MHD FREE CONVECTIVE FLOW PAST A POROUS PLATE · the stagnation point flow due to a shrinking sheet...
Transcript of MHD FREE CONVECTIVE FLOW PAST A POROUS PLATE · the stagnation point flow due to a shrinking sheet...
MHD FREE CONVECTIVE FLOW PAST A POROUS PLATE
P. Rama Krishna Reddy* M. C. Raju**
* Research scholar, Department of Mathematics, Shri Venkateshwara University, VenkateshwaraNagar,
Rajabpur, Gajraula, Amroha, Uttar Pradesh 244236, U. P., India. Email: [email protected]
** Research supervisor and corresponding author, Email: [email protected], Cell: 9848998649
Abstract:This manuscript is focused on unsteady magentohydrodynamic (MHD) free convective
flow of a double diffusive fluid past a moving vertical porous plate in the presence of thermal
radiation and first order homogeneous chemical reaction. The temperature of the plate is
assumed span wise cosinusoidally fluctuating with time in the presence of heat generation. The
second order perturbation technique is employed to investigate the non- linear partial differential
equations governing the fluid flow which are non- dimensionalized by introducing the similarity
transformations. The effects of magnetic intensity, radiation, porous permeability, Eckert
number, Schmidt number and heat generation/absorption parameters on velocity, concentration
and temperature. Also, the skin friction coefficients, the rate of heat transfer and rate of mass
transfer at the surface of the plate are computed numerically. The results show that within the
boundary layer, the velocity and temperature are found to decrease with the increasing values of
Prandtl number and radiation parameters however the trend is reverse with respect to porous
permeability and heat generation/absorption parameters.
Key words:
Fluctuatingtemperature,Chemical reaction,Suction,Heat generation, MHD, Double diffusive
fluid and porous medium.
List of symbols
cp Specfic heat at constant pressure (J kg-1 K-1)
C Species concentration (mol m-3)
C Dimensionless concentration
Cf Skin friction
C Concentration away from the wall(mol m-3)
wC Concentration at the wall(mol m-3)
D Chemical molecular diffusivity (m2s-1)
Ec Eckert number
International Journal of Pure and Applied MathematicsVolume 118 No. 5 2018, 507-529ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu
507
g Acceleration due to gravity (m/s2)
Gr Grashof number
Gm Solutal Grashof number
H0 Magnetic field(N/Am)
k Thermal conductivity( W m-1K-1)
kc Rosseland mean absorption coefficient (m-1)
K Permeability of the porous medium
K1 Rate of chemical reaction parameter
K* Permeability of porous medium (m2)
K Dimensionless permeability
l Wave length (m)
M Hartmann number
Nu Nusselt number
Pr Prandtl number
0Q
Heat generation or absorption coefficient
rq Radiative heat flux ( W m-2)
Re Raynolds number
R Radiation parameter
So Soret number
Sc Schmidt number
T Fluid temperature (K)
wT Wall temperature (K)
0T Mean temperature (K)
T Free stream dimensional temperature(K)
T Dimensionless time
u Dimensionless velocity
e Magnetic permeability (H m-1 or N A-1)
Dimensionless frequency of oscillation
Dimensionless chemical reaction parameter
Dimensionless temperature
Coefficient of thermal expansion
Coefficient of concentration expansion
Dimensionless heat generation/ absorption parameter
v Kinematic viscosity (m2s)
Density of the fluid ( kgm-3)
Electrical conductivity(S/m)
s Stefan-Boltzmann constant(Wm-2K-4)
International Journal of Pure and Applied Mathematics Special Issue
508
wvu Velocity component along zyx directions (m s-1)
1. Introduction:
The study of heat flow with heat and mass transfer has become industrially more
important as most of the engineering processes occur at a high temperature such as in nuclear
power plants,gas turbines and various propulsion devices for aircraft,missiles,satellites and space
vehicles.In all of these applications,the radiation effects can be quite significant at high
temperature.Further, in astrophysical regimes,the presence of planetary debris,cosmic
dusted,creates a suspended porous medium saturated with plasa fluids.As in other porous media
problems such as aeromechanics and insulation engineering, the conventional approach is to
simulate the pressure drop across the porous regime. Israel-Cookey et al. [1] examined the
influence of viscous dissipation and radiation on unsteady MHD free- convection flow past an
infinite heated vertical plate in a porous medium with time dependent suction. Chamka [2]
examined an unsteady simultaneous convection heat and mass transfer flow along a vertical
permeable plate embedded in a fluid saturated porous medium in the presence of mass blowing
or suction, magnetic field and heat absorption. Pal and Mandal [3] studied the effects of thermal
radiation onMHD Darcy Forchheimer convective flow past a stretching sheet in porous medium.
