MHD Boundary Layer Flow of Heat and Mass Transfer in the presence of Heat Generation/Absorption, radiation and chemical reaction of a temperature dependent Viscous Fluid Past a time dependent permeable Vertical Plate under Oscillatory Suction

MHD Boundary Layer Flow of Heat and Mass Transfer in the presence of Heat Generation/Absorption, radiation and chemical reaction of a temperature dependent Viscous Fluid Past a time dependent permeable Vertical Plate under Oscillatory Suction

Adebile E.A1 and Sogbetun L.O2 Department of Mathematical Sciences Federal University of Technology, Akure

Abstract

In this research, the researchers studied and made analysis on the MHD boundary layer flow of a variable viscous fluid over a vertical porous plate in a porous medium of time dependent permeability in the presence of radiation and chemical reaction under oscillatory suction velocity taking into account the heat generation/absorption and reaction parameter effects. A time dependent suction was assumed and the radiative flux was described using the differential approximation for radiation. The governing system of partial differential equations was linearised using asymptotic techniques. Computational results and graphical representations showing the effects of the governing model parameters were made and found to be in good agreement with those in the literature.

Keywords: heat generation/absorption, reaction parameter, MHD, vertical porous plate, viscous fluid, suction velocity

1. Literature review

   Interest on MHD flow through a porous plate by researchers has tremendouslyincreased over the years. This is due to the fact that heat and mas transfer through porous media occur in many engineering, geophysical and biological applications. For instance, permeable porous plates are used in the filtration processes and also for heated body to keep its temperature constant and to make the heat insulation of the surfaces more effective. Many investigators have considered works on the unsteady oscillatory free convective flow through porous media because of its importance in chemical engineering; turbo machinery and aerospace technology.

   Kumar et al [2] in his work,examined an unsteady oscillatory laminar free convective fluid through a porous medium along a porous hot vertical plate with time dependent suction in the presence of heat source/sink. In another development, Soundalgekar et al [3],analysed free convective effects on the oscillatory flow past an infinite vertical porous plate with constant suction. Singh et al 43], and Venkateswarlu and Rao [5] in their researches studied the effects of permeability variation and oscillatory suction velocity on free convection and mass transfer flow of a viscous fluid past an infinite vertical porous plate in the presence of a uniform transverse magnetic field. Okedoye et al [6 ], on the other hand, researched on the unsteady magnetohydrodynamic heat and mass transfer in MHD flow of an incompressible, electrically conducting, viscous fluid past an infinite vertical porous plate along with porous medium of time dependent permeability under oscillatory suction velocity normal to the plate.

   All these aforementioned references did not considered flows involving effects of heat generation/absorption and reaction parameter on the oscillatory suction velocity in the presence of temperature dependent viscosity while such flows are encountered in various fields.Adebile

   ' '

   ' '

   and Sogbetun[1] very recently investigated MHD flow of a fluid with temperature dependent viscosity but considered the steady situationuner the influence of constant suction velocity.In the present study, we extend the previos work by investiging the effects of heat generation//absorption and reaction parameter on the unsteady MHD boundary layer flow of a variable viscous fluid over a vertical porous plate in a porous medium of time dependent permeability in the presence of radiation and chemical reaction under oscillatory suction

   ' '

   ' '

   velocity. The permeability of the porous medium is considered to be

   K t'

   K0 1

   ein t

   and the

   suction velocity is assumed to be constant.

2. Nomenclature

v t'

