Heat Generation, Thermal Radiation and Chemical Reaction Effects on MHD Mixed Convection Flow over an Unsteady Stretching Permeable Surface

7/29/2019 Heat Generation, Thermal Radiation and Chemical Reaction Effects on MHD Mixed Convection Flow over an Unst

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Heat Generation, Thermal Radiation and Chemical Reaction Effects on MHD Mixed Convection Flow over an Unsteady Stretching PermeableSurface

Insan Akademika

Publications

INTERNATIONAL JOURNAL

OF BASIC AND APPLIED SCIENCE

P-ISSN: 2301-4458

E-ISSN: 2301-8038

Vol. 01, No. 02

Oct 2012www.insikapub.com

363

Heat Generation, Thermal Radiation and Chemical Reaction Effects on MHD

Mixed Convection Flow over an Unsteady Stretching Permeable Surface

Md. Shakhaoath Khan1, Ifsana Karim

2, and Md. Haider Ali Biswas

3

1,2,3Mathematics Discipline; Science Engineering and Technology School

Khulna University, Khulna-9208, Bangladesh

[email protected], [email protected], [email protected]

Key words Abstract

Magnetohydrodynamics;

Thermal radiation;

Heat generation;

Chemical reaction;

Mixed Convective flow;

Unsteady flow;

Stretching permeable

surface

The unsteady MHD mixed convective laminar boundary layer flow of an

incompressible viscous fluid over continuously stretching permeable surface in the

presence of thermal radiation, heat generation and chemical reaction is studied.

The unsteadiness in the momentum, temperature and concentration fields is

because of the time-dependent stretching velocity and surface temperature and

concentration. Similarity transformations are used to convert the governing time

dependent boundary layer equations into to a system of nonlinear ordinary

coupled differential equations containing Magnetic parameter, Thermalconvective parameter, Mass convective parameter, Suction parameter, Radiation

parameter, Eckert number, Prandtl number, Heat source parameter, Chemical

reaction parameter, Schmidt number, Soret number and Unsteadiness parameter.

TheNactsheim-Swigert shooting technique together with Runge-Kutta six order

iteration schemes has been used for numerical procedure. Comparisons with

previously published work are performed and are found to be in excellent

agreement. The effects on the velocity, temperature and concentration

distributions as well as skin-friction coefficients, Nusselt number and Sherwood

number of the various important parameters are discussed graphically.

2012 Insan Akademika All Rights Reserved

1 Introduction

The study of convective heat and mass transfer fluid flow over stretching surface in the presence of thermal

radiation, heat generation and chemical reaction is gaining a lot of attention. This study has many

applications in industries, many engineering disciplines. These flows occur in many manufacturing processes

in modern industry, such as hot rolling, hot extrusion, wire drawing and continuous casting. For example, in

many metallurgical processes such as drawing of continuous filaments through quiescent fluids and

annealing and tinning of copper wires, the properties of the end product depends greatly on the rare of

cooling involved in these processes. Sakiadis (1961) was the first one to analyze the boundary layer flow on

continuous surfaces. After that, Crane (1970) studied the boundary layer flow past a stretching plate. A few

researchers give attention to consider the unsteady flows over stretching surface. Wang (1990) studiedunsteady boundary layer flow of a finite liquid film by restricting the motion to a specified family of time


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Khan et al.

364 Insan Akademika Publications

dependence. Andersson et al. (1996) investigated the unsteady stretching flow in the case of power-law fluid

film whereas Andersson et al. (2000) extended Wangs unsteady thin film stretching problem to the case of

heat transfer. Recently, Ishaket al. (2009) presented the heat transfer characteristics caused by an unsteady

stretching permeable surface with prescribed wall temperature. Sharidan et al. (2006) analyzed a similarity

analysis to investigate the unsteady boundary layer over a stretching sheet. Wang (2009) studied viscousflow due to stretching sheet with surface slip and suction.

The thermal radiation and heat generation effects on MHD convective flow is new dimension added to the

study of stretching surface has important applications in physics and engineering particularly in space

technology and high temperature processes such as it plays an important role in controlling the heat transfer

process in polymer processing industry. The effect of radiation on heat transfer problems have been studied

by Hossain and Takhar (1996), Takhar et al. (1996). Seddeek (2002) analyzed the effects of radiation and

variable viscosity on a MHD free convection flow past a semi-infinite flat plate with an aligned magnetic

field. In many chemical engineering processes, chemical reactions take place between a foreign mass and the

working fluid which moves due to the stretch of a surface. Kandasamy et al. (2006) analyzed effects of

chemical reaction, heat and mass transfer on boundary layer flow over a porous wedge with heat radiation in

the presence of suction or injection. Muhaimin et al. (2009) studied the effect of chemical reaction, heat andmass transfer on nonlinear MHD boundary layer past a porous shrinking sheet with suction. Rajesh (2011)

investigates chemical reaction and radiation effects on the transient MHD free convection flow of dissipative

fluid past an infinite vertical porous plate with ramped wall temperature.