Turkyilmazoglu and Pop [4] discussed heat and mass transfer effects on unsteadyMHDfree
convection flow past an impulsively started infinite vertical plate with Soret and heat source
effects. Ram et al. [5] examined the effects of porosity on unsteadyMHDfree convective
boundary layer flow along a semi- infinite vertical plate with time dependent suction by taking
into account the effects of dissipation. Parsa et al. [6] presented steady laminar MHD boundary
layer flow past a stretching surface with uniform free stream and internal heat generation or
absorption in an electrically conducting fluid. The transient MHDfree convective flow of a
viscous, incompressible, electrically conducting, gray, absorbing-emitting, but not scattering,
optically thick fluid medium which occupies a semi-infinite porous region adjacent to an infinite
hot vertical plate moving with constant velocity was presented by Ahmed and Kalita [7]. The
influence of thermal radiation and Darcian drag forceon unsteady MHD thermal-convection flow
past a semi-infinite vertical plate immersed in a semi-infinite saturated porous regime with
variable surface temperature in the presence of transversal uniform magnetic field have been
discussed by Ahmed et al.[8]. Singh et al. [9] analysized the unsteady MHD free convection,
incompressible, electrically conducting and fluctuating flow past an impulsively started
isothermal vertical plate by taking into account the effect of viscous dissipation. Raju and Varma
[10] considered an unsteady MHD free convective, thermal diffusive and visco- elastic fluid with
heat and mass transfer in porous medium bounded by an infinite vertical porous plate with heat
absorption and dissipation. Ziyauddin et al. [11] investigated MHDheat transfer flow of a
International Journal of Pure and Applied Mathematics Special Issue
509
micropolar fluid past a wedge with heat generation or absorption by considering the radiation
effect along with viscous dissipation, Joule heating, and Hall and ion-slip effects. Su [12]
examined unsteady MHDmixed convective flow and heat transfer over an impulsively stretched
permeable vertical surface with partial velocity and thermal slip conditions in the presence of
thermal heat generation and suction/injection. Hazarika and Ahmed [13] analyzed an unsteady
MHD flow of two-dimensional, laminar, incompressible, Newtonian, electrically-conducting and
radiating fluid along a semi-infinite vertical permeable moving plate with periodic heat and mass
transfer by taking into account the effect of viscous dissipation in presence of chemical reaction.
Sahoo et al. [14] consideredMHD unsteady free convection flow past an infinite vertical plate
with constant suction and heat sink. Singh and Kumar [15] studied fluctuating heat and mass
transfer on unsteady MHDfree convection flow of radiating and reacting fluid past a vertical
porous plate in slip- flow regime. Mishara et al. [16] examined heat and mass transfer in the
MHDflow of avisco-elstic fluid in a rotating porous channel with radiative heat and chemical
reaction. Hussaini et al. [17] discussed MHD unsteady radiating memory convective flow with
variable suction. Pal et al. [18] investigated steady magneto hydro dynamic (MHD) boundary
layer flow of a Casson nanofluid over a vertical stretching surface with combined effects of
thermal radiation, Ohmic dissipation, thermophoresis and Brownian motion on heat and mass
transfer. Vajravelu et al. [19] studied non-linear convection effects on the flow past a flatporous
plate. Chand and kumar [20] described the effects of Hall current and slip conditions on heat and
mass transfer in the unsteady flow of the viscoelastic fluid (Walter’s liquid B’ model) past an
infinite porousflat plate subjected to uniform suction through a porous medium. Singh and
Pathak [21] focused the effects of Hall current and rotation on MHD free convection flow in a
vertical rotating channel filled with theporous medium. Srinivsasacharya and Reddy [22]
investigated on mixed convection heat and mass transfer from a vertical plate embedded in a
power-law fluid- saturatedporous mediumwith radiation and chemical reaction effects.
Chaudhary and Kumar [23] described steady two- dimensional, laminar, viscous incompressible
stagnation point flow past a porous medium with heat generation of an electrically conducting
fluid over a stretching surface in the presence of a magnetic fluid. Pal and Mandal [24] examined
mixed convection-radiation interaction on stagnation-point flow of nanofluids over a
stretching/shrinking sheet embedded in a porous medium in the presence of viscous dissipation
and internal heat generation. Kumar and Sood [25] investigated non-linear convection effects on
the stagnation point flow due to a shrinking sheet in the presence of the porous medium.
Soundalgekar [26] analyzed the viscous dissipative heat on the two-dimensional unsteady free
convective flow past an infinite vertical porous platewhen the temperature oscillates in time and
there is constant suction at the plate.
