Embed Size (px)
Citation preview
MHD HEAT AND MASS TRANSFER FLOW OF A
REACTIVE, RADIATIVE AND ABSORBING FLUID
THROUGH A VERTICAL POROUS PLATE
G Nagesh1* Prof R Sivaprasad2
1Research Scholar, Department of Mathematics, S K University, Ananthapuramu-515001, A.P., India
3Professor, Department of Mathematics, Sri Krishnadevaraya University, Ananthapuramu-515003,
A.P., India
Email: [email protected] & [email protected]
Abstract: In this paper an attempt is made to the unsteady, two-dimensional, laminar, boundary-layer flow
of a viscous, incompressible, electrically conducting and heat-absorbing fluid along a semi-infinite vertical permeable moving plate in the presence of a uniform transverse magnetic field, thermal radiation, radiation
absorbing and first order chemical reaction. The plate is assumed to move with a constant velocity in the direction of fluidflow while the free stream velocity is assumed to follow the exponentially increasing small perturbation law. The governing equations of this problem are solved using perturbation technique and the solutions for velocity, temperature, concentration, skin friction coefficient, Nusselt number and Sherwood number are obtained. The effects of various thermo physical parameters such as Schmidt number, magnetic parameter, chemical reaction parameter, Grashof number, thermal Grashof number, Prandtl number, heat absorption and radiation absorption parameter over the velocity, temperature, concentration, skin friction coefficient, Nusselt number and Sherwood number are discussed through graphs and tables.
Keywords: Radiation absorption, Chemical reaction, Heat source, Magneto hydrodynamic (MHD),
porous medium, skin-friction.
1. INTRODUCTION
The study of magneto hydrodynamics (MHD) plays an important role in
agriculture, engineering and petroleum industries. MHD has won practical applications, for instance, it may be used to deal with problems such as cooling of nuclear reactors by
liquid sodium and induction flow water which depends on the potential difference in the
fluid direction perpendicular to the motion and goes to the magnetic field and also study of MHD of viscous conducting fluids is playing a significant role, owing to its practical
interest and abundant applications, in astro-physical and geo-physical phenomenon.
Astro-Physicists and geo-physicists realized the importance of MHD in stellar and planetary processes. The main impetus to the engineering approach to the electromagnetic
fluid interaction studies has come from the concept of the magneto hydro dynamics, direct
conversion generator, ion propulsion study of flow problems of electrically conducting fluid, particularly of ionized gases is currently receiving considerable interest. Such
studies have made for years in convection with astro-physical and geo-physical problems
such as Sun spot theory, motion of the interstellar gas etc. Recently, some engineering problems need the studies of the flow of an electrically conducting fluid, in ionized gas
are called plasma. Many names have been used in referring to the study of plasma
phenomena.
MHD double diffusive and chemically reactive flow through porous medium
bounded by two vertical plates was studied by Ravi Kumar et al. [1]. MHD free
convective flow through a porous medium past a vertical plate with ramped wall temperature was studied by Sinha et al.[2]. Effect of heat transfer on MHD blood flow
through an inclined stenosed porous artery with variable viscosity and heat source was discussed by Tripatihi et al. [3]. Steady MHD Mixed Convective flow in presence of
inclined magnetic field and thermal radiation with effects of chemical reaction and Soret
embedded in a porous medium was studied by Sharmilaa et al. [4]. Rama Krishna reddy [5] studied MHD free convective flow past a porous plate. Joule heating and thermal
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 295
diffusion effect on MHD fluid flow past a vertical porous plate embedded in a porous
medium was studied by Obulesu et al, [6]. Free convection arises in the fluid when
temperature changes cause density variation leading to buoyancy force sacting on the fluid elements. The study of heat and mass transfer to chemical reacting MHD free
convection flow with radiation effects on a vertical plate has received a growing interest
during the last decades. Accurate knowledge of the overall convection heat transfer has vital importance in several fields such as thermal insulation, drying of porous solid
materials, heat exchanges, stream pipes, water heaters, refrigerators, electrical conductors
and industrial, geophysical and astrophysical applications, such as polymer reduction, manufacturing of ceramic, packed-bed catalytic reactors, food processing, cooling of
nuclear reactors, enhanced oil recovery, under ground energy transport, magnetized
plasma flow, high speed plasma wind, cosmic jets and stellar system. For some industrial applications such as glass production, furnace design, propulsion systems, plasma physics
and space craftre-entry aerothermodynamics which operate at higher temperatures and
radiation effect can also be significant.