v0 1

ein t

where v0

0 and 1 is a positive

u : Velocity along x coordinate T ' : Non dimensional fluid temperature

v : Velocity along y coordinate C ' : Non dimensional species concentration

g : Acceleration due to gravity T : Fluid temperature

U ' : Non dimensional fluid velocity : Reaction parameter

*

Tw : Ambient temperature : Stefan- Boltzmann constant

C : Species concentration B : Coefficient of mass expansion

Cw : Ambient species concentration

B0 : Transverse magnetic field

B : Coefficient of thermal expansion

: Skin-friction coefficient : Electrical conductivity

k : Thermal conductivity Sc : Schmidt number

c p : Specific heat at constant pressure D : Molar diffusivity

A : Pre-exponential factor

Grc : Mass grashof number

v0 : Normal velocity at the plate Gr : Thermal grashof number

o

o

k * : Mean absorption coefficient : Delta, 0 1

M : Hartmann number : Epsilon, 0 o 1

Nu : Nuselt number Sh : Sherwood number

: Angular velocity t :Time

Pr :Prandtl number : Fluid viscosity

2.0 Mathematical Formulation

A magnetohydrodynamic flow of viscous, incompressible, electrically conducting fluid past an infinite vertical plate in a porous medium under suction velocity is considered. The x- axis is taken along the plate in the direction of the flow and y- axis normal to it. A uniform magnetic field is applied normal to the direction of the flow. It is assumed that the magnetic Reynold number is less than unity so that the induced magnetic field is neglected in comparison to the applied magnetic field. We further assumed that all the fluid properties are constant except that of the influence of density variation with temperature. Thus, the basic flow in the medium is entirely due to buoyancy force caused by temperature difference between the wall and the

medium. Initially at t 0 , the plate as well as fluid is assumed to be at the same temperature and

T

T

w

w

the concentration of species is very low so that the Soret and Dofour effect are neglected [6].

When t

0 , the temperature of the plate is instantaneously raised (or lowered) to '

and the

w

w

concentration of species is raised (or lowered) to C ' . Under the above assumptions and taking

the usual Boussinesqs approximation into account, the governing equations for momentum, energy and concentration are presented below:

dv'

0

dy'

(2.1)

'

'

'

'

C v' C

t ' y'

2C '

D

y'2

A C ' C

(2.2)

T ' ' T ' k 2T ' ' '

p

p

v t T T

t ' y'

c y'2

(2.3)

U ' U ' 1 U '

U ' B2U '

v'

g * T ' T

g C ' C 0

t ' y'

y' y '

k ' 1

eiwt (2.4)

The boundary conditions are:

U ' 0 T '

w

w

1 eiwt T T

T C '

w

w

1 eiwt C C

C at y ' 0

w

w

U ' 0 ,

T ' T

C ' C

as y '

(2.5)

The suction velocity from equation (2.1) is assumed to be v'

1 is positive constant.

Introducing the following non dimensional quantities:

v0 (1

eiwt )

here v0

0 and

v y '

v 2t ' U '

4 n'

T ' T

y 0

t 0 U w

v

v

2

2

f v0 0

Tw T

C ' C

k 'v 2 A

g * T T

v

v

D

D

C k 0

G w Sc

v

v

Cw C

2 2 r 3

0 0

c B ' g C C

v

v

Pr P

M 0

and

G w

v

v

2

2

kt v0 0

rc 3

0

Below are the non-dimensional governing equations for momentum, energy and concentration of the unsteady state and their boundary conditions:

1 C iwt C 1 2C

1 e

4 t y

C

Sc y 2

(2.6)

1 1 e

4 t

iwt

y

1 2

Pr y 2

(2.7)

1 U 1

4 t

eiwt U G

y r

Grc C y

U

y k 1

1

eiwt

M 2 U

(2.8)

The relevant boundary conditions in dimensionless form are:

U 0,

1 eiwt ,

C 1 eiwt

on y 0

U 0, 0,

C 0 as y (2.9)

The fluid viscosity was assumed to obey the Reynolds model [7]

e (2.10)

Where , is a parameter depending on the nature of the fluid. Using equation (2.10) in equation (2.8), we obtain

1 U 1

4 t

eiwt U G

y r

U

Grc C y e y

1

k 1 eiwt

M 2 U

(2.11)

3.0 Method of Solution

To solve equations (2.6), (2.7) and (2.11), we seek an asymptotic expansion about for our dependent variables of the form

U y, t

U 0 y

iwt 2

U e

U e

0

0

1

…..

(3.1a)

y, t 0 y

iwt 2

e

e

0

0

1

…..

(3.1b)

C y, t

C0 y

iwt 2

C e

C e

0

0

1

…..