In the chemical engineering processes, the chemical reaction occurs between a foreign
mass and the fluid in which the plate is moving. These processes take place in many industrial
areas that is manufacturing of ceramics or glassware, polymer production, and food
processing.Prakash and Ogulu [27] analylized the problem of unsteady two-dimensional
boundary layer flow of a viscous incompressible, electrically conducting fluid along a semi-
International Journal of Pure and Applied Mathematics Special Issue
510
infinite vertical plate in the presence of thermal and concentration buoyancy effects. The effects
ofchemical reactionand heat generation on unsteady MHD free convection heat and mass transfer
flow past an infinite hot vertical porous plate are obtained by Singh and Kumar [28]. Ibrahim and
Mankinde [29] investigated radiation effect onchemically reaction MHD boundary layer flow of
heat and mass transfer past a porous vertical flat plate. The influence of heat absorption,
radiation, Joule heating, viscous dissipation, chemical reactionand thermal diffusion on unsteady
MHD Couette flow of a dusty viscoelastic fluid in a vertical irregular porous channel with
convective boundary and varying mass diffusion was discussed by Sivaraj and Kumar[ 30]. Rout
and Pattanayak [31] have examined and reported chemical reaction and radiation effects on
MHD flow past an exponentially accelerated vertical porous plate in the presence of heat
generation with variable temperature embedded in a porous medium. Venkateswarlu and Satya
naraya [32] examined radiation absorption andchemical reactioneffectson MHD free convection
heat transfer flow of a nanofluid bounded by semi-infinite flat plate in a rotating system. Mishra
et al. [33] dealt with heat and mass transfer characteristic of MHD visco-elastic fluid in a rotating
porous channel with radiative heat and chemical reaction. Garg et al. [34] investigated on
chemically reacting, radiating and rotating MHD convective flow of visco-elastic fluid through
porous mediumin vertical channel. Makinde et al. [35] studied the effects of buoyancy forces,
homogeneous chemical reaction, thermal radiation, partial slip, heat source, Thermophoresis and
Brownian motion on hydromagnetic stagnation point flow of nanofluid with heat and mass
transfer over a stretching convective surface. Srinivasacharya and Jagadeeshwar [36]
investigated the boundary layer flow, heat and mass transfer towards the exponentially stretching
sheet in a viscous fluid. Loganathan and Ganesan [37] analyzed the effects of radiation on the
flow past an impulsively started infinite vertical plate in the presence of mass transfer. Nadeem
et al. [38] investigated the boundary layer flow behavior of a Jeffery fluid due to an
exponentially stretching surface and explained the effects of thermal radiation for two cases of
heat transfer analysis known as Prescribed exponential order surface temperature (PEST) and
Prescribed exponential order heat flux (PEHF).
In this paper, we have extended the work of Ramet al. [39] and investigated the effects of
thermal radiation and chemical reaction on free convective heat and mass transfer unsteady flow
with heat generation and constant suction normal to an infinite hot vertical porous plate when the
plate temperature is Cosinusoidallyfluctuating with time.The present study of heat generation or
absorption in moving fluids is important in view of several physical problems involving
exothermic or endothermic chemical reaction. The governing partial equations are transformed
into a set of ordinary differential equations by using a similarity transformation which is further
solved with the help of second order perturbation scheme. To the best of our knowledge,this kind
of work has not yet been investigated.
2. Mathematical Formulation
Consider the flow of a conducting fluid past an infinite hot porous plate lying vertically on zyx plane.The plate is assumed to be infinite in lengthand taken along the fluid in x
International Journal of Pure and Applied Mathematics Special Issue
511
direction, therefore all physical quantities are independent of x . A magnetic field of uniform
strength B0 is applied normally.Let wvu ,, be the components of the velocity in the
zyx ,, directions respectively.Due to suction at the surface of the plate with constant
velocity V , w is independent of z and assumed as zero.Further, we assume that the
magnetic Reynolds number is very small so that the induced magnetic field is negligible in
comparison to the applied magnetic field. The fluid is considered hereto be gray,
absorbing/emitting radiation but a non-scattering medium. Noexternal electrical field is applied
and effect of polarization of ionized fluid is negligible,therefore, electrical field is assumed to be
zero. There exists a first order chemical reaction between the fluid and species concentration, the
heat generated during chemical reaction cannot be neglected. Followed by Ram et al. [39] and by
considering Boussinesq’s approximation, the flow field is governed by the following set of
equations:
Continuity equation:
0
y
v
(1)
Momentum Equation:
u
ku
B
z
u
y
uCCgTTg
y
uv
t
u*
2
022
*
22)()(
(2)
Energy equation:
)((0
22
22
TTQ
y
q
z
T
y
T
y
Tv
t
Tc r
p
(3)
Species diffusion equation:
))()2222
22
1
*22
z
T
y
TDCC
z
C
y
CD
y
Cv
t
C
(4)
The temperature of the plate is considered to vary spanwisecosinusoidally fluctuating with time
and assumed to be of the form
twl
zTTTtzTw
cos, 00
(5)
The corresponding initial and boundary conditions are as follows:
0u ,
twl
zTTTT
cos00 , wCC at y =0
0u ,
TT ,
CC yas (6) For the case of an optimality thin gray gas, local radiative heat flux in the energy equation
International Journal of Pure and Applied Mathematics Special Issue
512
44
4
TTk
y
qes
r
(7)
If temperature differences within the flow are sufficiently small,then equation (7) can be linearized by
expanding
4
T
into the Taylors series about
T
,which after neglecting higher order
terms,takes the form
4334
4
TTTT (8)
In the view of Eqs.(7) and (8), Eq.(3) reduces to
)(16 (0
322
22
TTQTTTk
z
T
y
T
y
Tv
t
Tc
sep
(9)
The non-dimensional parameters as follows
l
yy
, tt , v
uu
,l
zz
, l
kk
, twtv
l 2
, l
zz
,
1
2kl ,
TT
TT
C
C
0w
,C
C
(10)
Using the transformation (10) and Eq.(5), the momemtum Eq.(2), energy Eq. (9) and
concentration Eq.(4) reduce to the following dimensionless form
k
uuM
z
u
y
uCGGR
y
uR
t
umree
2
2
2
2
22
(11)
r
e
r
eP
RR
zyPyR
t
2
2
2
2
21
(12)
2
2
2
2
02
2
2
21
zySK
zySyR
tc
C
e
(13)
Where v
VlRe
is the Reynolds number,
3
0
V
TTvgGr
is the thermal Grashof number,
3
0
V
CCvgGm
is the mass / Solutal /modified Grashof number,
v
lBM
2
02
is the magnetic parameter/ Hartmann number,
International Journal of Pure and Applied Mathematics Special Issue
513
2
3216
kV
TvkR se
is the radiation parameter,
k
cP
p
r
is the Prandtl number,
D
vSc
is the Schmidt number,
vc
lQ
p
2
0
is the heat generation parameter,
CC
TTDS
0
00
is the Soret number.
