Aligned magnetic field, radiation and chemical reaction effects on unsteady dusty viscous flow with...

Chemical and Process Engineering Research www.iiste.org

ISSN 2224-7467 (Paper) ISSN 2225-0913 (Online)

Vol.27, 2014

37

Aligned Magnetic Field, Radiation and Chemical Reaction Effects

on Unsteady Dusty Viscous Flow with Heat

Generation/Absorption

J.V. Ramana Reddy1 Dr.V.Sugunamma

2* P.Mohan Krishna

3 Dr.N.Sandeep

4

1,3Research Scholars, Dept.of Mathematics, S.V.University, Tirupati, India.

2Associate Professor, Dept.of Mathematics, S.V.University, Tirupati, India.

4Assistant Professor, Fluid Dynamics Division, VIT University, India.

Abstract

We analysed the laminar convective flow of a dusty viscous fluid of non conducting walls in

presence of aligned magnetic field with volume fraction, radiation, heat absorption along with

chemical reaction. The governing equations of the flow are solved by Perturbation

Technique. Further, the effects of all physical parameters on the velocities of fluid phase and

dust phase, temperature and concentration are analysed and discussed through graphs.

Key Words: Dusty Fluid, Laminar flow, MHD, Chemical Reaction, Viscous flow.

1. Introduction

The Dusty fluid is a mixture of fluid and fine dust particles .The influence of dust

particles on convective flow of dusty viscous fluids in presence of magnetic field and

chemical reaction has its importance in many areas like environmental pollution, cooling

effects of air conditioners, magneto hydrodynamic generators, pumps, accelerators and flow

meters .This type of flow has uses in nuclear reactors, geothermal systems and filtration. The

possible presence of dust particles in combustion MHD generators and their effect on

performance of such devices leads to study of volume fraction of dust particles in non

conducting walls in the presence of aligned magnetic field.

The study of convective flow of dusty viscous fluid under the influence of different

physical conditions has been carried out by many researchers. Saffman [1] has discussed the

stability of laminar flow of a dusty gas. Ezzat et al. [2] studied space approach to the hydro

magnetic flow of a dusty fluid through a porous medium by using Laplace transformation

technique. Sandeep and Sugunamma [3] discussed the effect of inclined magnetic field on

unsteady free convection flow of a dusty viscous fluid between two infinite flat plates filled

by a porous medium. Chakrabarti [4] analysed the boundary layer in a dusty gas. Datta and

Mishra [5] have investigated the boundary layer flow of a dust fluid over a semi infinite flat

plate .Mohan Krishna et al. [6] studied the Magnetic field and chemical reaction effects on

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38

convective flow of a dusty viscous fluid. In this study they used transverse magnetic field.

Anurag Dubey and Singh [7] discussed effect of dusty viscous fluid on unsteady laminar free

convective flow through porous media with thermal diffusion. Sandeep et al. [8] analysed the

effect of radiation and chemical reaction on transient MHD free convective flow over a

vertical plate through porous media .Mishra et al [9] have studied the two-dimensional

transient conduction and radiation heat transfer with temperature dependent thermal

conductivity. Attia [10] studied the unsteady couettee flow with heat transfer on dusty fluid

with variable physical properties.

Some researchers like Anjali Devi and Jothimani [11] have discussed the heat transfer

in unsteady MHD oscillatory flow. Further, Malashetty et al. [12] have investigated the

convective magnetohydrodynamic two phase flow and heat transfer of a fluid in an inclined

channel. Palani and Ganesan [13] have discussed the heat transfer effects on dusty gas flow past

a semi infinite inclined plate. Ibrahimsaidu at al. [14] analysed the MHD effects on convective

flow of dusty viscous fluid with volume fraction of dust particles in the absence of aligned

magnetic field, radiation, heat absorption and chemical reactions. In continuation of this

study and with the help of above cited papers we have studied the laminar convective flow of

a dusty viscous fluid with non conducting walls in the presence of aligned magnetic field

with volume fraction, radiation, heat absorption along with chemical reaction. The governing

equations of the flow are solved by Perturbation Technique. Further we analysed effects of all

physical parameters on the fluid phase and dust particles phase.