Cheng and Minkowycz [7] have presented similarity solutions for free thermal convection
from a vertical plate in a fluid-saturated porous medium. The problem of combined
thermal convection from a semi-infinite vertical plate in the presence or absence of a porous medium has been studied by many authors. Nakayama and Koyama [8] have
studied pure, combined and forced convection in Darcian and non-Darcian porous media.
Lai and Kulacki [9] have investigated coupled heat and mass transfer by mixed convection from an isothermal vertical plate in a porous medium. Hsieh et al [10]has
presented non-similar solutions for combined convection in porous media. Chamkha [11]
has investigated hydro magnetic natural convection from aiso-thermal inclined surface adjacent to a thermally stratified porous medium. Hall current effects on MHD convective
flow past a porous plate with thermal radiation, chemical reaction and heat generation
/absorption was studied by Obulesu et al. [12]. Mohammed Ibrahim et al.[13]have investigated Heat source and chemical effects on
MHD convection flow embedded in a porous medium with Soret, viscous and Joules
dissipation. Chamkha et al [14] has presented MHD flow of uniformly stretched vertical permeable surface in the presence of heat generation / absorption and chemical reaction.
Chandra Reddy et al. [15] have presented MHD Natural Convective Heat
Generation/Absorbing and Radiating Fluid Past a Vertical Plate Embedded in Porous Medium – an Exact Solution. Chemical reaction on unsteady MHD convective heat and
mass transfer past a semi-infinite vertical permeable moving plate with heat absorption
was studied by Mythreye et al, [16]. Radiation absorption effect on MHD dissipative fluid past a vertical porous plate embedded in porous media was studied by Obulesu et al. [17].
Obulesu, Dastagiri and Siva prasad [18] have presented Radiation absorption and
chemical reaction effects on MHD radiative heat source/sink fluid past a vertical porous plate. Radiation absorption effect on MHD, free convection, chemically reacting
viscoelasticfluid past an oscillatory vertical porous plate in slip flow regime was studied
by Raju et al, [19]. Effect of Inclined Magnetic Field and Radiation Absorption on Mixed Convection Flow of a Chemically Reacting and Radiating Fluid Past a Semi Infinite
Porous Plate was studied by Obulesu et al. [20]. K Raghunath et al. [21] have studied
Heat and mass transfer on Unsteady MHD flow of a second grade fluid through porous medium between two vertical plates. Raghunath K et al. [22] Discussed Hall Effects on
MHD Convective Rotating Flow of Through a Porous Medium past Infinite Vertical
Plate. Raghunath k, et al. [23] have discussed Heat and mass transfer on MHD flow of Non-Newtonian fluid over an infinite vertical porous plate.
2. MATHEMATICAL FORMULATION
Consider unsteady two-dimensional flow of a laminar, incompressible, viscous,
electrically conducting and heat absorbing fluid past a semi-infinite vertical permeable
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 296
moving plate embedded in a uniform porous medium and subjected to a uniform
transverse magnetic field in the presence of thermal and concentration buoyancy effects.