The corresponding boundary conditions are given by
;01),cos(1,0 yattzu
;0,0,0 yatu
(14)
3. SOLUTION OF THE PROBLEM
To investigate the effects of various parameters like Grashof number, Prandtl number, porous
permeability parameter and Schmidt number etc, on velocity and temperature fields in the
boundary layer generated on the surface, the solution of partial differential equations(11-13)
along with boundary condition(14) is obtained using perturbation technique (Chamkha[2],
Kumar and Singh [15], Ram et al/ [5]) through which it is assumed the components of velocity,
temperature and concentration respectively as follows: )(2
2
2)(
10
tzitzi eueuuu
)(2
2
2)(
10
tzitzi ee (15)
)(2
2
2)(
10
tzitzi ee
Substituting equation (15) into set of equations (11)-(13) and equating the like powers of ε, the
following set of equations are obtained:
Zeroth order equations
00
2
0
1
0
11
0 ReRe mr GGNuuu (16)
0Re 0
1
0
11
0 rr PP (17)
CCcec SSSkRS 00
211
00
1
0
11
0 4 (18)
The corresponding boundary conditions reduce to the following form;
00 u , 10 , 10 ,at y=0
International Journal of Pure and Applied Mathematics Special Issue
514
00 u , 00 , 00 , at y= (19)
First order equations
11
2
1
1
1
11
1 Re)(Re mr GGuwiNuu (20)
0)(Re 1
21
1
11
1 iwPP rr (21)
CCCcec SSiwSSkRS 0
11
11
2
1
21
1
11
1 (22)
The corresponding boundary conditions reduce to
01 u , 11 , 01 at y=0
01 u , 01 , 01 at y= (23)
Second order equations
22
2
2
21
2
11
2 Re)24(Re mr GGuwiNuu (24)
0)24(Re 2
21
2
11
2 iwPP rr (25)
The corresponding boundary conditions reduce to
02 u , 02 , 02 at y=0
02 u , 02 , 02 at y= (26)
Solving the above differential equations along with the corresponding boundary conditions, the
following solutions are obtained: ym
e 1
0
(27)
yme 3
1
(28)
02 (29)
ymymekek 17
110 1
(30)
ymymekek 39
221
(31)
02 (32)
ymymymekekekku 7113
21430
(33)
ymymymekekekku 9315
21651
(34)
02 u (35)
4. Results and discussion:
The set of non-dimensionless governing equations (16) - (18) and subject to the corresponding
appropriate boundary conditions (19 ) are solved analytically by using regular perturbation
technique. The solutions are carried out for different flow parameters on the flow quantities and
the computed results are presented in figures1-11.
International Journal of Pure and Applied Mathematics Special Issue
515
The effect of Reynolds number on temperature is shown in figure 1, for all values of
Reynolds number the temperature profiles have significant appearance near the plate but as the
values of Reynolds number are increased temperature decreased and reaches ambient
temperature far away from the plate.The effect of Prandtl number on temperature is presented in
Fig.2. It is noticed that the temperature of fluid decreases as Prandtle number increases.The
effect of radiation parameter on temperature is presented in Fig.3. It is noticed that temperature
of the fluid decreases as radiation parameter increases.The effect ofheat absorption /generation
parameter on temperature is presented in Fig.4. From this figure, it observed that temperature
increases for increasing values of absorption parameter but reverse trend in the presence of
generation parameter.The effect of Schmidt number on concentration is presented in the Fig.5.
This figure witnesses that, concentration decreases for increasing values of Schmidt number.The
effect of chemical reaction parameter on concentration is presented in Fig.6, from which it is
noticed that the concentration decreases on increasing the values of chemical reaction parameter.
The effect of Soret number on concentration is presented in Fig.7, which depicts that
concentration increases for increasing values of Soret number. The effect of Solutal Grashof
number on velocity is presented in the Fig.8.It is noticed that the velocity increases on increasing
the values of Solutal Grashof number.The effect of Hartmann number on velocity is presented in
the Fig.9, that shows that the velocity decreases as expected on increasing thevalues of Hartmann
number. The effect of Grashaf number on velocity is shown in the Fig.10, it is noted that the
velocity increases on increasing the values of thermal Grashaf number. This is due the fact that
the presence of buoyancy force that enhances the velocity for both the cases of thermal and
solutal Grashof numbers. The effect of Reynolds number on velocity is presented in the Fig.11
through which it is noticed that the velocity increases on increasing the values of Reynolds
number. The effect of permeability of the porous medium on velocity is presented in the Fig.12,
from this figure it is noticed that velocity increases for increases values of permeability
parameter.