In the view of the above discussions the aim of the present study is to analyze the effects of heat generation,

thermal radiation and chemical reaction on MHD mixed convective boundary layer flow of an

incompressible viscous fluid over an unsteady permeable stretching surface in presence of suction/ injection.

The conservation equations of mass, momentum, energy and concentration were transformed into a two-

point boundary value problem. These nonlinear equations along with the appropriate boundary conditions are

then solved by employing Nactsheim-Swigert shooting technique together with Runge-Kutta six order

iteration schemes (1965). Comparisons with previously published works of Grubka and Bobba (1985), Ishak

et al. (2009) and Dulal Pal (2011) are performed and excellent agreement between the results is obtained.

2 Mathematical Formulation

An unsteady two dimensional MHD mixed convective laminar boundary layer flow of a viscous

incompressible and electrically conducting fluid over a stretching permeable surface under the influence of

thermal radiation, heat generation and chemical reaction which issues from a thin slot is considered. The

physical configuration has shown in Figure 1.

Fig. 1: Physical configuration and coordinate system


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The unsteadiness in the momentum, temperature and concentration fields is because of the time-dependent

stretching velocity ( ),wu x t and surface temperature ( ),wT x t and concentration ( ),wC x t .The x -axis is taken

along the stretching surface in the direction of the motion with the slot as the origin, and the y-axis is

perpendicular to the sheet in the outward direction towards the fluid of ambient temperature T . The flow isassumed to be confined in a regiony > 0. We assume that the velocity is proportional to its distance from the

slit. A strong magnetic field is applied in the y- direction and the uniform magnetic field strength (magnetic

induction)0B can be taken as ( )00, ,0 .B B= Under these assumptions along with the boundary layer

approximations and considering the viscous dissipation, the governing boundary layer equations for

momentum, heat and mass transfer in the presence of thermal radiation, heat generation and chemical

reaction take the following form of the governing equations is given by:

u v0

x y

+ =

...(1)

*( ) ( )2

0

u u u uu v g T T g C C B u

t x y y y

+ + = + +

...(2)

( )

22

r

2

p p p p

Q qT T T k T 1 uu v T T

t x y C y C C y c y

+ + = + +

o

...(3)

( )2 2

m Tm r2 2

m

D kC C C C T u v D k C C

t x y y T y

+ + = +

...(4)

And the boundary condition for the model is:

( ) ( ) ( ), , , , , ,w w w wu u x t v v T T x t C C x t = = = = aty 0=

...(5)

, ,u 0 T T C C

= as y

...(6)

where x and y represent coordinate axes along the continuous surface in the direction of motion and

perpendicular to it, respectively. u and v are the velocity components along x andy directions, respectively

and tis the time. pc is the specific heat at constant pressure, kis the thermal conductivity of the fluid, v is the

kinematic viscosity, is the dynamic viscosity and is density of fluid, Qo heat generation constant, mD is

the coefficient of mass diffusivity, T is the thermal diffusion ratio, mT is the mean fluid temperature and

rK is the rate of chemical reaction. ( ) ( ), and ,w wT T x t C C x t = = temperature and concentration respectively of

the stretching surface, andT C

is the temperature and concentration respectively far away from the

stretching surface with and .w w

T T C C

> > The term ( ) ( )1

2wu

xwv f 0

= represents the mass transfer at the

surface withw

v 0> for injection andw

v 0< for suction. The flow is caused by the stretching of the sheet

which moves in its own plane with the surface velocity ( , ) / ( )w

u x t ax 1 ct = , where a (stretching rate) and c are

positive constants having dimension time-1

(with ,ct 1 c 0< ). It is noted that the stretching rate / ( )a 1 ct

increases with time since a> 0. The surface temperature and concentration of the sheet varies with the

distancex from the slot and time tin the form:


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andw wbx bx

T T C C 1 ct 1 ct

= + = +

...(7)

where b is constant with b 0 . It should be noted that when t = 0 (initial motion), equations (1)-(4) describe

the case of steady flow over a stretching sheet. The particular form of ( ) ( ) ( ), , , and ,w w wu x t T T x t C C x t = =

presented in this paper has been chosen in order to devise a similarity transformation (Ishak et al.(2009)),

which transform the governing partial differential equations (1)-(4) into a set of highly nonlinear ordinary

differential equations.