2. Mathematical Formulation

Consider an unsteady laminar flow of a dusty, incompressible, Newtonian, electrically

conducting, viscous fluid of uniform cross section h , when one wall of the channel is fixed

and the other is oscillating with time about a constant non-zero mean. Initially at 0t , the

channel wall as well as the fluid is assumed to be at the same temperature 0T and

concentration0C . When t>0 , the temperature of the channel wall is instantaneously raised to

wT and concentration raised to wC which oscillates with time and is thereafter maintained

constant. Let the fluid flow is along the x- axis at the fixed wall and y- axis is perpendicular

to it. The aligned magnetic field is applied to the flow along y>0direction with the first order

chemical reaction. Here the dust particles are solid, spherical, non-conducting, and equal in

size and uniformly distributed in the flow region. The density of dust particles is constant and

the temperature between the particles is uniform throughout the motion. The interaction





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between the particles, chemical reaction between the particles and liquid has been considered.

The volume occupied by the particles per unit volume of the mixture, (i.e., volume fraction of

dust particles) and mass concentration have been taken into consideration.

The governing equations of the flow are given by

2*

0 02

1(1 ) (1 ) ( ) (C )

u p ug T T g C

t x y

2 2

20 0 0( ) sincKN KN Hv u u

(1)

2*

0 0 0 02( ) (C ) ( )

v p uN m g T T g C KN u v

t x y

(2)

2

02

1(T )r

p p p

qT k T QT

t C y C y C

(3)

2

02(C )l

C CD K C

t y

(4)

The boundary conditions of the problem are given by

int int

0; ( , ) v(y, t) 0, T(y, t) C(y, t) 0 for 0 y 1

0; ( , ) v(y, t) 0, T(y, t) C(y, t) 0 0

( , ) v(y, t) 1 , T(y, t) C(y, t) 1 1

t u y t

t u y t at y

u y t e e at y

(5)

Where u(y,t) is the velocity of the fluid and v(y,t) is velocity of the dust particles, m is the

mass of each dust particle, 0N is the number density of the dust particle, T is the

temperature, 0T is the initial temperature,

wT is the raised temperature, C is the

concentration, 0C is the initial concentration,

wC is the raised concentration, is the volume

fraction of the dust particle, f is mass concentration of dust particle, is the volumetric

coefficient of the thermal expansion, K is the Stoke’s resistance coefficient, is the

electrical conductivity of the fluid, C is the magnetic permeability,

0H is the magnetic field

induction, is the aligned magnetic field angle, pC is the specific heat at constant pressure,

k is the thermal conductivity, K l is chemical reaction parameter,

1K is dimensionless

chemical reaction parameter.

The Problem is simplified by writing the equations in the following non dimensional form.

Here the characteristic length is taken to be h and characteristic velocity is v .





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2* * * * * * * *0 0

2 2

0 0

, , , , , , ,w w

T T C Cx y h p t uh vhx y p t u v T C

h h h T T C C

(6)

Substituting the above non dimensional parameters of equation (6) in the governing equations

(1) – (4) then we get (after removing asterisks)

2

1 22( )

u p uGrT GcC v u Mu

t x y

(7)

2

2( )

v p uf GrT GcC u v

t x y

(8)

2

2

1(1 )

PrH

T TR Q T

t y

(9)

2

12

1C CK C

t Sc y

(10)

Where

3 * 3 2

0 0 01 2 12 2 2

1

23 *2 2 2 2 0

0 1*

1

( ) ( ), , , , , ,

(1 ) 1

16sin , ,Pr , , .

3

w wH

lc

p

g T T h g C C h KNf m QhGr Gc Q

Kh k

mN K hT f kR M h H f Sc K

kk C D

Here Gr is Thermal Garshof number and Gc is Mass Garshof number, M is Magnetic

parameter, f is Mass concentration of dust particles, is Concentration resistance ratio, Pr

is Prandtal number, Sc is (Schmidt number), K l is Chemical reaction parameter.