International Journal of Pure and Applied Mathematics Special Issue
516
Fig.1. Effect ofReynolds number on temperature
Fig.2. Effect of Prandtl number on temperature
0 0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
y
Re=1,3,5,7
Re=7
R=0.5
Pr=7=0.5
w=.5
z=0
e=0.1
t=pi/2
0 0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
y
Re=10R=0.5Pr=7
=0.5
=.5z=0;
=0.1
t=/2
Pr=0.71,7.0
International Journal of Pure and Applied Mathematics Special Issue
517
Fig.3. Effect of Radiation parameter number on temperature
Fig.4. Effect of radiation absrption/generation parameter on temperature
0 0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
y
R=0.2,0.5,1,2
Pr=0.71Re=10
=0.5
=0
t=/2
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
y
=0.1
=0.5
=1.0
=1.5
= -0.1
= -0.5
=-1.0
= -1.5
Pr=0.71Re=10,
=0
t=/2
International Journal of Pure and Applied Mathematics Special Issue
518
Fig.5. Effect of Schmidt number on concentration
Fig.6. Effect of Chemical reaction parameter on concentration
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
y
Sc=0.16,0.22,0.61,0.78,0.96
Re=10R=0.5Pr=7.0
=5z=0
=0.1
t=/2S
0=0.1
Kc=0.2
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
y
Re=10R=0.5Pr=7.0
=5z=0
=0.1
t=/2S
0=0.1
Sc=0.22
Kc=0.2,0.4,0.6,0.8,1.0
International Journal of Pure and Applied Mathematics Special Issue
519
Fig.7. Effect of Soret number on concentration
Fig.8. Effect of Solutal Grashof number onvelocity
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
y
Re=10R=0.5Pr=7.0
=5z=0
=0.1
t=/2Kc=0.2Sc=0.22
S0=0.1,0.3,0.5,0.7,1.0
0 1 2 3 4 50
1
2
3
4
5
6
y
u
Gm=5,6,7,8
Gr=5Re=5M=2Pr=7
=1z=0K
p=0.1
=0.5Sc=0.22Kc=0.2S
0=2
R=0.5
International Journal of Pure and Applied Mathematics Special Issue
520
Fig.9. Effect of Hartmann number on velocity
Fig.10. Effect of Grashof number on velocity
0 1 2 3 4 50
1
2
3
4
5
6
7
8
y
u
M=1,2,3,4
Re=5Gr=10Gm=10Pr=7
=1z=0R=0.5K
p=0.1
Ql=0.5
Sc=0.22Kc=0.2S
0=2
0 1 2 3 4 50
2
4
6
8
10
12
14
y
u Gr=5,10,15,20
Gm=15Re=5M=2Pr=7
=1z=0K
p=0.1
Ql=0.5
Sc=0.22Kc=0.2S
0=2
R=0.5
International Journal of Pure and Applied Mathematics Special Issue
521
Fig.11. Effect of Reynolds number on velocity
Fig.12. Effect of Permeability of the porous medium onvelocity
Table 1 :Rate of heat transfer coefficient(Nusselt number)
rP R Re ω χ Nu
0.71 2 5 0.5 0.5 9.0103
7.0 2 5 0.5 0.5 36.2816
0 1 2 3 4 50
2
4
6
8
10
12
14
16
y
u
Re=5,6,7,8
Gr=10
0 1 2 3 4 50
1
2
3
4
5
6
7
y
u
Kp=0.1,0.3,0.5,1.0
Gr=5Gm=8Re=5M=2Pr=7
=1z=0
=0.5Sc=0.22Kc=0.2S
0=2
R=0.5
International Journal of Pure and Applied Mathematics Special Issue
522
0.71 3 5 0.5 0.5 10.5655
0.71 5 5 0.5 0.5 11.8847
0.71 2 10 0.5 0.5 18.0568
0.71 2 15 0.5 0.5 27.0952
0.71 2 20 0.5 0.5 36.1316
0.71 2 5 0.6 0.5 9.0103
0.71 2 5 0.7 0.5 9.0055
0.71 2 5 0.5 0.6 9.0055
0.71 2 5 0.5 0.7 9.0007
0.71 2 5 0.5 0.8 8.9959
Table 2 :Rate of mass transfer coefficient( Sherwood number)
Sc S0 Re Kc Sh
0.16 1 1 0.2 0.8351
0.22 1 1 0.2 0.8364
0.60 1 1 0.2 1.8567
0.78 1 1 0.2 2.6760
0.16 1 1 0.2 0.8351
0.16 2 1 0.2 1.5901
0.16 3 1 0.2 2.3452
0.16 4 1 0.2 3.1002
0.16 1 3 0.2 1.4516
0.16 1 4 0.2 1.7745
0.16 1 5 0.2 2.2138
0.16 1 6 0.2 2.6645
0.16 1 1 0.3 0.5288
0.16 1 1 0.4 0.4449
0.16 1 1 0.5 0.4119
0.16 1 1 0.6 0.3965
Table 3: Skin friction coefficient
Sc S0 Re Gr Gm M Kc Kp Cf
0.16 1 1 2 2 1 0.2 0.5 5.3505
0.22 1 1 2 2 1 0.2 0.5 4.9752
0.60 1 1 2 2 1 0.2 0.5 3.5759
0.78 1 1 2 2 1 0.2 0.5 2.8403
0.16 2 1 2 2 1 0.2 0.5 6.5608
0.16 3 1 2 2 1 0.2 0.5 7.7711
0.16 4 1 2 2 1 0.2 0.5 8.9814
0.16 5 1 2 2 1 0.2 0.5 10.1916
