
(2)
∂∂
−+
∂∂
+∂∂
=∂∂
+∂∂
yq
cu
kcyu
cyT
yTv
xTu r
ppp ρυ
ρυα 12
2
2
2 (3)
( )∞−−∂∂
=∂∂
+∂∂ CCk
yCD
yCv
xCu 12
2
(4)
The boundary conditions of velocity, temperature and
concentration are
ww CCTTvu ==== ,,0,0 at 0=y ∞∞∞ === CCTTUu ,,
as ∞→y (5)
where u, v,T and C are the fluid x-component of velocity,
y-component of velocity, temperature and concentration
respectively,υ - the kinematics viscosity, ρ - the density, σ - the
Electric conductivity of the fluid, B0- the magnetic induction, Tβ
and Cβ - the coefficients of Thermal and concentration expansions
respectively,
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
449 Copyright © 2016. Vandana Publications. All Rights
Reserved.
α - the thermal diffusivity, pc - the specific heat at constant
pressure, rq - the radiative heat flux, Dm - the mass diffusivity
g- the gravitational acceleration, wT -the temperature of the hot
fluid at the surface of the plate ,
wC - the species concentration at the plate surface, ∞T - the
temperature of the fluid far away from the plate,
∞C - the free stream concentration, k′ - the permeability of the
porous medium, k- the thermal conductivity , K1- chemical reaction
rate and γ- the acute angle.
Continuity equation (1) is identically satisfied by the stream
function ( )yx,ψ , defined as
xv
yu
∂∂
−=∂∂
=ψψ , (6)
By using Rosseland approximation, the radiative heat flux rq is
given by
yT
kqr ∂
∂−=
4
*
*
34σ
(7)
where *σ is the Stefan – Boltzman constant and *k is the mean
absorption coefficient. It should be noted that by using Rosseland
approximation , the present analysis is limited to optically thick
fluids. If the temperature differences within the flow are
sufficiently small then equation (7) can be linearized by expanding
4T in a Taylor series about the free stream temperature
∞T which after neglecting the higher order terms takes the
form
434 34 ∞∞ −≅ TTTT (8)
To transform equations (2) - (4) in to a set of ordinary
differential equations, the following similarity transformations
and dimensionless variables are
introduced.x
Uyυ
η ∞= , ( )ηυψ fxU∞= ,
( )ηfUu ′= ∞ , [ ]ffxUv −′= ∞ ηυ
21
( )
−−
=∞
∞
TTTT
w
ηθ , ( )
−−
=∞
∞
CCCC
w
ηφ ,
( )2∞
∞−=U
xTTgG wTr
β,
( )
2∞
∞−=U
xCCgG wcc
β
∞
=U
xBM
ρσ 20 ,
ρυ
=Pr ,
*
3*
316
kkTR ∞= σ ,
mDSc υ= , ( )∞
∞
−=
TTcUE
wpc
2
,∞
=U
xkKr 1 ,
( )∞∞
−=
TTkcUxNwp
υ,
∞′=
Ukxυδ , (9)
Substituting the equations (6) to (9 ) into the equations (2) to
(5) we obtain
( ) 0coscos21
=++′−++′′+′′′ δδγφγθ MfGcGrfff
(10)
( ) ( ) 0Pr)(PrPr211 22 =′+′′+′+′′+ fNfEcfR θθ (11)
021
=−′+′′ φφφ ScKrScf (12)
The transformed boundary conditions are
1,1,0,0 ===′= φθff at 0=η
0,0,1 ===′ φθf as ∞→η (13)
where prime ( ´ ) denotes differentiation with respect to η. η
is the similarity parameter, ( )ηf - the dimensionless stream
function, ( )ηθ - the dimensionless temperature, ( )ηφ - the
dimensionless concentration, ψ – the stream function, rG - the
local thermal Grahsof number, cG - the local solutal Grahsof
number, M- the magnetic field parameter, δ - the permeability
parameter, γ - the inclination parameter , R- radiation parameter,
, Pr - the Prandtl number , Ec - the Eckert number, N - viscous
dissipation parameter , Sc - the Schmidt number , Kr - the chemical
reaction parameter .