(3.1c)

Corresponding to the species equation we have

C

C

0

0

0

0

'' ScC '

Sc C0 0

(3.2)

C0 0 1,

C0 y

0 as y

C

C

1

1

1

1

'' ScC '

Sc 1 iw

4

C ScC '

(3.3)

0

0

1

1

C1 0 1,

C1 y

0 as y

Corresponding to the energy equation we have

0

0

0

0

0

0

'' Pr ' Pr 0

(3.4)

0 0 1 0 0 as y

1

1

1

1

'' Pr ' Pr

1 iw

4 1

Pr '

(3.5)

0

0

1 0 1

1 0 as y

Corresponding to the momentum equation we have

U

U

U

U

e

e

' '

y 0 0

1 M 2 U k0

Gr 0

Grc C0

(3.6)

U

U

0

0

'

'

U 0 0 0, U 0 0,

U

U

U

U

e

e

' '

y 1 1

1. iwU

   4 1

   1 M 2 U k0

   Gr 1

   Grc C1 0

   (3.7)

   1

   1

   U1 0 0, U1 0

   Solving equations (3.2)-(3.5) and substitute the results into (3.1b) and (3.1c), we have

   C y eny

   8

   8

   a exy

   a10

   eny eiwt

   (3.8)

   y,t

   emy

   a3e

   my iwt

   y

   y

   a e e

   a e e

   5

   (3.9)

   Where

   x 1 S S 2 4S iw

   n 1 S S 2 4S a

   Scn

   2 c c c 4

   2 c c

   c 10 iw

   2

   2

   n Scn Sc 4

   a8 1 a10 m

   1

   1

2. Pr

P P 2

r 4Pr

P

P

2

2

4P iw a

Pr m

a 1 a

2

2

2 r r r 4 5 iw 3 5

m Pr m Pr 4

To solve equations (3.6)-(3.7), we make use of the following transformation: Let o where 1. Thus we assumed the following:

U0 U00

U01

…

h..o.t

(3.10)

U1 U10

U11

…

h..o.t

(3.11)

Substitute equations (3.12)-(3.14) into equations (3.6)-(3.7) and compile the order of . We have

d 2U dU 1

00 00

M 2 U G G C

(3.12)

dy 2

dy k0

00 r 0

rc 0

U 00 (0) 0 U 00 ( ) 0

d dU d 2U dU 1

01

01

00

01

01

M 2 U 0

(3.13)

dy dy

dy 2

dy k0

U 01 (0) 0 U 01 ( ) 0

d 2U dU 1 1 dU

10

10 iw

M 2 U G

G C 00

dy2

dy 4 k0

10 r 1

rc 1 dy

(3.14)

U10 0 0, U10 0

d dU d dU d 2U dU 1 1 dU

10 eiwt 10

11

11 iw

M 2 U

01

dy 0 dy dy 1 dy

dy2

dy 4 k0

11 dy

U11 0 0,

U11 0

(3.15)

Based on the solutions obtained in equations (3.12) and (3.15), we calculate equation (3.1a) to be

U y, t

y

y

a11e

my

my

y

y

a13 e

ny

ny

my

my

a14 e

y

y

xy

xy

a15 e

ny

ny

m y

m y

a17 e

2my

a e

a e

y

y

18

n m y

a e

a e

19

a20 e

m

m

y

y

a21e

a22 e

n

n

y

y

m

m

y

y

y

y

m y

m y

y

y

m y

m y

a23 e

a25 e

a26 e

a27 e

a28 e

a29 e

y

y

y

y

y

y

a30 e

a32 e

a33 e

2my

a e

a e

2 y

2 y

x

x

y

y

34

x m y

a e

a e

n

n

y

y

35

n m y

a e

a e

y

y

36

a37 e

eiwt

a38 e

a39 e

a40 e

a41e

a42 e

a43 e

(3.16)