International Journal of Pure and Applied Mathematics Special Issue
523
0.16 1 0.5 2 2 1 0.2 0.5 2.8327
0.16 1 0.6 2 2 1 0.2 0.5 3.0782
0.16 1 0.7 2 2 1 0.2 0.5 3.5637
0.16 1 0.8 2 2 1 0.2 0.5 4.1045
0.16 1 1 3 2 1 0.2 0.5 5.9934
0.16 1 1 4 2 1 0.2 0.5 6.6363
0.16 1 1 5 2 1 0.2 0.5 7.2792
0.16 1 1 6 2 1 0.2 0.5 7.9220
0.16 1 1 2 3 1 0.2 0.5 7.3829
0.16 1 1 2 4 1 0.2 0.5 9.4153
0.16 1 1 2 5 1 0.2 0.5 11.4477
0.16 1 1 2 6 1 0.2 0.5 13.4801
0.16 1 1 2 2 2 0.2 0.5 4.5206
0.16 1 1 2 2 3 0.2 0.5 3.6090
0.16 1 1 2 2 4 0.2 0.5 2.4587
0.16 1 1 2 2 5 0.2 0.5 1.8016
0.16 1 1 2 2 1 0.3 0.5 4.8098
0.16 1 1 2 2 1 0.4 0.5 4.5280
0.16 1 1 2 2 1 0.5 0.5 4.3562
0.16 1 1 2 2 1 0.6 0.5 4.2382
0.16 1 1 2 2 1 0.2 0.6 5.5390
0.16 1 1 2 2 1 0.2 0.7 5.6997
0.16 1 1 2 2 1 0.2 0.8 5.8384
0.16 1 1 2 2 1 0.2 0.9 5.9593
Table 1 shows the effects of important physical parameters on tangential Nusselt number. It is
indicated that Nusselt increases with increasing values of Prandtl number, Radiation parameter,
Reynolds number, dimensionless frequency oscillation and Nusselt number decreases with
increase the value of χ. Table 2 shows the effects of pertinent physical parameters on Sherwood
number. Sherwood number increases with the increasing values of Schmidt number, Soret
number, Reynolds number but reverse effect in the presence of chemical reaction. Table 3 shows
the effects of physical parameters on tangential skin friction coefficient. It is indicated that skin
friction coefficient decreases for increasing values of Schmidt number, Hartmann number,
permeability of the porous medium and reverse trend is noticed in the presence of Soret number,
Reynolds number, thermal Grashof number, and Solutal Grashof number.
5. Conclusions:
The problem of unsteady MHD free convective heat generation and absorption and mass transfer
flow of thermal radiation and chemical reaction fluid a past a vertical porous plate has been
International Journal of Pure and Applied Mathematics Special Issue
524
investigated .The present study may also be more useful in the study of the effects of magnetic
fields on blood circulation,cardiovascular events, crude oil transportation,etc. The conclusions of
the study are as follows:
Temperature decreases for increasing values of Reynolds number,Prandle number,Radiation
parameter whereas it decreases for increasing values of radiation absorption parameter but mixed
effect is noticed in the presence of heat generation/absorption parameter.Concentration decreases
for increasing values Schmidt number and Chemical reaction parameter but reverse effect is seen
in the presence of Soret number. Velocity decreases for increasing values of Hartmann
number,Reynolds number and Permeability of the porous medium but reverse effect is
noticed in the presence of thermal Grashof number, solutal Grashof number. Skin friction
coefficient increases for increasing values of Soret number,Raynolds number,Grashof
number,Solutal Grashof number,Permeability of porous medium ofcourse reverse trend is
noticed in the case of Schmidt number,Hartmann number,Rosseland mean absorption coefficient.
Nusselt number increases for increasing values of Prandtl number,Radiation parameter,Raynolds
number and opposite impact is noticed in the case of Dimensionless frequency of
oscillation,Dimensionless heat generation/absorption parameter. Similarly Sherwood number
increases for increasing values of Schmidt number,Soret number,Raynolds number and opposite
reaction is noticed in the case of Rosseland mean absorption coefficient.