The Rosseland approximation (Rohsenow et al., 1998)isexpressed for radiative heat flux and leads to the

form as:

4

*

4

3r

Tq

y

=

...(8)

Where s is the Stefan-Boltzmann constant and *k is the mean absorption coefficient. The temperature

difference with in the flow is sufficiently small such that 4T may be expressed as a linear function of the

temperature, then the Taylors series for 4T about T

after neglecting higher order terms:

4 3 44 3 .T T T

= ...(9)

In order to attains a similarity solution to equations (1) to (4) with the boundary conditions (5) and (6) the

following dimensionless variables are used:

( )

( )

( )

, , , ( )

1 1

( ) , ( )w w

a ay x y t xf

ct ct

T T C C

T T C C

= =

= = = = ...(10)

Therefore,

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

bx bxT x y t T C x y t C

ct ct

= + = +

...(11)

where ( ), ,x y t is the stream function defined by:

( )/( ), ( ).

1 1

ax au f v f

y ct x ct

= = = =

...(12)

Which automatically satisfies the continuity Eq. (1). It must be noted that expression (11) on which the

analysis is based are valid only for ( )since1t c ct 1< < . Now from the above transformations the non

dimensional, nonlinear, coupled ordinary differential equations are obtained as:

//// / // / / /

2

T M

ff ff f f Mf 0

2

+ + + + =

...(13)


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( )/

/ / / / / /

1 2

r r c1 N P f f Q E f 0

2

+ + + + + =

...(14)

// / / / / /

02 x

c c c r c eS f S f S S S R + + + =

...(15)

where the notation primes denote differentiation with respect to and the parameters are defined as ca

= is

the parameter that measures the unsteadiness,2

w

B xM

u

o

= is the Magnetic Parameter,

2 2

( )r wT

e w

G g x T T

R u

= = is

the Thermal convective parameter,*

( )m wM 2 2

e w

G g x C C

R u

= = is the Mass convective parameter,

rp

= is the

Prandtl number,( )

2

wc

p w

uE

c T T

=

is the Eckert number,*

*

3

r

16 TN

3 K k

= is the Radiation parameter, ,p w

Q xQ

C u

o

= is

the Heat source parameter,c

m

SD

= is the Schmidt number,

( )

( )m T w

r

m w

D T TS

T C C

=

is the Soret number and

2

r

w

k

u

=

is the Chemical reaction parameter.

The transformed boundary conditions:

/

/

f f , f 1, 1, 1, at 0

f 0 0, 0, as

= = = = =

= = =

o

...(16)

where ( )f 0 f with f 0 and f 0= < >o o o

corresponding to injection and suction respectively.

Physical significance for this type of flow and heat transfer situation are the local skin-friction coefficient,

local Nusselt number and the local Sherwood number can be defined in dimensionless form as:

( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1

/ / / /2 2 20 , 1 0 and 0x x xf e u e r h e

C R f N R N S R

= = + = ...(17)

where( )

2

1xe

axR

ct=

is the Local Reynolds number based on the surface velocity.

It is to be noted that the present problem reduces to steady-state flow for 0 = in absence of Magnetic

parameter, Thermal convective parameter, Mass convective parameter, Radiation parameter, Eckert number,

Schmidt number, Soret number, Heat source parameter and Chemical reaction parameter then the closed-

form solutions for flow and thermal fields in terms of Kummers functions are respectively given by Ishaket

al. (2009);

( ) ( )( )

( )

2

2

1, 1, / 1,

1, 1, /

r r r

r r r

M P P Pef e

M P P P

+

= =+

...(18)

Where ( ), ,M a b z denotes the confluent hypergeometric function according to Abramowitz and Stegun(1965).

Now using equation (16) we have from equation (18) as; ( ) ( )1

0 with 0 0 1 and 1f fo

= = > < < >


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correspond to injection and suction, respectively. Also the skin friction coefficient ( )// 0f and local Nusselt

number ( )/ 0 are given by:

( ) ( ) ( )( )

2

/ / /

21, 1, / 0 ,1, 1, /

r r r

r

r r r

M P P Pf PM P P P

+ = = +

...(19)

3 Numerical Simulation

The non dimensional, nonlinear coupled ordinary differential equations (13) to (15) with boundary condition

(16) are solved numerically using standard initially value solver the shooting method. For the purpose of this

method, the Nactsheim-Swigert shooting iteration technique together with Runge-Kutta six order iteration

scheme is taken and determines the temperature and concentration as a function of the coordinate .