The Corresponding non-dimensional boundary conditions are:

int int

0; ( , ) v(y, t) 0, T(y, t) C(y, t) 0 for 0 y 1

0; ( , ) v(y, t) 0, T(y, t) C(y, t) 0 0

( , ) v(y, t) 1 , T(y, t) C(y, t) 1 1

t u y t

t u y t at y

u y t e e at y

(11)

3. Solution of the Problem

To solve the equations (7-10) we use the below equations introduced by Soundalgekar and

Bhat





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.

int

0 1

int

0 1

int

0 1

int

0 1

( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

u y t u y e u y

v y t v y e v y

T y t T y e T y

C y t C y e C y

(12)

pp

x

is constant

After substituting equations (12) in equations (7) – (10), we obtain

0 1 2 0 1 0 0 0( ) ( ) ( ) ( ) ( ) ( )u y M u y v y p GrT y GcC y (13)

1 1 2 1 1 1 1 1( ) ( ) ( ) ( ) ( ) ( )u y M in u y v y GrT y GcC y (14)

0 0 0 0 0( ) ( ) ( ) ( ) ( )v y u y u y p GrT y GcC y

(15)

1 1 1 1 1( inf) ( ) ( ) ( ) ( ) ( )v y u y u y GrT y GcC y

(16)

0 0(1 ) ( ) ( ) 0HR T y Q T y (17)

1 1(1 ) ( ) Pr ( ) 0HR T y Q in T y (18)

0 1 0( ) K ( ) 0C y Sc C y (19)

1 1 1( ) ( ) 0C y Sc K in C y (20)

The corresponding boundary conditions becomes

0 1 0 1 0 1 0 1

0 1 0 1 0 1 0 1

( ) ( ) ( ) ( ) 0, ( ) ( ) ( ) ( ) 0 0

( ) ( ) ( ) ( ) 1, ( ) ( ) ( ) ( ) 1 1

u y u y v y v y T y T y C y C y at y

u y u y v y v y T y T y C y C y at y

(21)

On solving equation (17) and (19) with the help of boundary conditions (21), we get

20

2

sin L( )

sin L

yT y (22)

00

0

sin( )

sin

h L yC y

h L (23)

Substituting the equations (22) and (23) in equations (13) and (15), we get

020 1 2 0 1 0

2 0

sinsin L( ) ( ) ( ) ( )

sin L sin

h L yyu y M u y v y p Gr Gc

h L (24)

020 0 0

2 0

sinsin L( ) ( ) ( )

sin L sin

h L yyv y u y u y p Gr Gc

h L

(25)

Substituting equation (25) in (24), we obtain





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2 020 0

2 0

sinsin L( ) ( )

sin L sin

h L yyu y A u y p Gr Gc

h L (26)

Where

2 2

1

MA

By solving equation (26) with the help of boundary conditions (21), we get

020 12 2 2 2 2

2 2 0 0

sinsin Lsin( ) (cos 1)

sin sin L sin

h L yyp hAy Gr Gcu y hAy B

A hA L A L A h L

(27)

The first and second order partial derivatives of 0 ( )u y are given by

020 12 2 2 2 2

2 2 0 0

coscos Lcos( ) (sin )

sin sin L sin


A hA L A L A h L

(28)

020 12 2 2 2 2

2 2 0 0

sinsin Lsin( ) (cos )

sin sin L sin


A hA L A L A h L

(29)

Substituting the above equations (27) and (29) in equation (25), we obtain

0 20 2 1 32 2 2 2 2

0 0 2 2

024 5

2 0

sin sin Lsin( ) (cos )

sin sin sin L

sinsin L

sin L sin

h L y yp hAy Gc Grv y B hAy B B

A hA L A h L L A

h L yyB p B Gr Gc

h L

(30)

By solving equations (18) and (20) with the boundary conditions (21), we obtain

31

3

sin( )

sin

L yT y

L (31)

11

1

sin( )

sin

hL yC y

hL (32)

Substituting equations (31) and (32) in equations (14) and (16), we obtain

3 11 1 2 1 1 1

3 1

sin sin( ) ( ) ( ) ( )

sin sin

L y hL yu y M in u y v y Gr Gc

L hL (33)

3 11 1 1

3 1

sin sin( inf) ( ) ( ) ( )

sin sin

L y hL yv y u y u y Gr Gc

L hL

(34)

Substituting equation (34) in equation (33), we obtain

2 3 11 1 1 1

3 1

sin sin( ) ( ) ( )

sin sin

L y hL yu y B u y v y Gr Gc

L hL (35)