Appendix
Where 2
4Re22
1
rrre PPPRm
2
)(4Re 222
3
iwPPPRm
rrre
2
4Re22
7
cccec SkSRSm
2
4Re 222
9
ccccec iwSSkSRSm
2
4Re2
13
NRm e
2
4Re2
15
wiNRm e
Pr
Re2 R
REFERENCES:
1. Cookey,C.I.,Ogulu, A.,Omubo-pepple,V.B,. “Influence of viscous dissipation and radiation
on unstudyMHD free- convection flow past an infinite vertical heated plate in an optically
thin environment with time-dependent suction”,Int.J.Heat Mass Transf.(46),2305-2311(2003)
2. Chamkha,A.J.,“Unsteady MHD convective heat and mass transfer past a semi-infinite
permeable moving plate with heat absorption”,Int.J.Eng.Sci.42(2),217-230(2004)
International Journal of Pure and Applied Mathematics Special Issue
525
3. Pal,D.,Mandal,H., “The influence of thermal radition on hydromagnetic Darcy- Forchheimer
mixed convection flow past a stretching sheet embedded in a Meccanicaporous medium”.
Meccanica 46(4),739-753(2011)
4. Turkyilmazoglu ,M.,Pop,I., “Soret and heat source effects on the unsteady radiative MHD
free convection flow from an impulsively started infinite vertical plate”,Int.J.Heat Mass
Transf.55(25),7635-7644(2012)
5. Ram,P.,Kumar,A.,Singh,H., “Effects of porosity on unsteady MHD flow past a semi-
infinite moving vertical plate with time dependent suction” , Indian J.Pure
Appl.Phys,51(7),461-470(2013)
6. Parsa,A.B,.Rashidi,M.M.,Hayat,T., “MHD boundary layer flow over a stretching surface
with internal heat generation or absorption” , Heat Transf.Asian Res.42(6),500-514(2013)
7. Ahmed, S. and Kalita, K., 2013, “Magnetohydrodynamic transient flow through a porous
medium bounded by a hot vertical plate in the presence of radiation: A theoretical analysis,”
J. Eng. Phy. andThermophysics, 86, pp. 30-39.
8. Ahmed, S., Batin, A. and Chamkha, A. J., 2014, “Finite difference approach in porous media
transport modeling for MHD unsteady flow over a vertical plate: Darcian Model,” Int. J.
Numerical Methods for Heat and Fluid Flow, 24(5), 1204- 1223 (2014).
9. Singh,H.,Ram,P.,Kumar,V., “Unsteady MHD free convection past an impulsively started
isothermal vertical plate with radiation and viscous dissipation”,FDMP10(4),521-550(2014)
10. RajuM.C.,Varma,S.V.K., “ Soret effects due to natural convection in a non- Newtonian fluid
flow in porous medium with heat and mass transfer”,Journal of Naval Architecture and
Marine Enginnering, Vol.11(2),147-156,doi.org/10.3329/jname.v6i1.2654(2014)
11. Ziya Uddin., Manojkumar,Harmand, S.,“Influence of thermal radiation and heat
generation/absorption on MHD heat transfer flow of a micropolar fluid past a wedge with
Hall and Ion slip currents”,18(2),S489-S502 (2014)
12. Su,X.,“Analytical computationof unsteady MHD mixed convective heat transfer over a
vertical stretching plate with partial slip conditions”, Indian J. Pure Appl. Phys, 53(10), 643-
651(2015)
13. Hazarika,N.J, Ahmed, S: “Mathematical Analysis for Optically Thin Radiating/ Chemically
Reacting Fluid in a Darcian Porous Regime”. Global J. Pure and Applied
Mathematics,13(6), 1777-1798 (2017).
14. Sahoo,P.K.,Dutta,N.,Biswal,S., “MHD unsteady free convection flow past an infinite vertical
plate with constant suction and heat sink”,Indian J.pure and applied Mathematics,34(1),145-
155,(2003)
15. Singh,K.D.,Kumar,R.,“Effects of chemical reactions on unsteady MHD free convection heat
and mass transfer for flow past a hot vertical porous plate with heat generation/absorption
through porous medium”. Indian J.Phys,84(1),93-106(2010)
16. Mishra,S.R.,Dash,G.C.,Acharya,M., “Mass and heat transfer effect on MHD flow of a visco-
elastic fluid through porous medium with oscillatory suction and heat source”, Int.J.Heat
Mass Transf.57(2),433-438(2013)
International Journal of Pure and Applied Mathematics Special Issue
526
17. Hussaini,S.A., Ramana Murthy,M.V., Rafiuddin, S., HarisinghNaik,“MHD Unsteady
Radiating Memory Convective Flow with Variable Suction”. Int.Jof Science, Engineering
and Technology Research. 3(11), 2278-7798 (2014)
18. Pal, D., Roy, N., Vajravelu, K., “Effects of thermal radiation and Ohmic dissipation on MHD
Casson nanofluid flow over a vertical non-linear stretching surface using scaling group
transformation”. Int. J.Mech. Sci. 114, 257–267 (2016)
19. Vajravelu, K.,Cannon,J.R.,Leto,J.,Semmoum,R.,Nathan,S.,Draper,M.,Hammock,D.,“Non-
linear convection at a porous flat plate with application to heat transfer from a
dike”,J.Math.Anal.Appl. 277(2),609-623 (2003)
20. Chand, K.,Kumar,R.,“Hall effect on heat and mass transfer in flow of oscillating visco-elastic
fluid through porous medium with wall slip conditions”,Indian J.Pure Appl.Phys,50(3),149-
155(2012)
21. Singh,K.D., Pathak,R.,“Effects of rotation and hall current on mixed convection MHD flow
a porous medium filled in a vertical channel in the presence of thermal radiation”, Indian
J.Pure Appl.Phys,50(2), 77-85(2012)
22. Srinivsasacharya,D,.Reddy,G.S,.“Mixed convection heat and mass transfer over a vertical
plate in a power-law fluid- saturated porous medium with radiation and chemical reaction
effects”,HeatTransf.Asian Res.42(6),485-499(2013)
23. Chaudhary,S., Kumar,P., “Magneto hydrodynamic stagnation point flow past a porous
stretching surface with heat generation”, Indian J. Pure Appl. Phys, 53(5), 291-297 (2015)
24. Pal, D., Mandal, G.:“Mixed convectionradiation on stagnation-point flow of nanofluids over
a stretching/shrinking sheet in a porous medium with heat generation and viscous
dissipation”. J. Pet. Sci. Eng.126, 16–25 (2015)
25. Kumar,R., Sood,S.,“Numerical analysis of stagnation point non-linear convection flow
through porous medium over a shrinking sheet” Int.JAppl.Comput.Math.,1-
15(2016),doi:101007/s40819-016-0150-2
26. Soundalgekar, V. M., 1972, “Viscous dissipative effects on unsteady free convective flow
past a vertical porous plate with constant suction,” Int. J. Heat and Mass Transfer, 15, pp.