III. SOLUTION OF THE PROBLEM
The governing boundary layer equations (10) to (12) subject to
the boundary conditions (13) are solved numerically by using
Runge-Kutta fourth order method along with shooting technique.
First of all higher order non-linear differential equations (10) to
(12) are converted into simultaneous linear differential equations
of first order and they are further transformed into initial value
problem by applying the shooting technique(Jain
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
450 Copyright © 2016. Vandana Publications. All Rights
Reserved.
et al. [27]). The resultant initial value problem is solved by
employing Runge-kutta fourth order technique. Numerical results are
reported in Figures 2 - 34 for various values of the physical
parameters of interest. From the process of numerical computation ,
the skin-friction coefficient, the Nusselt number and Sherwood
number which are respectively proportional to ( )0f ′′ ,
( )0θ ′− and ( )0φ′− are also sorted out and numerical values
are presented in a tabular form .
IV. RESULTS AND DISCUSSION
The governing equations (10)-(12) subject to the boundary
conditions are solved as described in section 3. As a result of the
numerical calculations, the dimensionless velocity , temperature
and concentration are obtained and their behaviour have been
discussed for variations in governing parameters. The results are
presented in Figures from 2 – 34. Numerical results for the skin –
friction, Nusselt number and Sherwood number are reported in Table
1 and Table 2. Fig. 2 and Fig. 3 show the effects of thermal
Grashof number Gr and solutal Grashof number Gc on the velocity
respectively. As shown the velocity increases as Gr and Gc
increases. Physically the thermal Grashof number G r > 0 means
heating of the fluid or cooling of the boundary surface and for
cooling of the fluid or heating of the boundary surface Gr < 0
and the absence of free convection current corresponds to Gr = 0.
The effect of magnetic parameter on velocity of the fluid is
illustrated in Fig. 4. A decrease in the velocity is noticed on
increasing the magnetic field parameter M, due to the fact that
magnetic field exhorts a retarding force on free convective flow
which retards the flow. i.e., the velocity boundary layer becomes
thinner and thinner on increasing M. Fig. 5 represents the effect
of porocity parameter δ on the velocity. An increase in the
velocity of the fluid is observed. The variation of the velocity on
increasing the inclination parameter γ is depicted in Fig. 6. It is
noticed that the velocity of the fluid is decreasing. Fig. 7
demonstrates the effect of dissipation parameter due to porous
medium on the velocity with an increasing N, we find that there is
a decrease in the velocity. Fig. 8 shows the effect of radiation
parameter on the velocity. On increasing the radiation parameter,
there is an increase in the velocity. Velocity boundary layer was
not effected by the Prandtl number as reported in the Fig. 9. The
effects of Eckert number on velocity of the fluid in the boundary
layer is represented in the Fig. 10. No change in the velocity is
found for small values (0.01, 0.001) of Eckert number Ec which are
of practical interest, by taking the values greater than 1 increase
in the velocity is observed. A slight change in the velocity is
observed on increasing the Schmidt number as shown in Fig. 11. The
effects of chemical reaction parameter on momentum boundary layer
is presented in the Fig. 12. It is found that the velocity slightly
decreases on increasing the chemical reaction parameter.
The effects of thermal and solutal Grashof numbers Gr and Gc are
illustrated in Fig. 13 and Fig. 14. A decrease in temperature of
the fluid is found on increasing Gr and Gc. The effect of magnetic
field parameter M on dimensionless temperature is depicted in Fig.
