Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 7 (2017), pp. 3811-3832
© Research India Publications
http://www.ripublication.com
MHD Flow of a Newtonian Fluid Through a Porous
Medium in Planer Channel
1P. Gurivi Reddy T.S.Reddy2 S.V.K.Varma3
1Reader in mathematics, S. B. V. R. Degree College, Badvel, A.P., India .
2Department of Mathematics, Global College of Engineering, Kadapa -516101, A.P India
3Department of Mathematics, S.V.University, Tirupati -517502, A.P, India.
Abstract
This manuscript is an investigation on the effects of slip condition, chemical
reaction, radiation and unsteady MHD periodic flow of a viscous,
incompressible, electrically conducting fluid through a porous medium in in
two cases viz. Case–I: Uniform plate Temperature and Uniform Concentration
and Case–II: Constant heat and mass flux.. The governing equations have
been solved by perturbation technique. The skin friction and rate of heat
transfer and mass transfer are also derived. The effects of various physical
parameters like magnetic parameter, Reynolds number, Grashof number,
modified Grashof number, permeability parameter, chemical reaction
parameter, and Schmidt number are analyzed though graphs. Shear stress
increases with the increase in magnetic parameter or permeability parameter
and it shows a reverse effect in the case of slip parameter or Schmidt number
in both cases of the study.
Keywords: MHD, Chemical reaction, Radiation, Periodic flow, Planer
channel and Slip flow regime.
1. INTRODUCTION:
The phenomenon of slip-flow regime has attracted the attention of a large number of
scholars due to its wide ranging applications. The problem of the slip flow regime is
very important in this era of modern science, technology and vast ranging
3812 P. Gurivi Reddy, T.S.Reddy and S.V.K.Varma
industrialization. In many practical applications, the particle adjacent to a solid
surface no longer takes the velocity of the surface. The particle at the surface has a
finite tangential velocity, it slips along the surface. The flow regime is called the slip
flow regime and its effect cannot be neglected. The fluid slippage phenomenon at the
solid boundaries appear in many applications such as micro channels or Nano
channels and in applications where a thin film of light oils is attached to the moving
plates or when the surface is coated with special coating such as thick monolayer of
hydrophobic mechanical device where a thin film of lubricant is attached to the
surface slipping over one another or when the surfaces are coated with special coating
to minimize the friction between them. The effect of the fluid slip page at the wall for
Coutte flow are considered by Marques et al. [2] under steady state conditions and
only for gases. The closed form solutions for steady periodic and transient velocity
field under slip condition have been studied by Khaled and Vafai [3]. The effect of
slip condition on MHD steady flow in a channel with permeable boundaries has been
discussed by Makinde and Osalusi [4]. Anil Kumar et.al.[5] discussed the perturbation
technique to unsteady MHD periodic flow of viscous fluid through a planer channel.
The effect of slip condition on unsteady MHD oscillatory flow of a viscous fluid in a
planer channel is investigated by Mehmood and Ali [6].
Mageto-hydrodynamics is attracting the attention of the many authors due to its
applications in geophysics and astrophysics, it is applied to study the stellar and solar
structures, interstellar matter, radio propagation through the ionosphere etc. In
engineering in MHD pumps, MHD bearings etc. Since some fluids can also emit and
absorb thermal radiation, it is of interest to study the effect of magnetic field on the
temperature distribution and heat transfer when the fluid is not only an electrical
conductor but also when it is capable of emitting and absorbing thermal radiation.
This is of interest because heat transfer by thermal radiation is becoming of greater
importance when we are concerned with space applications and higher operating
temperatures. Soundalgekar and Takhar [7] first, studied the effect of radiation on the
natural convection flow of a gas past a semi-infinite plate using the Cogly-Vincentine-
Gilles equilibrium model. For the same gas Takhar et al. [8] investigated the effects of
radiation on the MHD free convection flow past a semi-infinite vertical plate. Later,
Hossain et al. [9] studied the effect of radiation on free convection from a porous
vertical plate. Muthucumarswamy and Kumar [10] studied the thermal radiation
effects on moving infinite vertical plate in presence of variable temperature and Mass
diffusion. An analytical solution for unsteady free convection in porous media has
been studied by Magyari et al. [11]. Chamkha et al. [12] studied the effects of Hydro
magnetic combined heat and mass transfer by natural convection from a permeable
surface embedded in a fluid saturated porous medium. Mzumdar and Deka [13]
studied MHD flow past an impulsively started infinite vertical plate in presence of
thermal radiation. The growing need for chemical reactions in chemical and
hydrometallurgical industries require the study of heat and mass transfer with
chemical reaction. The effect of chemical reaction on heat and mass transfer in a
laminar boundary layer flow has been studied under different conditions by several
authors [14-47].
MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel 3813
In this paper the influence of slip on MHD periodic flow of viscous fluid in presence
of radiation and chemical reaction in a planer channel is discussed in two cases viz.,
Case (I) with uniform temperature and concentration, Case (II) with heat and mass
flux have been studied. The coupled non-linear partial differential equations are
solved using a perturbation scheme. And a series of solutions are expressed in terms
and hyperbolic sine and cosine functions for velocity, temperature and concentration
fields for both cases of this study.
2. NOMENCLATURE :
u* : Velocity along x-axis k : Permeability of porous medium
C* : Concentration Nu : Nusselt number
C0 : Fluid concentration at the boundary y = 0 Pe : Peclet number
Cw : Fluid concentration at the boundary y = a T*
: Fluid temperature
T0 : Fluid temperature at the boundary y = 0 Sc : Schmidt number
Tw : Fluid temperature at the boundary y = a Da : Darcy number
P* : Fluid pressure t* : Time
: Kinematic viscosity coefficient B0 : Electromagnetic induction
H0 : Intensity of magnetic field Cp : Specific heat at constant pressure
: Thermal conductivity q* : The radiative heat flux
D : Mass diffusivity E : Eckert number
Gm : Modified Grashof number Gr : Thermal Grashof number
g : Gravitation due to acceleration M : Magnetic parameter
Kc : Non- dimensional rate of chemical reaction *
cK : Rate of chemical reaction
Re :Reynolds number n : Frequency of the periodic flow
u : Non- dimensional velocity F : Radiation parameter
qr : Radiative flux
k : Non- dimensional permeability coefficient of a porous medium
GREEK SYMBOLS
: Thermal conductivity ρ : Fluid density
ν : Kinematic viscosity θ : Dimensionless temperature
σ : Electrical conductivity μ : Dynamic viscosity
T : Coefficient of volume expansion e : Conductvity of the fluid
: The mean radiation absorption coefficient τ : Non-dimensional shear stress
C : Coefficient of volume expansion with concentration
3814 P. Gurivi Reddy, T.S.Reddy and S.V.K.Varma
3. FORMULATION OF THE PROBLEM:
Here we consider an unsteady MHD flow of viscous, incompressible, electrically non-
conducting fluid in a planer channel in the presence of transverse applied magnetic
field. The *x -axis is taken along the plate in vertical upward direction against to the
gravitational field and the *y -axis is taken normal to it. Initially, one plate is heated at
constant temperature Tw and another one is heated to the temperature 0T . A transverse
magnetic field of uniform strength 0B is applied normal to the direction of the flow.
The fluid has small electrical conductivity and the electromagnetic force produced is
very small, so the induced magnetic field is neglected and there is no applied voltage
which implies the absence of an electrical field. Viscous and Joules dissipations are
neglected in energy equation. The fluid considered here is gray, absorbing/emitting
radiation but a non-scattering medium. There is a first order chemical reaction
between the diffusing species and the fluid. Then under usual Boussinesq’s
approximation the fully developed flow is governed by the following set of equations
as described by Reddy et al. [25, 26]:
0
2
2* * 2 ** * * * * *
0 0* * **
1( ) ( )T C
Bu P u u u g T T g C Ct x ky
(1)
2
* 2 * *
* **
1
P P
T T qt C C yy
(2)
2
* 2 ** * *
0* *( )C
C CD K C Ct y
(3)
The relevant boundary conditions are given as follows
Case I: Uniform plate Temperature and Concentration:
** * * * * *
0*
UU h T T C C at yW Wy
* * * * * *
0 0 0 U T T C C at y a (4)
Case II: Constant heat and mass flux:
** * ** *
, 0* * *
qU T q C WU h at yky y y
* * * * * *
0 0 0 U T T C C at y a
(5)
MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel 3815
Consider the fluid which is opptically thin with a relatively low denisity and radiative
heat flux is given by