The governing equations for this investigation are based on the balances of mass, linear momentum, energy and concentration species. Taking into consideration the assumptions
made above, these equations can be written in Cartesian frame of reference, as follows
0*
*
y
v (1)
p
CTK
uu
BCCgBTTg
y
u
x
p
y
uV
t
u **)*()*(
*
*
*
*1
*
**
*
*2
0**
2
2
(2)
)*()*(*
*1
*
*
*
**
*
* *1*1
2
2
CC
C
RTT
C
Q
y
q
Cy
T
y
TV
t
T
pp
r
p
(3)
)*(*
*
*
**
*
* *
2
2
CCK
y
CD
y
CV
t
CC
(4)
Where x*, y*, and t* are the dimensional distances along and perpendicular to the plate and dimensional time, respectively. U* and V* are the components of dimensional
velocities along x* and y* directions, respectively, is the fluid density, is the
kinematic viscosity, CP is the specific heat at constant pressure, is the fluid electrical
Conductivity, B0 is the magnetic induction, K* is the permeability of the porous medium, T is the dimensional temperature,Q0 is the dimensional heat absorption coefficient, C is
the dimensional concentration, is the fluid thermal diffusivity, D is the mass
diffusivity, g is the gravitational acceleration, and βT and βC are the thermal and
concentration expansion coefficients, respectively. The magnetic and viscous dissipations
are neglected in this study. Under the above assumptions, the appropriate boundary conditions for the distributions of velocity, temperature and concentration are given by
0at)(*,)(*,*****
yeCCCCeTTTTuu tn
ww
tn
wwp
yCCTTeUuu tn as**),1(***
0 (5)
Where up*, Cw and Tw are the wall dimensional velocity, concentration and temperature,
respectively. Up*, C∞ and T∞ are the free stream dimensional velocity, concentration and temperature, respectively. U0and n* are constants.It is known from Eq.(1) that the
suction velocity at the plate surface is a function of time only and it is assumed in the
following form:
V* = - V0(1+ ɛAent) (6)
WhereA is a real positive constant, and A are small less than unity and V0 is a scale
of suction velocity which has non-zero positive constant. Outside the boundary layer, Eq. (2) gives
*
2
0**
**
*1
U
BU
Kdt
dU
x
p
p
(7)
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 297
On introducing the following non-dimensional quantities,
2
0
0
2
0
2
2
0
2
00
***
2
00
**
2
0
2
0
**
*
**
*
0
*
0
*
0
00
*,
*,
*,,
)(,
)(,
,,Pr,*
,*
,,,*
,*
,*
v
vnn
Lvh
vtt
KvK
vU
CCgGc
vU
TTgGr
v
BM
DSc
K
C
CC
CCC
TT
TT
U
uU
U
UU
yvy
vU
uu
pwcwT
p
ww
p
p
)(
)(,,
4,
**2
0
**2
1
0
1
2
0
2
1
2
0
TTKv
CCRR
vC
Kv
IF
v
KKr
w
w
p
C
(8)
The governing equations (2) to (4) can be rewritten in the non-dimensional form as
follows
)()1(2
2
uUNCGcGry
u
dt
dU
y
uAe
t
u nt
(9)
RCFyy
Aet
nt
12
2
)1Pr(Pr (10)
ScKrCy
C
y
CAeSc
t
CSc nt
2
2
)1( (11)
QFFKMNwhere 1,/1
The corresponding boundary conditions are given by
yasCeUu
yateCeUu
nt
ntnt
p
0,0,1
0,1,1,
(12)
3. SOLUTION OF THE PROBLEM
The equations (9) to (11) are coupled, non-linear partial differential equations and these cannot be solved in closed form. However, these equations can be reduced to a set of
ordinary differential equations, which can be solved analytically. So this can be done,
when the amplitude of oscillations (ε <<1) is very small, we can assume the solutions of
flow velocity u, temperature field θ and concentration in the neighborhood of the plate
as:
)()(),(
)()(),(
)()(),(
10
10
10
yCeyCtyC
yeyty
yueyutyu
nt
nt
nt
(13)
Substituting equations (13) into equation (9)–(11) and equating the coefficients at the
terms with the same powers of ε, and neglecting the terms of higher order, the following
equations are obtained.