S

S

c

c

c

c

Where

1 1

2

1

m 2 Pr

1 4 1 M 2

k0

P

P

2

2

r 4Pr

1 1 1 4 1 M 2 n

2 k0

1

2 Sc

2 4S

1

2 Pr

iw

P

P

2

2

r 4Pr 4

1 1 1

2

4 iw 1 M 2

4 k

0

1 1

2

a13

1 4 iw 1

4 k0

Gr

M 2 x

a14

1 2

S

S

2 Sc c

Grc

4Sc

iw

4

a11

a13

a14

m2 m

1 M 2

k0

n2 n

1 M 2

k0

m a eiwt

m a eiwt a

a eiwt

1 2a m 2

a17

5 5 11

18

5 13

a

a

m 2 m

1 M 2

k0

4m 2 2m

1 M 2

k0

a

a

n a eiwt

m a eiwt

n m a n

a a eiwt

a19

5 5 14

20

3 11

n m 2 n

m

m 1 M 2

k0

a a me iwt

2 1 M 2

k0

a

a

n a na eiwt

a21

13 3

22

14 3

m 2 m

1 M 2

k0

n 2 n

1 M 2

k0

a15

a17

a18

a19

a20

a21

a22 a25

2

Grt a3

iw 1 M 2

4 k0

a26

Grt a5

a13 m

a27

Grc a8

m 2 m

iw 1 M 2

4 k0

x 2 x

iw 1 M 2

4 k0

a28

Grc a10

a14 n

a29

a11

n 2 n

iw 1 M 2

4 k0

2 iw 1 M 2

4 k0

a23

a25

a26

a27

a28

a29

1 a eiwt m a

a32

5 23

m 2 m

iw 1 M 2

4 k0

m a a eiwt m

a ma

eiwt m a a

a33

25 5 26 3 25 21

m 2 m

iw 1 M 2

4 k0

a eiwt 2a m 2 a

a eiwt

1 m x a x

a34

5 26 18

a35

5 27

4m 2 2m

iw 1 M 2

4 k0

x m 2 x m

iw 1 M 2

4 k0

n m a n

n m a

na eiwt a

1. a a eiwt

   m a a

   a36

   28 28 5 19

   a37

   29 5 29 17

2. m 2 n m

iw 1 M 2

4 k0

m 2 m

iw 1 M 2

4 k0

a a eiwt 2 a 2 a eiwt

a38

23 3

a39

25 3

2 iw 1 M 2

4 k0

4 2 2

iw 1 M 2

4 k0

x a xa eiwt n a na eiwt a

a40

27 3

a41

28 3 22

x 2 x

a

iw 1 M 2

4 k0

a eiwt a

n 2 n

a

iw 1 M 2

4 k0

a42

29 3 20

a43

15

2 iw 1 M 2

4 k0

2 iw 1 M 2

4 k0

a30

a32

a33

a34

a35

a36

a37

a38

a39

a40

a41

a42

a43

Skin-friction coefficient at the plate is

U

y y 0

a11

a13 m

a14 n

a15

a17

m a18 2m

a19 n m

a20

a21 m

a22 n

a23

a25

a26 m

a27 x

a28 n

a29

a30

a38

a32

m a39 2

a33 m

a40 x

a34 2m

a41 n

a35 x

m a42

a36 n

m a43

a37

m eiwt

(3.17)

5

5

Heat transfer coefficient

Nu at the plate is

y

y

3

3

Nu m

y 0

a a m ei t

(3.18)

Mass transfer coefficient

Sh at the plate is

S

S

n

n

C

y

y

h a8 x

y 0

a10

n ei t

(3.19)

4.0 Discussion of Results

In order to investigate the effect of various varying parameters on the flow behaviour and the temperature distribution within the boundary layer, computational results were obtained for

various values of

M ,Gr

,Grc , Sc , k0 , , , ,

and t with fixed values for Pr and Sc . These

parameters were assigned the following values M

0.5,Gr

1.2,Grc

1.0, k0

0.2,

0.2,

0.5,

0.2,

0.2 and t

2 except where stated otherwise while the values of Pr and

Sc were taken to be 0.71 and 0.6 respectively for plasma. It should be noted that

0, 0 and

0 represent destructive, no and generative chemical reactions respectively. Also,

0, 0 and 0 indicates heat absorption, no heat generation/absorption and heat

generation respectively. The figures are presented in 3-dimensional figures.

From equation (2.10), we could see that increase in viscosity parameter leads to decrease in viscosity.

Numerical values of skin friction are showed in Table 4.1. We observed that an increase in viscosity parameter, mass Grashof number or the thermal Grashof number increases skin friction whereas increase in magnetic parameter or reaction parameter leads to a decrease in skin friction coefficient.

Table 4.2 displayed the effects of viscosity parameter on the dimensionless velocity. It is discovered thatincrease in viscosity parameter bring about increase in the fluid velocity near the plate only while reverse is the case as we move away from the plate. The fluid velocity increased and reached its maximum value at very short distance from the plate and then decreased to zero.

Figures 4.1 – 4.10, show the effect of the varying parameters on the velocity field. It is observed that maximum velocity occurs in the body of the fluid close to the surface. Figures 4.1-4.2 highlight the effects of delta and epsilon on the fluid velocity. It could be seen that increase in delta or epsilon increases velocity. We observed in figure 4.3 that increase in destructive chemical reaction parameter reduces the velocity field while increase in the generative chemical reaction parameter increases the velocity. We displayed the effect of on velocity in figure 4.4; it is shown that increases in brings about increase in velocity field.

