MHD Flow of a Newtonian Fluid Through a Porous Medium · PDF fileMHD Flow of a Newtonian Fluid...

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

mailto:[email protected]

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


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)


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)


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)


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)


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)


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


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


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


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


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


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


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


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


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


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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