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 298
Zero order terms:
00000 Gm - Gr - -N=uN - Cuu (14)
00100 R- =F-Pr C (15)
0=ScK 000 rCCScC (16)
First order terms:
110111 Gc - Gr -A - n)-(N n)u(N- Cuuu (17)
RC-PrA - = )Pr(Pr 101111 Fn (18)
0111 ScA)( CCnKrScCScC (19)
The corresponding boundary conditions are
yasCCuu
yatCCuUu p
0,0,0,0,1,1
01,1,1,1,0,
101010
101010
(20)
Solving equations (14) – (19) under the boundary conditions (20), the following solutions
are obtained
)exp( 10 ymC (21)
)exp()exp( 22110 ymbymb (22)
)exp()exp()exp(1 3524130 ymbymbymbu (23)
)exp()exp( 47161 ymbymbC (24)
)exp(
)exp()exp()exp(
511
41029181
ymb
ymbymbymb
(25)
)exp()exp()exp(
)exp()exp()exp(1
617516415
3142131121
ymbymbymb
ymbymbymbu
(26)
Substituting equations (21)–(26) in equation (13) we obtain the velocity temperature and concentration field
))exp()exp()exp(
)exp()exp()exp(1(
))exp()exp()exp(1(
617516415
314213112
352413
ymbymbymb
ymbymbymbe
ymbymbymbu
nt
(27)
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 299
))exp(
)exp()exp()exp((
)exp()exp(
511
4102918
2211
ymb
ymbymbymbe
ymbymb
nt
(28)
))exp()exp(()exp( 47161 ymbymbeymC nt (29)
Skin Friction:
The non-dimensional skin friction at the surface is given by
0
yy
u
ntebmbmbmbmbmbm
bmbmbm
)(
)(
176165154143132121
534231
(30)
NusseltNumber :
The rate of heat transfer in terms of the Nusselt number is given by
ntebmbmbmbmbmbmNu )()( 11510492812211 (31)
Sherwood Number :
The rate of mass transfer on the wall in terms of Sherwood number is given by
0
yy
CSh
ntebmbmmSh )( 74611 (32)
4. RESULTS AND DISCUSSION
In order to get a physical insight into the problem numerical calculations are carried out
for the velocityu, the temperatureT and concentration C,in terms of the parameters Magnetic field parameter (M),Permeability parameter (K), Grash of number (Gr),
modified Grash of number (Gc), Schmidt number (Sc), Prandtl number (Pr), Heat
absorption parameter (Q), Chemical reaction parameter (Kr), Radiation parameter (F) and Radiation absorbtion parameter respectively. Throughout the computations we employ the
Prandtl number Pr= 0.71, Grashofnumber G r= 3, modified Grashof number Gc= 1,
Schmidt number SC=0.22, Magnetic field parameter M=2, Radiation parameter F=0.5, Heat absorption parameter Q=1, Chemical reaction parameter Kr=1, Permeability
Parameter K= 1, Radiation absorbtion parameter R=2, the frequency of oscillations n
=0.1, scale of free stream velocity Up = 0.5, t=1,ε = 0.2, and A=0.5.
0
yy
Nu
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 300
4.1. Velocity Profiles
Figures 1 to 4 display the effects of a magnetic field parameter (M), permeability
parameter (K), Grashofnumber (Gr) and modified Grashofnumber (Gc) on velocity distributions respectively. From Figure 1, it is observed that an increase of magnetic field
parameter leads to decrease in velocity fields. It is because that the application of
transverse magnetic field will result a resistive type force (Lorentz force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity. In Figure 2, we
represent the velocity profiles for different values of permeability parameter (K). The
flow field sufferan increase in the velocity at all points in the presence of permeability parameter(K). In Figures 3 and 4, velocity profiles are displayed with the variation in
Grashof number(Gr) and modified Grashof number (Gc). From these figures are noticed
the velocity gets increase by the increase of Grashof number(G) and modified Grashof number (Gc).