15. As we increasing the values of M decrease in the temperature is
noticed. For different values of the porosity parameter δ the
temperature profile is plotted in Fig. 16. It is observed that the
thermal boundary layer thickness decreases as porosity parameter
increases. Fig. 17 illustrates the effect of inclination parameter
γ on temperature . A slight change in the temperature is seen on
increasing γ . Fig. 18 displays the effect of dissipation parameter
N on temperature of the fluid. Greater viscous dissipative heat
causes rise in the temperature. The reason behind this is a rise in
the value of viscous dissipation helps to improve the thermal
conductivity of the fluid. The influence of radiation parameter R
on temperature is shown in Fig. 19. It is noticed that temperature
increases on increasing R due to increase in the radiation
parameter provides more heat to the fluid that causes an
enhancement in the temperature. The effect of Prandtl number on
temperature of the fluid is presented in Fig. 20. It is clear that
the dimensionless temperature decreases with an increase in Prandtl
number Pr. The Prandtl number is inversely proportional to the
thermal diffusivity of the fluid and due to this thermal boundary
layer reduces. so, temperature gradient vanishes quicker for higher
values of Pr . Fig. 21 displays the the effect of Eckert number on
temperature of the fluid and it is noticed that temperature
increases on increasing the Eckert number Ec . Fig. 22 shows the
effect of Schmidt number Sc on the the thermal boundary layer
thickness, a slight change in temperature on increasing Sc is
noticed. Fig. 23 depicts the influence of chemical reaction
parameter Kr on temperature of the fluid . It is found that
temperature of the fluid is slightly rised on increasing Kr. Fig.
24 and Fig. 25 represent the effects of thermal and mass Grashof
numbers Gr and Gc on concentration respectively. It is noticed that
concentration decreases on increasing Gr and Gc. The effect of
magnetic parameter M on concentration of the fluid is illustrated
in Fig. 26. Decrease in concentration boundary layer thickness is
observed on increasing the magnetic parameter. For different values
of porosity parameter δ graphs for the concentration of the fluid
are plotted in Fig. 27. A slight change in the thickness of the
concentration boundary layer is found from the figure on increasing
porosity parameter. Fig. 28 demonstrates the effect of inclination
parameter γ on the concentration field. slight increase in the
concentration is observed on increasing γ . The effects of
dissipation parameter on the concentration boundary layer is
reported in Fig. 29. Rising the dissipation heat causes reducing
the thickness of the concentration boundary layer. Fig. 30 displays
the effect of radiation parameter R on the concentration of the
fluid. A Decrease in thickness of the concentration boundary layer
is noticed on increasing R . Fig. 31 shows that Prandtl number Pr
does not effect the thickness of the concentration boundary layer.
The effect
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
451 Copyright © 2016. Vandana Publications. All Rights
Reserved.
of Eckert number on the concentration of the fluid is presented
in the Fig. 32. A Decrease in the concentration is found on
increasing the Eckert numbered. Fig. 33 demonstrates the effect of
Schmidt number Sc on concentration and it is seen that
concentration decreases on increasing the Schmidt number. Effect of
the chemical reaction parameter on the concentration is reported in
the Fig. 34. The concentration boundary layer becomes thinner on
increasing the chemical reaction parameter Kr. This is due to the
fact that the chemical reaction in this system results in
consumption of the chemical and hence results in decrease of
concentration.
V. CONCLUSIONS
From the present study we arrive at the following significant
observations. By comparing the present results with that of Sing h
et al. [ 26], it is found that there is a good agreement.
Increasing thermal Grashof number increases the velocity, but
reduces the temperature and concentration. Increasing mass grashof
number increases the velocity, but reduces the temperature and
concentration. Increasing porosity parameter increases the
velocity, but reduces the temperature and concentration. Increasing
radiation parameter increases the velocity and temperature , but
reduces the
concentration. Increasing Eckert number increases the velocity
and temperature, but reduces the concentration. Increasing magnetic
parameter increases the velocity, but reduces the temperature and
concentration. Increasing dissipation parameter increases
temperature and concentration, but decreases the velocity
.Increasing Prandtl number slightly increases the concentration,
but reduces the velocity and temperature. Increasing Schmidt number
results a decrease in the velocity and concentration, but its
influence is slight in the temperature. Increasing inclination
parameter increases the temperature and concentration, but reduces
the velocity. Increasing chemical reaction parameter enhances the
temperature , but reduces the velocity and concentration. Skin
friction coefficient and the Sherwood number increases with the
increase in the Radiation parameter while Nusselt number decreases.
Skin fraction coefficient and Nusselt number decreases where as
sherewood number increases on increasing chemical reaction
parameter.
.