On solving equation (35), with the help of boundary conditions (21), we get





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3 11 6 2 2 2 2

3 3 1 1

sin sinsin( )

sin sin sin

L y hL yhBy Gr Gcu y B

hB L B L L B hL

(36)

The first and second order partial derivatives of 1( )u y are

3 11 6 2 2 2 2

3 3 1 1

cos coscos( )

sin sin sin


hB L B L L B hL

(37)

3 11 6 2 2 2 2

3 3 1 1

sin sinsin( )

sin sin sin


hB L B L L B hL

(38)

Substituting the above equations (36) and (38) in equation (34), we obtain

311 7 6 82 2 2 2

1 1 3 3

3 19

3 1

sinsinsin( )

sin sin sin

sin sin

sin sin

L yhL yhBy Gc Grv y B B B

hB L B hL L B L

L y hL yB Gr Gc

L hL

(39)

Substituting the equations (27) and (36) in equation (12), we obtain the expression for

velocity of the fluid phase as

0212 2 2 2 2

2 2 0 0

int3 16 2 2 2 2

3 3 1 1

sinsin Lsin( , ) (cos 1)

sin sin L sin

sin sinsin

sin sin sin

h L yyp hAy Gr Gcu y t hAy B

A hA L A L A h L

L y hL yhBy Gr GcB e

hB L B L L B hL

(40)

Substituting the equations (30) and (39) in equation (12), we obtain expression for the

dust phase as

02 12 2 2

0 0

02 23 4 52 2

2 2 2 0

317 6 82 2 2 2

1 1 3 3

9

sinsin( , ) (cos )

sin sin

sinsin L sin L

sin L sin L sin

sinsinsin

sin sin sin

si

h L yp hAy Gcv y t B hAy B

A hA L A h L

h L yy yGrB B p B Gr Gc

L A h L

L yhL yhBy Gc GrB B B

hB L B hL L B L

B Gr

int

3 1

3 1

n sin

sin sin

eL y hL y

GcL hL

(41)

Substituting the equations (22) and (31) in equation (12), we obtain the expression for

temperature as

int32

2 3

sinsin L( , )

sin L sin

L yyT y t e

L (42)





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Substituting the equations (23) and (32) in equation (12), we get we obtain the expression

for concentration as

int0 1

0 1

sin sin( , )

sin sin

h L y hL yC y t e

h L hL (43)

4. Results and Discussion

In order to study the behavior of fluid velocity ( )u ,dusty velocity ( )v , temperature

( )T and concentration ( )C fields, a comprehensive numerical computation is carried out for

various values of the parameters that describe the flow characteristics, and the results are

reported in terms of graphs as shown in Figures (1) – (15).

The variation of fluid velocity for different values of aligned magnetic field angle ( )

is shown in Figure 1. It is observed that the velocity profiles of fluid phase and dust phase

decreases with an increase in aligned magnetic field angle. In general increase in magnetic

field causes the decrease in fluid velocity, because of induced forces acting opposite to flow.

In present case an increase in aligned angle causes the increase in magnetic field .Therefore

our results are exactly coincident with transverse magnetic field case at π/2 which is

clearly shown in Fig. 2. Fig 3 depicts the increase in radiation parameter causes the decrease

in fluid and dust phase velocities. Fig.4 shows the effect of heat generation/absorption

parameter on velocity profiles of fluid and dust phase. It is observed that an increase in heat

generation/absorption parameter causes the increase in velocities of the fluid and dust phase.

From Fig.5 it is interesting to note that an increase in chemical reaction parameter causes the

increase in velocity of fluid phase but due to chemical reaction with dust particles the dust

phase velocity decreases initially and then follows the fluid phase.

Velocity profiles for different values of mass Grashof number ( )Gc and thermal

Grashof number ( )Gr are shown in figures 6 and 7 respectively. It is evident that an increase

in mass Grashof number decreases the velocity of the fluid phase, but it is reversed in dust

phase. But in case of thermal Grashof number the above results are completely differs. The

effect on velocity profiles for different values of Prandtl number (Pr) are shown in figure 8.It

is evident that an increase in Prandtl number causes decrease in fluid phase velocity. But it

helps to the dust phase velocity. Figure 9 represents velocity profiles for different values of

Schmidt number ( )Sc . It is clear that the velocity of fluid and dust phases increases with an

increase in Schmidt number. From fig.10 it is clear that the increase in volume fraction of the

dust particles increases the velocity of the fluid and dust phase.