*2 * *
0*4 ( )
q T Ty
(6)
Where is the the mean radiation absorption coefficient
On introducing the following non dimensional quantities,
Case I:
* ** * * *
0
* *
0
, , , , ,W
T Tx y u t vx y u ta a v a T T
* * *
0
* *
0
, ,W
C C aPC PvC C
* * 2 * * 22 2 *2 0 0
2
( ) ( )4, , ,T W C W
r m ag T T a g C C aa KF G G D
K v v a
* 2 2
0, , , ,C PC C e e
K a Bva va va CK S R P Mv D
(7)
Case II:
* ** * * *
0, , , ,T Tx y u t vx y u t
qaa a v ak
* *
0
*,
W
C CCaq
3 *2 2 32 4
, , ,C WTr m
g a qa g a qF G GK v v
* *
2, a
aP kP Dv a
* 2 2
0, , , ,C PC C e e
K a Bva va va CK S R P Mv D
(8)
The non-dimensional form of the governing equations (3.1) to (3.3) reduce to
22 2( )e r m
u P uR K M u G G Ct x y
(9)
22
2eP Ft y
(10)
2
2
C C CC CS K S Ct y
(11)
3816 P. Gurivi Reddy, T.S.Reddy and S.V.K.Varma
Where
22
*
1
a
aKD K
,The corresponding boundary conditions are
Case I:
uu hy
, 1, 1C at 0y
0,u 0 , 1C at 1y
(12)
Case II:
uu hy
, 1, 1
Cy y
at 0y
0u , 0 , 1C at 1y
(13)
For purely periodic flow, let the Pressure gradient be of the form, i tP e
x
where
is a constant and is the frequency of oscillations
4. SOLUTION OF THE PROBLEM:
In order to solve the governing equations (9) to (11) subject to the boundary
conditions (12) and (13), the following perturbation technique is used. The governing
equations (9) to (11) are expanded in small perturbation parameter ( 1) .
Let 2
1( , ) ( ) ( ) ( ) ................................i tou y t u y e u y o (14)
2
1( , ) ( ) ( ) ( ) ................................i toy t y e y o (15)
2
1( , ) ( ) ( ) ( ) ................................i toC y t c y e c y o (16)
Using equations (14) to (16) into the equations (9) to (11) and we get
Zero-th order terms:
2
0 1 0 0 0'' r mu m u G G C (17)
2''
0 0 0F (18)
2
0 5 0' 0'C m C (19)
MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel 3817
First order terms:
2
1 2 1 1 1/'' r mu m u G G C (20)
2
1 3 1'' 0m (21)
2
1 4 0' 0'C m C (22)
The corresponding boundary conditions are
Case I:
0 10 1 0 0 0 1, , 1, 0, 1, 0
du duu h u h C Cdy dy
at 0y
0 1 0 0 0 10, 0, 0, 0, 0, 0 u u C C at 1y (23)
Case II:
0 0 01 1 10 1, , 1, 0, 1, 0
du Cdu Cu h u hdy dy y y y y
at 0y
0 1 0 0 0 10, 0, 0, 0, 0, 0 u u C C , at 1y (24)
Solving equations (17) to (22) under the boundary conditions (23) and (24), the
following solutions are obtained.
Case I:
( ) cosh( ) sinh (cosh sinh ) (cosh sinh )1 2 2 2 3 1 4 5 2 5
u y c m y c m y k Ny k Ny k m y k m yo (25)
0 1( ) cosh sinhy Fy k Fy (26)
0 5 2 5( ) cosh sinhC y m y k m y (27)
1 3 1 4 1 5( ) cosh sinhu y c m y c m y k (28)
1( ) 0y (29)
1( ) 0C y (30)
Finally the expressions for velocity, Temperature and concentration are given by
1( , ) cosh sinhy t Fy k Fy (31)
5 2 5( , ) cosh sinhC y t m y k m y
(32)
3818 P. Gurivi Reddy, T.S.Reddy and S.V.K.Varma
1 2 2 2 3 1
4 5 2 5
3 1 4 1 6
cosh sinh (cosh sinh )
( , ) (cosh sinh )
( cosh( ) sinh( ) )i t
c m y c m y k Fy k Fyu y t k m y k m y
e c m y c m y k
(33)
Case II:
( ) cosh( ) sinh (cosh sinh ) (cosh sinh )5 5 50 2 6 2 8 1 9 2 u y c m y c m y k Ny k Ny k m y k m y (34)
0 1
1
1( ) cosh sinhy Fy k Fy
k F (35)
0 5 2 5
2 5
1( ) cosh sinhC y m y k m y
k m (36)
1 3 1 4 1 5( ) cosh sinhu y c m y c m y k (37)
1( ) 0y (38)
1( ) 0C y (39)
Finally the expressions for velocity, Temperature and concentration are given by
1
1
1( , ) cosh sinhy t Fy k Fy
k F (40)
5 2 5
2 5
1( , ) cosh sinhC y t m y k m y
k m (41)
cosh( ) sinh (cosh sinh )5 2 6 2 8 1( , )
(cosh sinh ) ( cosh( ) sinh( ) )5 59 2 3 1 4 1 6
c m y c m y k Ny k Nyu y t i tk m y k m y e c m y c m y k
(42)
5. NUSSELT NUMBER:
From temperature field, the rate of heat transfer in terms of Nusselt number is given in
non-dimensional form as follows:
For Case (I): 10
Nu k Fy y
(43)
MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel 3819
For Case (II): 1
0
Nuy y
(44)
6. SHERWOOD NUMBER:
From concentration field, the rate of mass transfer in terms of Sherwood number is
given in non-dimensional form as follows:
For Case (I): 520
CSh k m
y y (45)
For Case (II) : 1
0
CSh
y y (46)
7. SKIN-FRICTION:
From velocity field, the rate of change of velocity at the plate in terms of Skin-friction
is given in non-dimensional form as follows:
For
(47)
Case(I):
For Case (II): [ ( )] [ ( )]56 2 1 8 2 9 4 10
u i tc m k k F k k m e c my y (48)
8. RESULTS AND DISCUSSIONS:
Case (I): In order to point out the effects of various parameters on flow
characteristics, the following discussion is set out. The values of Prandtl number are
chosen Pr = 7 (water) and Pr= 0.71 (air). The value of the Schmidt number is chosen
to represent the presence of species by Carbon dioxide (1.00). Profiles for velocity,
temperature and concentration fields in addition to skin-friction are presented in
figures 1-18 for Gr>0 and Gm>0. Figure 1 shows the effect of wall slip h on the
velocity field. From this figure it is observed that velocity increases as the slip
parameter h increases. Velocity profiles for different values of time t, Magnetic
parameter M, Reynolds number Re, and permeability parameter k are presented in
figures from 2-5 respectively. It is observed that velocity decreases with increasing
values of time t or magnetic parameter M or Reynolds number Re or permeability
[ ( )] [ ( )]52 2 1 3 2 4 4 10
u i tc m k k F k k m e c my y
3820 P. Gurivi Reddy, T.S.Reddy and S.V.K.Varma
parameter k. The effects of thermal Grashof number Gr, modified Grashof number
Gm, radiation parameter F, Schmidt number Sc and chemical reaction parameter Kc are
shown in figures from 6-10 respectively. It is observed that velocity increases with
increasing values of thermal Grashof number Gr or modified Grashof number Gm
while decreases with increasing values radiation parameter F or Schmidt parameter Sc
or chemical reaction parameter Kc.
Temperature profiles for different values of radiation parameter F and Schmidt
parameter Scare shown in figures 11 and 12 respectively. From these figures it is seen
that velocity increases with the decrease in radiation parameter F or Schmidt
parameter Sc. Concentration profiles for different values of chemical reaction
parameter Kc are presented in figure 13. It is noticed that concentration increases a
chemical reaction parameter Kc decreases. A variation in shear stress τ is shown in
figures 14-18 for different values of wall slip parameter h, magnetic parameter M,
permeability parameter k, radiation parameter F and chemical reaction parameter Kc.
It is found that skin friction decreases with the increase in wall slip parameter h or
magnetic parameter M or permeability parameter k or radiation parameter F or
chemical reaction parameter Kc.
Case (II): In order to know the influence of different physical parameters, viz., wall
slip parameter h, magnetic parameter M, permeability parameter k, radiation
parameter F, chemical reaction parameter Kc, Reynolds number Re, thermal Grashof
number Gr, modified Grashof number Gm, Schmidt number Sc and time t on the
physical flow field, computations carried out for velocity, temperature, concentration
and skin-friction and they are represented through figures 19-36 for Gr> 0 and Gm> 0.
The values of Prandtl number are chosen Pr = 7 (water) and Pr= 0.71 (air). The value
of the Schmidt number is chosen to represent the presence of species by Carbon
dioxide (1.00). Figure 19represents the effect of wall slip h on the velocity field. From
this figure it is observed that velocity increases as an increase in slip parameter h.
Velocity profiles for different values of time t, Magnetic parameter M, Reynolds
number Re, and permeability parameter k are presented in figures from 20-23
respectively. It is noticed that as t or M or Re or k increases velocity also increases.