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
452 Copyright © 2016. Vandana Publications. All Rights
Reserved.
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
453 Copyright © 2016. Vandana Publications. All Rights
Reserved.
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
454 Copyright © 2016. Vandana Publications. All Rights
Reserved.
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
455 Copyright © 2016. Vandana Publications. All Rights
Reserved.
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
456 Copyright © 2016. Vandana Publications. All Rights
Reserved.
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
457 Copyright © 2016. Vandana Publications. All Rights
Reserved.
Table1. showsthe variations of skin friction rate, nusselt
number and sherwood number for different parameters.
• Increasing radiation parameterR increases skinfriction
coefficient and Sherwood number, but reduces the Nusselt number
• Increasing chemical reaction parameter Kr increasesSherwood
number, but reduces the skin friction coefficient and Nusselt
number.
• Increasing dissipation parameter N, increases Sherwood number,
but reduces skin friction coefficient and nusselt number.
• Increasing inclination parameter γ decreases skin friction
coefficient, Nusselt number as well as Sherwood number.
• Increasing buoyancy parameters Gr and Gc increases skin
friction coefficient, Nusselt number and Sherwood number.
• Increasing magnetic parameter decreases skin friction
coefficient, Nusselt number and Sherwood number.
Table 2. shows the comparison of present work with that of Singh
et al.[26]. From the table it is clear that there is a good
agreement of present results with that of Singh et al.[26].
Table1: Variation of ( )0f ′′ , ( )0θ ′− and ( )0φ′− for
different values of Gr, Gc, M, Pr, R,
Kr, γ and N
Table 2:comparison of present results for ( )0f ′′ , ( )0θ ′−
and ( )0φ′− at the plate for different values of Gr, Gc , M,
Pr, Ec, Sc, N and γ for Kr = 0 , R = 0 with that of Singh et al.
[26].
Gr Gc M Pr R Kr γ N ( )0f ′′ - ( )0θ ′ - ( )0φ ′ 0.2 0.4 0.6 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1
0.1 0.1 0.1 0.2 0.4 0.6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
1 1 1 1 1 1 1 1.5 2.0 1 1 1 1 1 1 1 1 1 1 1
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.9 7.7 0.7 0.7 0.7 0.7
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
1 1 1 1
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1.5 2.0 0.5
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1
1.5 2.0 2.0 0.5 0.5 0.5 0.5 0.5
45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o 45o
45o 45o 45o 60o 90o 45o 45o 45
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.2 0.4 0.6
0.282752 0.38481
0.486086 0.375073 0.375073 0.470329 0.231404 0.183974 0.156977
0.231404 0.231172 0.225703 0.231611 0.231736 0.231821 0.22875
0.226445 0.224545 0.231611 0.202367 0.131566 0.231847 0.229454
0.227616
0.26876 0.272716 0.276285 0.272143 0.272143 0.275448 0.266625
0.262643
0.2601 0.266625 0.271239 0.391472 0.262536 0.260061 0.258401
0.26239
0.262376 0.262179 0.262536 0.261605 0.259349 0.25805
0.249072 0.240241
0.391036 0.393195 0.395309 0.392854 0.392854 0.394765 0.389937
0.387701 0.386416 0.389937 0.389927 0.389705 0.389946 0.389952
0.389956 0.497741 0.591098 0.674002 0.389946 0.38933 0.38783
0.389957 0.497765 0.591131
Gr Gc M Pr Ec Sc N γ P.k.sing h et.al [ 26] Present work ( )0f
′′ - ( )0θ ′ - ( )0φ ′ ( )0f ′′ - ( )0θ ′ - ( )0φ ′
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
458 Copyright © 2016. Vandana Publications. All Rights
Reserved.
REFERENCES [1] S. Ostrach, An analysis of Laminar Free
Convection Flow and Heat Transfer about a Flat Plate Parallel to
the Direction of the Generating Body Force. Technical Note,
NACAReport, Washington, 1952. [2] K. A. Yih, Free Convection effect
on MHD Coupled Heat and Mass Transfer of Moving Permeable Vertical
Surface. Int. Commun, Heat Mass Transfer, 26, 95-104, 1999. [3] M.