Furthermore, we investigate the effect of Hartmann number M on the fluid velocity in figure 4.5. We discovered that increase in Hartmann number M reduces the velocity field as a result of an opposing force (Lorentz force). Also, figure 4.6 shows that increase in permeability increases the velocity. The effectsof mass and thermal Grashof numbers on the velocity field are shown in fugures 4.7 and 4.8 respectively. We discovered that velocity increases as either mass or thermal Grashofnumber increases. Figure 4.9 and 4.10 show that increase in and t increase the velocity.

Figure 4.11 show the effect of chemical reaction parameters on concentration field. It is observed that for a generative chemical reaction, there exist oscillations in the field away from the surface. This brings about the presence of minimum and maximum concentration in the field which however less than the surface concentration. For destructive chemical reaction, the boundary

layer reduces as the reaction parameters increases. Also there is reduction in concentration field as reaction parameter increases positively.

In figure 4.13, we display the concentration field as a function of

y,t

, it could be seen that

concentration decreases as the flow progresses and decreases faster as we move away from the

boundary. While concentration is displayed as a function of ,t in figure 4.12. Oscillation is

observed along axis with a steep decrease in the field as y increases.

We displayed in figure 4.14- 4.16, the temperature profile for various values of parameters under consideration.

It could be seen from figure 4.14 that heat absorption resulted in decreases in the fluid body temperature, while heat increases the fluid body temperature which leads to presence of extremes temperature in the body of the fluid greater than the surface temperature.

In figure 4.16 and 4.15, we show the temperature profile as a function of

y,t

and ,t

respectively. It is observed that the temperature decreases as y increases, and oscillatory field along t and

axis which is more pronounced at the initial stage continuous as y increases.

Table 3.10: Skin-friction coefficient when 0 Table 3.11: Velocity U y distribution for various values of

Gr Grc 0.1 k0 0.1 0.0 0.2									y	U 0.2 U 0.1 U 0.0 U 0.1 U 0.2
-2.0	1.0	0.5	-0.5 0.2		-0.2	-0.2664	-0.2900	-0.3373	0	0	0	0	0	0
-1.0	1.0	0.5	-0.5 0.2		-0.2	0.0740	0.0779	0.0857	2	0.0945	0.0943	0.0941	0.0939	0.0937
0.0 1.0 0.5 -0.5 0.2 -0.2 0.4143 0.4458 0.5087									4	-0.0085 -0.0086 -0.0086 -0.0086 -0.0087
1.2	1.0	0.5	-0.5	0.2	-0.2	0.8228	0.8873	1.0164	6	-0.0261	-0.0261	-0.0261	-0.0261	-0.0261
1.2	-1.0	0.5	-0.5	0.2	-0.2	-0.0059	-0.0043	-0.0011	8	-0.0134	-0.0134	-0.0134	-0.0134	-0.0134
1.2	0.0	0.5	-0.5	0.2	-0.2	0.4084	0.4415 0.5076 10 -0.0008 -0.0008 -0.0008 -0.0008 -0.0008
1.2	1.0	0.0	-0.5	0.2	-0.2	0.8405	0.9067 1.0392
1.2	1.0	1.0	-0.5	0.2 -0.2		0.7759 0.8359 0.9560
1.2	1.0	0.5	-1.0	0.2 -0.2		0.7981 0.8612 0.9872

1.2 1.0 0.5 1.0 0.2 -0.2 0.7169 0.7776 0.8988

0.3

0.2

0.1

0

-0.1

-0.2

-0.1

0

0

0.1 2

4

0.3

0.2

0.1

0

-0.1

-0.2

-0.1

0

0

0.1 2

4

y

y

6

delta

6

y

y

0.2 10 8

epsilon

0.2 10 8

Fig 3.1: Velocity field as function of y, Fig 3.2: Velocity field as function of y,