The effects of thermal Grashof number Gr, modified Grashof number Gm, radiation
parameter F, Schmidt number Sc and chemical reaction parameter Kc are shown in
figures from 24-28 respectively. It is found that velocity increases with increasing
values of thermal Grashof number Gr or modified Grashof number Gm while it a
reverse effect with increasing values radiation parameter F or Schmidt parameter Sc or
chemical reaction parameter Kc.
Temperature profiles for different values of radiation parameter F is shown in figures
29. From this figure it is seen that velocity increases with the decrease in radiation
parameter F. Concentration profiles for different values of Schmidt parameter Sc and
chemical reaction parameter Kcare presented in figures 30 and 31 respectively. It is
observed that concentration increases with decrease in Schmidt parameter Sc or
chemical reaction parameter Kc. Shear stress is presented against time t for different
values of wall slip parameter h, magnetic parameter M, permeability parameter k,
radiation parameter F and chemical reaction parameter Kc in figures 32-36
MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel 3821
respectively. It is observed that Shear stress decreases with the increase in wall slip
parameter h or magnetic parameter M or permeability parameter k or radiation
parameter F or chemical reaction parameter Kc.
GRAPHS:
Case I:
Figure.1. Effect of wall slip h and time t on velocity field when 𝑆𝑐 = 1, 𝐺𝑟 = 1, 𝑅𝑒 =1, 𝑃𝑒 = 0.7, 𝑘 = 1, 𝐹 = 1, 𝜔 = 1, =1, 𝐾𝑐 = 1, 𝑀 = 1
Figure.2. Effect of Magnetic parameter M and Reynolds Number eR on velocity field
when 𝑆𝑐 = 1, 𝐺𝑟 = 1, 𝑃𝑒 = 0.7, 𝑘 = 1, 𝐹 = 1, 𝜔 = 1, =1, 𝐾𝑐 = 1, ℎ = 1,t=0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
y
u
h=0.0, 0.5, 1.0, 1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
y
u
t =
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
y
u
M = 0.0, 0.5, 1.0, 1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
y
u
Re = 1, 3, 5, 7
3822 P. Gurivi Reddy, T.S.Reddy and S.V.K.Varma
Figure.3. Effect of permeability parameter k and Grashof number 𝐺𝑟 on velocity field
when 𝑆𝑐 = 1, 𝑅𝑒 = 1, 𝑃𝑒 = 0.7, 𝑀 = 1, 𝐹 = 1, 𝜔 = 1, =1, 𝐾𝑐 = 1, ℎ = 1
Figure.4. Effect of Modified Grashof Number mG and Radiation parameter F on
velocity field when 𝑆𝑐 = 1, 𝐺𝑟 = 1, 𝑃𝑒 = 0.7, 𝑘 = 1, 𝑀 = 1, 𝜔 = 1, =1, 𝐾𝑐 = 1, ℎ = 1,t=0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
y
u
k = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
y
u
Gr = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
y
u
Gm = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
y
u
F = 0.5, 1.0, 1.5, 2.0
MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel 3823
Figure.5. Effect of Schmidt Number cS and chemical reaction parameter 𝐾𝑐 on
velocity field when 𝐺𝑚 = 1, 𝐺𝑟 = 1, 𝑃𝑒 = 0.7, 𝑘 = 1, 𝐹 = 1, 𝜔 = 1, =1, ℎ = 1, t=0,
M=1
Figure.6. Effect of Radiation parameter F on Temperature field and Schmidt number
cS on concentration field when 𝐺𝑚 = 1, 𝐺𝑟 = 1, 𝑃𝑒 = 0.7, 𝑘 = 1, 𝜔 =
1, =1, ℎ = 1, t=0, M=1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
y
u
Sc = 0.22, 0.78, 1.00, 2.01
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
u
Kc = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
F = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
C
Sc = 0.22, 0.78, 1.00, 2.01
3824 P. Gurivi Reddy, T.S.Reddy and S.V.K.Varma
Figure.7. Effect of the chemical reaction parameter cK on concentration field and k
on skin friction when 𝐺𝑚 = 1, 𝐺𝑟 = 1, 𝑃𝑒 = 0.7, 𝑘 = 1, 𝐹 = 1, 𝜔 = 1, =1, t=0, M=1
Figure.8. Effects of M and k on Skin-friction
Figure.9. Effects of F and KC Skin-friction
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
C
Kc = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
h = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
t
M = 0.0, 0.5, 1.0,1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
t
k = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2
0.21
0.22
0.23
0.24
0.25
0.26
0.27
t
F = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.21
0.22
0.23
0.24
0.25