Figure 1: Effect of Magnetic Parameter on Velocity
Figure 2: Effect of Permeability Parameter on Velocity
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 301
Figure 3: Effect of Grashof Number on Velocity
Figure 4: Effect of Modified Grashof Number on Velocity
4.2. Temperature Profiles
Figures 5 to 8 show the effects of material parameters such as Pr, F, R and Q on
temperature distribution. The effect of Prandtl number is very important in temperature
profiles. There is a decrease in temperatures due to increasing values of the Prandtl number (Pr) as shown in Figure 5. From Figure 6, it is clear that temperature decreases
with the increase in radiation parameter (F). In Figure7, the effect of heat absorption
parameter (Q) is shown on temperature profile. From this figure it is observed that temperature decreases with an increase in Q. In Figure 8, the effect of radiation
absorption parameter (R) is shown on temperature profile. From this figure it is observed
that temperature decreases with an increase in R.
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 302
Figure 5: Effect of Prandtl Number on Temperature
Figure 6: Effect of Radiation Parameter on Temperature
Figure 7: Effect of Heat absorption Parameter on Temperature
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 303
Figure 8: Effect of Radiation absorption Parameter on Temperature
4.3. ConcentrationProfiles
Figures (25) and (26) show the effect of Schmidt number (Sc) and chemical reaction parameter (Kr) on concentration profile. From Fig25-26, it is clear that concentration
decreases with the increase in Schmidt number and chemical reaction parameter.
Figure 9: Effect of Schmidt Number on Concentration
Figure 10: Effect of Chemical reaction Parameter on Concentration
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 304
Table – 1, shows numerical values of skin-friction for various of Grashof number (Gr),
modified Grashof number (Gc), Magnetic parameter (M), Porosity parameter (K). From table 1, we observe that the skin-friction increases with an increase in Grashof number
(Gr), modified Grashof number (Gc) and Porosity parameter (K)where as it decreases
under the influence of magnetic parameter.
Skin Friction (τ)
Gr Gc M K τ
3 4.2394
5 5.5383
8 7.4867
10 8.7856
2 4.8288
4 6.0078
6 7.1868
8 8.3658
2 4.2394
2.4 4.1858
2.8 4.1467
3 4.1315
0.5 4.1315
1 4.2394
1.5 4.986
2 4.3346
Table – 2 demonstrates the numerical values of Nusselt number (Nu) for different values
of Prandtl number (Pr), Radiation parameter (F), Heat absorption parameter (Q) and Radiation absorption parameter (R). From table 2, we notice that the Nusselt number
increases with an increase in Prandtl number, Radiation parameter and Heat source
parameter where as it decreases under the influence of Radiation absorption parameter.
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 305
Nusselt Number (Nu)
Pr F R Q Nu
0.11 0.3278
0.31 0.4098
0.51 0.5032
0.71 0.6080
1 0.9682
2 1.5272
3 1.9664
4 2.3359
1 1.3271
1.5 0.9675
2 0.6080
2.5 0.2484
0.5 0.1451
0.8 0.4389
1 0.6080
1.2 0.7613
Table – 3 shows numerical values of Sherwood number (Sh) for the distinction values of
Schmidt number (Sc) and Chemical reaction parameter (Kr). It can be noticed from Table - 3 that the Sherwood number enhances with rising values of Schmidt number and the
Chemical reaction parameter.
Sherwood Number (Sh)
Sc Kr Sh
0.2 0.9200
0.5 1.6031
0.9 2.3491
1 0.7421
1.2 0.7949
1.4 0.8436
2 0.9727
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 306
5. CONCLUSIONS
In this problem, is studied the MHD heat and mass transfer flow of a reactive, radiative
and absorbing fluid through a vertical porous plate. In the analysis of the flow the
following conclusions are made:
1. Velocity increases with an increase in Grashof number and as well as modified
Grashof number and Porosityparameter of the porous medium while, it decreases
in the existence of magnetic parameter. 2. Temperature decreases with an increase in Prandtl number, radiation parameter,
heat absorption parameter and increases with an increase in radiation absorption
parameter(R). 3. Concentration decreases with an increase in Schmidt number, chemical reaction
parameter.