0.3

0.2

0.1

0

-0.1

-2

-1

0

0

1 2

4

y

y

6

0.3

0.2

0.1

0

-0.1

-2

-1

0

1

phi

0

2

4

6

y

y

2 10 8

reaction parameter

2 10 8

Fig 3.3: Velocity field as function of

0.3

0.2

0.1

0

-0.1

0

Fig 3.4: Velocity field as function of

1

0

-1

-2

-4

1 -2

2

0

3 2

4

6

y

y

M 4 10 8

0

2

k0 4 14

2 0

4

8 6

12 10

y

Fig 3.5: Velocity field as function of

y, M

Fig 3.6: Velocity field as function of

y, k0

0.6

0.6

0.4

0.2

0

-0.2

-2

0.4

0.2

0

-0.2

-2

0

0

2

Grc

0

2

4

6

y

y

4 10 8

2

Grt

0

2

4

6

y

y

4 10 8

Fig 3.7: Velocity field as function of

y,Grc

Fig 3.8: Velocity field as function of

y,Grt

0.3

0.2

0.1

0

-0.1

-10

-5

0

0

5 2

4

6

y

y

w 10 10 8

0.3

0.2

0.1

0

-0.1

0

1

2

3

t

0

2

4

6

y

y

4 10 8

Fig 3.9: Velocity field as function of y,

Fig 3.10: Velocity field as function of

1.5

y, t

1 1

0.5

0

-0.5

-2

0.5

0

-0.5

0

1

-1

0

0

1 2

4

6

y

y

u 2 10 8

2

0

3 2

4

6

y

y

t 4 10 8

Fig 3.11: Concentration field as function of

1.5

1

0.5

0

-0.5

-10

Fig 3.12: Concentration field as function of

1

0.5

0

-0.5

-2

y, t

-5 -1

0

0

5 2

4

6

y

y

w 10 10 8

0

1

phi

0

2

4

6

y

y

2 10 8

Fig 3.13: Concentration field as function of

1.5

1.5

Fig 3.14: Temperature field as function of y,

1

1

0.5

0

-10

-5

0

0

5 2

4

6

y

y

w 10 10 8

0.5

0

0

1

2

3

t

0

2

4

6

y

y

4 10 8

Fig 3.15: Temperature field as function of

Numerical Method

Fig 3.16: Temperature field as function of

y, t

Applying Crank-Nicolson formula to equations (2.6), (2.7) and (2.11), we have

P3Ci 1, j 1

Q3Ci, j 1

R3Ci 1, j 1

D3i, j

(5.1)

P2 i 1, j 1

Q2 i, j 1

R2 i 1, j 1

D2i, j

(5.2)

P1U i 1, j 1

Q1Ui, j 1

R1Ui 1, j 1

D1i, j

(5.3)

With corresponding conditions (2.9) becoming

U0, j 0

0, j 1

eiwt j

C0, j 1

eiwt j

U , j 0

, j 0

C , j 0

(5.4)

Where

S3 1

r t P

fSc f 3

r

2 fSc

f2 t

f 4 y

Q3 1

r R

fSc 3

f2 t

f 4 y

r

2 fSc

D3i, j

P3Ci 1, j

S3Ci, j

R3Ci 1, j

1 R r ' f t

1 R r '

f t 1 R r '

P 2 Q 1

R 2

2 2 f Pr f 4 y 2 f Pr

1 R r ' t

2 f 4 y 2 f Pr

S2 1

f Pr f

D2i, j

P2Ci 1, j

S2Ci, j

R2Ci 1, j

d r ' t f

d r '

d r ' t f

d r ' t

P 1 2

Q 1 1 R

1 2

S 1 1 d

1 2 f

f 4 y

1 f 1 2 f

f 4 y

1 f f 2

D1i, j

P1U

i 1, j

S1U

i, j

R1U

i 1, j

t

f d3 d4

Thus, y and t are constant mesh sizes along y and t directions respectively. We need a scheme to find single values at next level time in terms of known values at an earlier time level.

A forward difference approximation for the first order partial derivatives of and U with

respect to t and y and a central difference approximation for the second order partial derivative

of and U with respect to t and y are used. We used the following transformations

1

f 4 f2 1

eiwt j d e

d 1 M 2 d

2

2

3

3

k0

Grt d 4

t

'

'

Grt C r 2 1

y

1

1

We have converted partial differential equations (2.6), (2.7) and (2.11) that hold everywhere in some domain into a system of simultaneous linear equations (5.1)-(5.3) to get approximate solutions. The corresponding code (programme) is written in Mathlab for calculating numerical solutions for concentration, temperature and velocity.