0.26
0.27
t
Kc = 0.5, 1.0, 1.5, 2.0
MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel 3825
Case II:
Figure.10. Effect of wall slip h and time t on velocity field when 𝑆𝑐 = 1, 𝐺𝑟 =1, 𝑅𝑒 = 1, 𝑃𝑒 = 0.7, 𝑘 = 1, 𝐹 = 1, 𝜔 = 1, =1, 𝐾𝑐 = 1, 𝑀 = 1
Figure.11. Effect of Magnetic parameter M and Reynolds Number eR on velocity
field when 𝑆𝑐 = 1, 𝐺𝑟 = 1, 𝑃𝑒 = 0.7, 𝑘 = 1, 𝐹 = 1, 𝜔 = 1, =1, 𝐾𝑐 = 1, ℎ = 1,t=0
Figure.13. Effect of permeability parameter k and Grashof number 𝐺𝑟 on velocity
field when 𝑆𝑐 = 1, 𝑅𝑒 = 1, 𝑃𝑒 = 0.7, 𝑀 = 1, 𝐹 = 1, 𝜔 = 1, =1, 𝐾𝑐 = 1, ℎ = 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
y
u
h = 0.0, 0.5, 1.0, 1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
y
u
t =
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
y
u
M = 0.0, 0.5,1.0, 1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
y
u
Re = 1, 3, 5, 7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
y
u
k = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
y
u
Gr = 0.5, 1.0, 1.5, 2.0
3826 P. Gurivi Reddy, T.S.Reddy and S.V.K.Varma
Figure.14. Effect of Modified Grashof Number mG and Radiation parameter F on
velocity field when 𝑆𝑐 = 1, 𝐺𝑟 = 1, 𝑃𝑒 = 0.7, 𝑘 = 1, 𝑀 = 1, 𝜔 = 1, =1, 𝐾𝑐 = 1, ℎ =1,t=0
Figure.15. Effect of Schmidt Number cS and chemical reaction parameter 𝐾𝑐 on
velocity field when 𝐺𝑚 = 1, 𝐺𝑟 = 1, 𝑃𝑒 = 0.7, 𝑘 = 1, 𝐹 = 1, 𝜔 = 1, =1, ℎ = 1, t=0,
M=1
Figure.16. Effect of Radiation parameter F on Temperature field and Schmidt number
cS on concentration field
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
y
u
Gm = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
y
u
F = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
y
u
Kc = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
y
u
Sc = 0.22, 0.78, 1.00, 2.01
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
F = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
C
Sc = 0.22, 0.78, 1.00, 2.01
MHD Flow of a Newtonian Fluid Through a Porous Medium in Planer Channel 3827
Figure.17. Effect of the chemical reaction parameter cK on concentration field and
h on Skin-friction
Figure.33. Skin-friction for different values of magnetic parameter M and
permeability parameter k
Figure.35. Skin-friction for different values of radiation parameter F and Kc
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
y
C
Kc = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
h= 0.0, 0.5, 1.0, 1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
t
M= 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.2
0.25
0.3
0.35
0.4
0.45
0.5
t
k = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.26
0.28
0.3
0.32
0.34
0.36
0.38
t
F = 0.5, 1.0, 1.5, 2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.29
0.3
0.31
0.32
0.33
0.34
0.35
0.36
0.37
t
Kc = 0.5, 1.0, 1.5, 2.0
3828 P. Gurivi Reddy, T.S.Reddy and S.V.K.Varma
9. CONCLUSIONS:
In this paper we have studied the effects of slip condition, chemical reaction and
radiation on MHD free convection periodic flow through a saturated porous medium
bounded by a vertical surface in a planer channel in two cases viz. Case–I: Uniform
plate Temperature and Concentration and Case–II: Constant heat and mass flux. In
the analysis of the flow the following conclusions are made.
1. In case (I) and (II) of the study, the velocity increases with an increase in slip
parameter h, Grashof number Gr, modified Grashof number Gm, Schmidt
number Sc, and Radiation parameter F, and it shows a reverse effect in the case
of magnetic parameter M, time t, permeability parameter k or Reynolds
number Re.
2. Temperature decreases with the increase in radiation parameter F for Case (I)
and (II).
3. Concentration decreases with an increase in chemical reaction parameter kc or
Schmidt number Sc in Case (I) as well as in Case (II) of the problem.
4. Shear stress increases with the increase in magnetic parameter M or
permeability parameter k and it shows a reverse effect in the case of slip
parameter h or Schmidt number Sc in both cases of the study.
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