4. As significant increase in seen in skin friction for Grashof number, modified Grashof number and porosityparameter while a decrease is seen in the presence of
magnetic parameter.
5. The rate of heat transfer increases with an increase Prandtl number, heat absorption parameter, radiation parameterand decreases with a radiation
absorption parameter.
6. The rate of mass transfer increases with an increase Schmidt number and Chemical reaction parameter.
APPENDIX
KMN
1 QFF 1
2
42
1
KrScScScm
2
4PrPr 1
2
2
Fm
2
4113
Nm
2
)(42
4
nKrScScScm
2
Pr)(4PrPr 1
2
5
nFm
2
)(4116
nNm
11
2
1
1Pr Fmm
Rb
12 1 bb
Nmm
GcGrbb
1
2
1
13
Nmm
Grbb
2
2
2
24 )1( 435 bbUb p
)(1
2
1
1
6nKrScScmm
AmScb
67 1 bb
)Pr(Pr
Pr
11
2
1
611
8Fnmm
RbbmAb
)Pr(Pr
Pr
12
2
2
229
Fnmm
bmAb
)Pr(Pr 14
2
4
710
Fnmm
Rbb
)(1 109811 bbbb
)(1
2
1
6831
12nNmm
bGcbGrbAmb
)(2
2
2
942
13nNmm
bGrbAmb
)(3
2
3
5314
nNmm
bAmb
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 307
)(4
2
4
710
15nNmm
bGcbGrb
)(5
2
5
1116
nNmm
bGrb
)1( 161514131217 bbbbbb
REFERENCES
[1]. Ravikumar V, Raju M.C., Raju G.S.S., Chamkha A.J. : MHD double diffusive and
chemically reactive flow through porous medium bounded by two vertical plates.
Int. J. Energy Technology, 2013,Vol.5(4),pp.1-8.
[2]. Sinha A., Ahmed N., Agarwalla S. : MHD Free convective flow through a porous
medium past a vertical plate with ramped wall temperature. Applied
Mathematical Sciences, Vol.11, 2017, No.10, pp.963-974.
[3]. Tripatihi B, Sharma B K: Effect of heat transfer on MHD blood flow through an
inclined stenosed porous artery with variable viscosity and heat source,
Romanian J.Biophys 2018 Vol.28(3),pp.89-102.
[4]. Sharmilaa K, Kannathal K: Steady MHD Mixed Convective flow in presence of
inclined magnetic filed and thermal radiation with effects of chemical reaction and soret embedded in a porous medium. IJSRSET, 2018 Vol 4(1)
[5]. Rama Krishna reddy P, Raju M.C,: MHD Free Convective Flow Past A Porous
Plate. International journal of pure and applied mathematics.2018, Vol 118 (5).pp.507-529.
[6]. Obulesu M, Siva Prasad R; Joule heating and thermal diffusion effect on MHD fluid flow past a vertical porous plate embedded in a porous medium, Bulletin of
Pure and Applied Sciences. Vol. 38E (Math & Stat.), No.1, 2019. P.117-134
Print version ISSN 0970 6577 Online version ISSN 2320 3226 DOI: 10.5958/2320-3226.2019.00011.0
[7]. P. Cheng, W.J. Minkowycz, J. Geophys. 82 (1977) 2040.
[8]. A. Nakayama, H. Koyama, J. Heat Transfer 109 (1987) 1041.
[9]. J.C. Hsieh, T.S. Chen, B.F. Armaly, Int. J. Heat Mass Transfer 36 (1985) 1485.
[10]. F.C. Lai, F.A. Kulacki, Int. J. Heat Mass Transfer 113 (1991) 252.
[11]. A.J. Chamkha, Int. J. Eng. Sci. 35 (1997) 975.