1253-1261.
27. Prakash,J., Ogulu,A.,“Unsteady two-dimensional flow of a radiating and chemically
reacting MHDfluid with time dependent suction”, Indian J.Pure Appl.Phys,44(11),805-
810(2006)
28. Singh,K,D,,Kumar,R .,“Fluctuating Heat and Mass Transfer on Unsteady MHD Free
Convection Flow of Radiating and Reacting Fluid past a Vertical Porous Plate in Slip- Flow
Regime.” J. Applied Fluid Mechanics.4(4),101-106(2011)
29. Ibrahim,S.Y.,Mankinde,O.D., “Radiation effect on chemically reacting MHD
boundarylayer flow of heat and mass transfer past a porous vertical flat plate”,Int.J.Physical
sciences,6(6),1508-1516(2011)
30. Sivaraj,R.,Kumar.,“Unsteady MHD dusty viscoelastic fluid coquette flow in an irregular
channel with varying mass diffusion”, Int.J.Heat Mass Transf.55(11),3076-3089(2012)
International Journal of Pure and Applied Mathematics Special Issue
527
31. Route,B.R., Pattanayak,H.B.,“Chemical reaction and radiation effects on MHD flow past an
exponentially accelerated vertical porous plate in the presence of heat generation with
variable temperature embedded in a porous medium”,Analysis of faculty engineering
hunedoara-Int.J.of Engg,VOL.4 253-259,(2013)
32. Venkateshwarlu,B., Satyanarayana, P. V., “ Chemical reaction and radiation absorption
effects on the flow and heat transfer of a nanofluid in a rotating system”, Applied Nano
science , 4(3), 351-360, (2015)
33. Mishara ,S.S.S.,Paikary ,P.,Dash, N., Ray,G.S, Mishra ,A ,Mishra ,A .P :“Heat and Mass
Transfer in the MHD Flow of a Visco-Elstic Fluid in a Rotating Porous Channel with
Radiative Heat and Chemical Reaction”. . Int. J Scientific & Engineering Research. 5(1),
1255-1274 (2014)
34. Garg,B.P., Singh,K.D., Neeraj“Chemically reacting, radiating and rotating MHD Convective
Flow of Visco-Elastic Fluid through Porous Medium in Vertical Channel”. Int. J Latest
Trends in Eng. and Tech. 5(2),314-326(2015)
35. Makinde,O. D.,Khan,W.A.,Khan,Z.H.,“Stagnation point flow of MHD chemically reacting
nanofluid over a stretching convective surface with slip and radiative heat”,J.of
proc.mech.engineering,doi:10.1177/0954408916629506 (2016)
36. Srinivasacharya,D., Jagadeeshwar,P.,“Slip viscous flow over an exponentially
stretchingporous sheet with thermal convective boundary conditions”,
Int.JAppl.Comput.Math.,doi10.1007/s40819-017-0311-y
37. Loganathan,P., and Ganesan,P.,“Effects of radiation on the flow past an impulsively started
infinite vertical plate” with,J.Eng.Phys.Thermophys,79(1),65-72(2006)
38. Nadeem,S.,Zaheer,S.,Fang,T., “Effects of thermal radiation the boundary layer flow of a
Jeffery fluid over an exponentially stretching surface”,Numer.Algorithm,57(2),187-
205(2011)
39. Ram, P., Singh, H., Rakesh kumar, Vikaskumar, Vimalkumar Joshi., “Free convective
boundary layer flow of radiating and reacting MHD fluid past a continusoidally Fluctuating
heated plate”, Int.JAppl.Comput.Math., doi10.1007/s40819-017-0355-z (2017)
International Journal of Pure and Applied Mathematics Special Issue
528
529
530