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The effect of radiation parameter ( )R on temperature profiles is shown in fig.11 and it

is clear that an increase in radiation parameter causes the decrease in dusty fluid

temperature.Fig.12 depicts the variation in temperature for different values of heat

generation/absorption parameter and it is observed that increase in heat generation/absorption

parameter causes the increase in fluid temperature. From figure 13 it is evident that fluid

temperature decreases with an increase in time.

The variations of concentration profiles for different values of the Schmidt

number ( )Sc and chemical reaction parameter1( )K are shown in Figs. 14 and 15 respectively.

It is observed that the concentration decreases gradually with increase in Schmidt number

( )Sc as well as chemical reaction parameter1( )K .

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

y

u

=pi/6

=pi/4

=pi/3

=pi/2

0 0.5 1-2

0

2

4

6

8

10

12

y

v

=pi/6

=pi/4

=pi/3

=pi/2

Figure 1: velocity profiles for different values of

When Pr =0.71,Gr =5,Gc =5, R =2,1M =8, Sc =2,Q =2,

1K =0.5, t =0.1.





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0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

y

u

M=2

M=4

M=6

M=8

0 0.5 1-2

0

2

4

6

8

10

12

y

v

M=2

M=4

M=6

M=8

Figure 2: velocity profiles for different values of M

When Pr =0.71,Gr =5,Gc =5, R =2, Sc =2,Q =2,1K =0.5, t =0.1.

0 0.5 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y

u

R=0.5

R=2

R=4

R=8

0 0.5 10

1

2

3

4

5

6

y

v

R=0.5

R=2

R=4

R=8

Figure 3: velocity profiles for different values of R

When Pr =0.71,Gr =5,Gc =5, Sc =2,Q =2,1M =3,

1K =0.5, t =0.1, =π/6





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0 0.5 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

y

u

Q=2

Q=4

Q=6

Q=8

0 0.5 10

1

2

3

4

5

6

7

y

v

Q=2

Q=4

Q=6

Q=8

Figure 4: velocity profiles for different values of Q

When Pr =0.71,Gr =5,Gc =5, Sc =2, R =2,1M =3,

1K =0.5, t =0.1, =π/6

0 0.5 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

y

u

K1=1

K1=2

K1=3

K1=4

0 0.5 10

1

2

3

4

5

6

y

v

K1=1

K1=2

K1=3

K1=4

Figure 5: velocity profiles for different values of 1K

When Pr =0.71,Gr =5,Gc =5, Sc =2, R =2,1M =3,Q =2, t =0.1, =π/6





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48

0 0.5 1-1

-0.5

0

0.5

1

1.5

y

u

Gc=2

Gc=4

Gc=6

Gc=8

0 0.5 10

1

2

3

4

5

6

7

8

y

v

Gc=2

Gc=4

Gc=6

Gc=8

Figure 6: velocity profiles for different values of Gc

When Pr =0.71,Gr =5,1K =0.5, Sc =2, R =2,

1M =3,Q =2, t =0.1, =π/6

0 0.5 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

y

u

Gr=4

Gr=8

Gr=12

Gr=16

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

y

v

Gr=4

Gr=8

Gr=12

Gr=16

Figure 7: velocity profiles for different values of Gr

When Pr =0.71,Gc =5,1K =0.5, Sc =2, R =2,

1M =3,Q =2, t =0.1, =π/6





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0 0.5 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y

u

Pr=0.71

Pr=1.0

Pr=7.0

Pr=11.4

0 0.5 10

1

2

3

4

5

6

7

y

v

Pr=0.71

Pr=1.0

Pr=7.0

Pr=11.4

Figure 8: velocity profiles for different values of Pr

When Gr =5,Gc =5,1K =0.5, Sc =2, R =2,

1M =3,Q =2, t =0.1, =π/6

0 0.5 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

y

u

Sc=2

Sc=2.5

Sc=3

Sc=4

0 0.5 10

1

2

3

4

5

6

y

v

Sc=2

Sc=2.5

Sc=3

Sc=4

Figure 9: velocity profiles for different values of Sc

When Pr =0.71,Gr =5,Gc =5,1K =0.5, R =2,

1M =3,Q =2, t =0.1, =π/6





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0 0.5 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