[12]. Obulesu M, Siva Prasad R; Hall Current Effects on MHD Convective Flow Past A Porous Plate with Thermal Radiation, Chemical Reaction and Heat
Generation /Absorption , To Physics Journal Vol 2 (2019) ISSN: 2581-7396.
[13]. Mohammed Ibrahim S, and Suneetha K; "Heat source and chemical effects on MHD convection flow embedded in a porous medium with Soret, viscous and
Joules dissipation," Aim Shams Engg Journal, Vol. 7, PP. 811-818 (2016).
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 308
[14]. Chamkha A.J.: MHD flow of uniformly stretched vertical permeable surface in
the presence of heat generation / absorption and chemical reaction, Int.
Commun.. Heat Mass transfer, 2003, Vol.30(3), pp.413-422.
[15]. Chandra Reddy P, Raju M.C,Raju G.S.S: MHD Natural Convective Heat
Generation/Absorbing and Radiating Fluid Past a Vertical Plate Embedded in
Porous Medium – an Exact Solution , Journal of the Serbian Society for Computational Mechanics / Vol. 12 / No. 2, 2018.
[16]. Mythreye A , Pramod J.P , Balamurugan K.S; Chemical reaction on unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable
moving plate with heat absorption. International Conference on Computational
Heat and Mass Transfer-2015, Procedia Engineering 127 ( 2015 ) 613 – 620.
[17]. Obulesu M, Siva Prasad R; Radiation Absorption Effect on MHD Dissipative
Fluid Past A Vertical Porous Plate Embedded in Porous Media, Bulletin of Pure
and Applied Sciences. Vol. 37E (Math & Stat.), No.1, 2018.P.184-199 Print version ISSN 0970 6577 Online version ISSN 2320 3226 DOI: 10.5958/2320-
3226.2018.00019.X.
[18]. Obulesu M, Dastagiri Y, Siva Prasad R;Radiation absorption and chemical
reaction effects on mhdradiative heat source/sink fluid past a vertical porous
plate,Journal of Engineering Research and Application ,ISSN : 2248-9622 Vol. 9,Issue 5 (Series -II) May 2019, pp 77-87.
[19]. Raju M.C., Vidyasagar B., Varma S.V.K., Venakataramana S.: Radiation
absorption effect on MHD, free convection, chemically reacting visco-ealsticfluid past an oscillatory vertical porous plate in slip flow regime, Int. J.
Engg. and Compt. Sci., 2014, Vol. 3(9), pp.8212-8221.
[20]. Obulesu M, Siva Prasad R;,Effect of Inclined Magnetic Field and Radiation Absorption on Mixed Convection Flow of a Chemically Reacting and Radiating
Fluid Past a Semi Infinite Porous Plate, .i-Manager's Journal on Mathematics;
Nagercoil Vol. 7, Iss. 4, (Oct/Dec 2018): 39-49. DOI:10.26634/jmat.7.4.15561
[21]. K Raghunath, R Sivaprasad, GSS Raju, Heat and mass transfer on Unsteady
MHD flow of a second grade fluid through porous medium between two vertical
plates, JUSPS-B Vol. 30(2), 1-11 (2018).
[22]. Raghunath K, R Sivaprasad, GSS Raju, Hall Effects on MHD Convective
Rotating Flow of Through a Porous Medium past Infinite Vertical Plate Annals of Pure and Applied Mathematics Vol. 16, No. 2, 2018, 353-263 ISSN: 2279-
087X (P), 2279-0888(online), DOI: http://dx.doi.org/10.22457/apam.v16n2a1.
[23]. Raghunath K, R Sivaprasad, GSS Raju, Heat and mass transfer on MHD flow of Non-Newtonian fluid over an infinite vertical porous plate, International Journal
of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13
(2018) pp. 11156-11163
Compliance Engineering Journal
Volume 11, Issue 1, 2020
ISSN NO: 0898-3577
Page No: 309