Sajid, T. Hayat and S. Asghar, Non-Similar Analytic Solution for
MHD Flow and Heat Transfer in a Third- Order Fluid over a
Stretching Sheet. InternationalJournal of Heat and Mass Transfer,
50, 1723-1736, 2007. [4] A. S. Gupta, Laminar Free Convection flow
of an Electrically Conducting Fluid FrombVertical Plate with
Uniform Surface Heat Flux and Variable Wall Temperature in the
Presence of a Magnetic field. Zeitschrift fur Angewandte Mathematik
und Physik , 13, 4, 324–333, 1962. [5] H. S. Takhar, A. A. Raptis,
and C. P. Perdikis, MHD asymmetric flow past a semi-infinite moving
plate. Acta Mechanica, 65 , 1–4, 287–290, 1987. [6] I. Pop, M.
Kumari, and G. Nath, Conjugate MHD flow past a flat plate. Acta
Mechanica, 106, 3-4, 215–220, 1994. [7] H. S. Takhar and G. Nath,
Similarity solution of unsteady boundary layer equations with a
magnetic field. Meccanica , 32 , 2, 157–163, 1997. [8] I. Pop and
T. Y. Na, A Note on MHD flow over a stretching permeable surface.
Mechanics Research Communications , 25 , 3, 263–269, 1998.
[9] H. S. Takhar, A. J. Chamkha and G. Nath, Unsteady flow and
heat transfer on a semi infinite flat plate with an aligned
magnetic field. International Journal of Engineering Science , 37,
13, 1723–1736, 1999. [10] A. Nakayama and H. Koyama, A general
similarity transformation for combined free and forced convection
flows within a fluid saturated porous medium. ASME J. Heat
Transfer, 109, 1041-1045, 1987. [11] P. Cheng and W. J. Minkowycz,
Free convection about a vertical flat plate embedded in a porous
medium with application to heat transfer from a dike. J.Geophys.
Res., 82 , 2040-2044, 1977. [12] O.D. Makinde, On MHD
boundary-layer flow and mass transfer past a vertical plate in A
porous medium with constant heat flux. Int. J. Numer. Methods Heat
Fluid Flow, 19 ,546-554, 2009. [13] A. Zeeshan and R. Ellahi,
Series solutions for nonlinear partial differential equations with
slip boundary conditions for non-Newtonian MHD fluid in porous
space. Applied Mathematics & Information Sciences. 7, 1,
253-261, 2013. [14] M.S. Alam, M.M. Rahman , M. A. Sattar, Effects
of variable suction and thermophoresis on steady MHD com bined free
forced convective heat and mass transfer flow over a semi-infinite
permeable inclined plate in the presence of thermal radiation, Int.
J. Therm. Sci. 47, 6, 758-765, 2008. [15] M. S. Alam , M. M. Rahman
, M. A. Sattar, On the effectiveness of viscous dissipation and
Joule heating on steady magnetohydrodynamic heat and mass transfer
flow over an inclined radiate isothermal permeable
0.2 0.4 0.6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.2 0.4 0.6 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
1 1 1 1 1 1 1 1.5 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.9 7.7 0.7 0.7 0.7 0.7
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02
0.04 0.06 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22
0.22 0.22 0.2 0.6 2.6 0.22 0.22 0.22 0.22 0.22 0.22
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.2 0.4 0.6 0.1 0.1 0.1
45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45
30 45 60
0.285508 0.386253 0.485873 0.286288 0.389048 0.491221 0.234691
0.186665 0.159254 0.234691 0.234353 0.227563 0.234713 0.234757
0.234800 0.234691 0.233772 0.229627 0.235164 0.236139 0.237083
0.257736 0.234691 0.204587
0.278131 0.283853 0.28932 0.278216 0.284125 0.289787 0.275034
0.269133 0.265511 0.275034 0.281863 0.439544 0.274521 0.273494
0.272468 0.275034 0.274413 0.27445 0.265999 0.24722 0.229222