y

u

=0.2

=0.4

=0.6

=0.8

0 0.5 1-1

0

1

2

3

4

5

6

7

8

9

y

v

=0.2

=0.4

=0.6

=0.8

Figure 10: velocity profiles for different values of

When Pr =0.71,Gr =5,Gc =5,1K =0.5, R =2,

1M =3,Q =2, t =0.1, =π/6, Sc =2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

y

T

R=0.5

R=2

R=4

R=6

Figure 11: Temperature profiles

for different values of R .When

Pr =0.71,Q =2, t =0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

y

T

Q=2

Q=4

Q=6

Q=8

Figure 12: Temperature profiles for

different values of Q .When

Pr =0.71, R =2, t =0.1





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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

y

T

t=0.4

t=0.8

t=1.2

t=1.6

Figure 13: Temperature profiles for

different values of t .When

Pr =0.71,Q =2, R =2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

y

C

Sc=2

Sc=2.5

Sc=3

Sc=4

Figure 14: Concentration profiles for different

values of Sc .When K =2,1K =0.5, t =0.1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

y

C

K1=1

K1=2

K1=3

K1=4

Figure 15: Concentration profiles for

different values of 1K .When

K =2, Sc =2, t =0.1.

Appendix:

0 1 1 1

2 3

2 2

1

2 11 2

1

1 2 2 2 2 2

2 0

2 3 4 2

5 6 2

3

, ,

Pr,

1 1

inB

in in

1 (1 cos hA)

11 , 1 ,

, 1

H H

L K Sc L K in Sc

Q Q inL L

R R

MA

fM in

f f

p Gr GcB

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Vol.27, 2014

52

References

[1] Saffman P .G, 1962. On the stability of laminar flow of a dusty gas. Journal of Fluid

dynamics, 13,120-128.

[2] Ezzat M. A, A .A El-Bary, M. M Morsey,20120. Space approach to the hydro magnetic

flow of a dusty fluid through a porous medium. Computers and Mathematics with

Applications. 59, 2868-2879.

[3] Sandeep N, V .Sugunamma. 2013. Effect of inclined magnetic field on unsteady free

convection flow of a dusty viscous fluid between two infinite flat plates filled by a porous

medium.Journal of Applied Mathematics and modelling.1, 1-9.

[4] Chakrabarti K .M,1974. Note on boundary layer in a dusty gas.AAIA Journal. 12, 1136-

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[5] Datta N and S. K Mishra, 1982. Boundary layer flow of a dust fluid over a semi infinite

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[6] Mohan Krishna P, V. Sugunamma and N. Sandeep,2013 . Magnetic field and chemical

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Sciences. 1,161-187.

[7] Anurag Dubey and U. R Singh,2012. Effect of dusty viscous fluid on unsteady laminar

free convective flow through porous medium along a moving porous hot vertical with

thermal diffusion. Applied Mathematical Sciences. 6, 6109-6124.

[8] Sandeep N, A. V .B Reddy, V. Sugunamma,2012. Effect of radiation and chemical

reaction on transient MHD free convective flow over a vertical plate through porous media.

Chemical and process engineering Research.,2,1-9.

[9] Mishra S.C, P. T Alukdhar, D. Trimas and F.Drust,2005.Two-dimensional transient

conduction and radiation heat transfer with temperature dependent thermal conductivity.

Int.com Heat and Mass transfer. 32,305-314.

[10] Attia H.A,2006.Unsteady MHD couettee flow and heat transfer of dusty fluid with

variable physical properties. Applied Mathematics and computation. 177,308-318.

[11]. Anjali Devi S.P and S. Jothimani,1996. Heat transfer in unsteady MHD oscillatory

flow, Czechoslovak Journal of Physics. 46, 825–838.

[12]. Malashetty M.S, J.C. Umavathi and Prathap Kumar,2001. Convective

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[13]. Palani G and P .Ganesan,2007. Heat transfer effects on dusty gas flow past a semi-

infinite inclined plate. Forschung im Ingenieurwesen.71, 223–230.

[14] Ibrahim Saidu, M .M Waziri, Abubakar Roko and Hamisu Musa,2010. MHD effects on

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