5.1 Tables and Graphical Presentations: Numerical Solution

To ensure the validity of our analytical solutions, we have compared our numerical solutions with the exact solutions for concentration, temperature and velocity for some variation parameters affecting fluid profile. These parameters are assigned the values

M 1.0,Gr

5.0,Grc

1.0,

0.1, R

0.5,

0.1,

0.1, n

0.5,t

1.0, A

0.3 in case 5.

The corresponding code (programme) is written in Mathlab for calculating both the exact and numerical solution. In table 5.1 – 5.3, the comparison between analytical values and numerical values for concentration, temperature and velocity are presented. The error differences are reasonable and passable. The closeness of curves corresponding to both exact and numerical

solutions in all the cases considered in figure 5.1-5.3 further confirm the accuracy of our method of solution.

Table 5.1: Comparison between exact and Table 5.2: Comparison between exact and

numerical values for concentration numerical values for temperature

y Analytical Numerical Error					y	Analytical Numerical Error
0	1	1	0		0	1	1	0
2	0.2529	0.2524	0.0005		2	0.1746	0.1739	0.0007
4	0.064	0.0637	0.0003		4	0.0305	0.0303	0.0002
6	0.0162	0.0161	0.0004		6	0.0053	0.0053	0
8	0.0041	0.0041	0		8	0.0009	0.0009	0
10	0.001	0.001	0		10	0.0002	0.0002	0

Table 5.3: Comparison between exact and

numerical values for velocity

1

numerical

0.9 analytical

y	Analytical	Numerical	Error
0	0	0	0
2	0.0837	0.1015	0.0178
4	0.0185	0.0222	0.0037
6	0.0042	0.005	0.0008
8	0.0010	0.0011	0.0001
10	0.0002	0.0003	0.0001

y	Analytical	Numerical	Error
0	0	0	0
2	0.0837	0.1015	0.0178
4	0.0185	0.0222	0.0037
6	0.0042	0.005	0.0008
8	0.0010	0.0011	0.0001
10	0.0002	0.0003	0.0001

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 2 4 6 8 10 12 14 16 18 20

Figure 5.1: Concentration profiles

Figure 5.2: Temperature profiles Figure 5.3: Velocity profile

1

numerical

0.25

numerical

0.9 analytical

0.8

0.2

analytical

0.7

0.6

0.15

0.5

0.4

0.1

0.3

0.2

0.05

0.1

0

0 2 4 6 8 10 12 14 16 18 20

0

0 2 4 6 8 10 12 14 16 18 20

REFERENCE

1. Adebile ,E .A.and Sogbetun MHD Flow of A Non-Newtonian Fluid With Temperature Dependent Viscosity Past A Vertical Plate In The Presence Of Radiative Heat Flux And Chemical Reaction. International Journal of Engineering Research & Technology (IJERT) Vol. 1 Issue 10,

2. A. Kumar, B. Chand and Kaushik, On Unsteady Oscillatory Laminar Free Convection Flow of an Electrically Conducting Fluid through Porous Medium along a Porous Hot Vertical Plate with Time Dependent Suction in the Presence of Heat Source/ Sink, J. of Acad Math, vol. 24, 339 354, 2002.

3. V. M. Soundalgekar, Free Convection Effects on the Oscillatory Flow Past an Infinite Vertical Porous Plate with Constant Suction 1, Proc. Royal Soc. London A 333, 25-36, 1973.

4. A.K. Singh, A.K. Singh and N.P. Singh, Heat and Mass Transfer in MHD Flow of a Viscous Fluid Past a Vertical Plate under Oscillatory Suction Velocity, Ind. J. of Pure Appl. Math, vol. 34, 429-442, 2003.

5. K. Venkateswarlu and J. A. Rao, Numerical Solution of Heat and Mass Transfer in MHD Flow of a Viscous Fluid Past a Vertical Plate under Oscillatory Suction Velocity, IE(I) Journal-MC, 206 -212, 2005.

6. Okedoye, A. M. and Bello, O. A., MHD Flow of a Uniformly Stretched Vertical Permeable Surface under Oscillatory Suction Velocity, J. of the Nigerian Association of Mathematical Physics vol. 13, 211-220, 20008.

[9]. Massoudi, M. and Phuoc, T. X., Flow of a Generalized Second Grade Non-Nowtonian Fluid with Variable Viscosity, Continum Mech. Thermodyn. 16, 529-538, 2004.