0.276465 0.275034 0.273119
0.262422 0.265106 0.26771 0.262467 0.265264 0.268009 0.261048
0.257986 0.256215 0.261048 0.261029 0.260676 0.261049 0.261052
0.261054 0.261048 0.279626 0.36543 0.261076 0.261134 0.26119
0.261678 0.261048 0.260222
0.282752 0.38481 0.486086 0.375073 0.375073 0.470329 0.231404
0.183974 0.156977 0.231404 0.231172 0.225703 0.231622 0.231644
0.231666 0.231611 0.226584 0.220376 0.231847 0.229454 0.227616
0.231611 0.202367 0.131566
0.26876 0.272716 0.276285 0.272143 0.272143 0.275448 0.266625
0.262643 0.2601 0.266625 0.271239 0.391472 0.26228 0.261767
0.261254 0.262536 0.262276 0.261971 0.25805 0.249072 0.240241
0.262536 0.261605 0.259349
0.391036 0.393195 0.395309 0.392854 0.392854 0.394765 0.389937
0.387701 0.386416 0.389937 0.389927 0.389705 0.389947 0.3899
0.389949 0.389946 0.575737 0.856657 0.389957 0.497765 0.591131
0.389946 0.38933 0.38783
-
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT):
2394-6962
459 Copyright © 2016. Vandana Publications. All Rights
Reserved.
surface in the presence of thermophoresis, Commun. Nonlinear.
Sci. Numer Simul. 14, 2132- 2143, 2009. [16] P. K. Singh, , Heat
and Mass Transfer in MHD Boundary Layer Flow past an Inclined Plate
with Viscous Dissipation in Porous Medium, International Journal of
Scientific & Engineering Research, 3, 61 , 2229-5518, 2012.
[17] R. Viskanta, R.J. Grosh. Boundary layer in thermal radiation
absorbing and emitting media. Int. J Heat Mass Transfer 5, 795–806
, 1962. [18] A. Raptis, Radiation and free convection flow through
a porous medium. Int.Commun. Heat Mass Transf, 25, 289-295, 1998.
[19] D. Hall, G. C. Vliet and T. L. Bergman, Natural convection
cooling of vertical rectangular channels in air considering
radiation and wall conduction. J. Electron. Pack, 121, 75-84, 1999.
[20] A. Y. Bakier, Thermal radiation effects on mixed convection
from vertical surfaces in saturated porous media. Int. Comm. Heat
and Mass Transfer, 28, 243-248, 2001. [21] A. Raptis and C.
Perdikis, Unsteady flow through a highly porous medium in the
presence of radiation. Transport in Porous Media, 57, 171-179,
2004. [22] C. G. Rao , Interaction of surface radiation with
conduction and convection from a vertical channel with
multiple discrete heat sources in the left wall, Numer. Heat
Transfer Part A: Appl., 52, 831-848, 2007. [23] A. K. Mishra, K.
Rajesh Menon and Shaima Abdullah Amer Shanfari, the Effect of
Radiation on Free Convection Heat and Mass Transfer Flow through
Porous Medium in a Vertical Channel with Heat Absorption/
Generation. International Journal of Advanced Research in Computer
Engineering & Technology , 4 , 7, 2015. [24] B. R. Sharma and
Nabajyoti Dutta, Influence of chemical reaction and thermal
radiation effects on MHD boundary layer flow over a moving vertical
porous plate, International Research Journal of Engineering and
Technology. 2 ,7 ,180-187, 2015. [25] S. Y. Ibrahim and O. D.
Makinde , Chemically reacting MHD boundary layer flow of heat and
mass transfer over a moving vertical plate with suction. Scientific
Research and Essays 5,19, 2875-2882, 4 October, 2010. [26] P.K.
Singh, Heat and Mass Transfer in MHD Boundary Layer Flow past an
Inclined Plate with Viscous Dissipation in Porous Medium,
International Journal of Scientific & Engineering Research, 3,
61, 2229-5518, 2012. [27] M. K. Jain, S. R. K. Iyengar and R.K.
Jain , Numerical methods for Scientific and engineering
computation. Wiley Eastern Ltd, New Delhi, India, 1985.