Extension of the iteration shell to above equation system of differential equation (16) is straightforward, there

are three asymptotic boundary condition and hence three unknown surface conditions // /( ), ( )f 0 0 and /( )0 .

4 Result and Discussions

The problem considering for MHD mixed convection fluid flow over an unsteady stretching permeable

surface with thermal radiation, heat generation and chemical reaction. The numerical values of velocity ( )/f ,

temperature ( ) and concentration ( ) due to steady ( )0 = and unsteady case ( )0 with the boundary

layer have been computed for different parameters as the Magnetic parameter ( )M , Thermal convective

parameter ( )T , Mass convective parameter ( )M , Suction parameter ( ) , Radiation parameter ( )rN , Eckert

number ( )cE , Prandtl number ( )rP , Heat source parameter ( )Q , Schmidt number ( )cS , Soret number ( )rS and

Chemical reaction parameter ( ) .The Unsteadiness parameter value 0.0 = is the steady-state solution. Also

the Skin-friction coefficient, Surface heat and mass transfer rates are plotted against Unsteadiness parameter

( ).

In order to assess the accuracy of the numerical results as the results for the reduced Nusselt number ( )/ 0

for different values of Suction parameter ( ) , Prandtl number ( )rP and Unsteadiness parameter ( ) the present

results compared with Grubka and Bobba (1985), Ishaket al.(2009) and Dulal Pal (2011). Comparison with

the existing results shows an excellent agreement, as presented in Table 1.


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Table 1. Comparison of results of the wall temperature gradient ( )/ 0 with previously published data for

the values ofT M c c r

M E Q S S 0.0. = = = = = = = =

rP Grubka andBobba (1985)Ishak

et al.(2009)

Exact solution

of Ishak

et al.(2009)

Dulal Pal (2011)* *A B Nr 0.0= = =

Present results

0.0 0.5 0.72 0.4570 0.457026833 0.457026833 0.457026986

1.00 0.5000 0.500000000 0.500000000 0.500000000

10.0 0.6452 0.645161289 0.645161290 0.645162987

1.0 0.01 0.0197 0.0197 0.019706354 0.019706795 0.019709754

0.72 0.8086 0.8086 0.808631350 0.808631352 0.808679760

1.00 1.0000 1.0000 1.000000000 1.000000000 1.000000000

3.00 1.9232 1.9337 1.923682594 1.923682561 1.923683398

10.0 3.7207 3.7207 3.720673901 3.720673903 3.720689456

100.0 12.2940 - 12.29408326 12.29408344 12.29409945

2.0 0.72 1.4944 1.494368413 1.494368414 1.494398634

1.00 2.0000 2.000000000 2.000000000 2.000000000

10.0 16.0842 16.08421885 16.08421882 16.08443212

1.0 0.5 1.00 0.8095 0.809511470 0.809516743

1.0 1.3205 1.320522071 1.320535432

2.0 2.2224 2.222355356 2.222367892

The physical representation is shown in Figures 2-25.

Figure 2 displays the dimensionless velocity distribution ( )/f for different values of where

T M r r c r cM 2.0, 4.0, 2.0,Q 1.0, 2.0,N 1.0,P 2.0,S 0.6,S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case It can

be observed that velocity profiles are increases as the increase.

Figure 3 exhibits the dimensionless velocity distribution ( )/f for different values of where

T M r r c r cM 2.0, 4.0, 2.0,Q 1.0, 0.5,N 1.0,P 2.0,S 0.6, S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it canbe found that velocity profiles are decreases as the increase.

Figure 4 represent the dimensionless velocity distribution ( )/f for different values of T where

M r r c r cM 2.0, 2.0, 2.5,Q 1.0, 0.5,N 1.0,P 2.0,S 0.6, S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can

be conclude that velocity profiles are increases as theT

increase.

Figure 5 depicts the dimensionless velocity distribution ( )/f for different values of M where

T r r c r cM 2.0, 2.0, 2.0,Q 1.0, 0.5,N 1.0,P 2.0,S 0.6, S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can be

observed that velocity profiles are increases as theM

increase.


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Figure 6 shows the dimensionless velocity distribution ( )/f for different values of M where

M T r r c r c2.5, 2.0, 2.0,Q 1.0, 0.5,N 1.0,P 2.0,S 0.6, S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can

be observed that velocity profiles are decreases as the Mincrease.

Figure 7 portrays the dimensionless temperature distribution ( ) for different values of where

T M r r c r cM 2.0, 4.0, 2.0,Q 1.0, 2.0,N 1.0,P 2.0,S 0.6,S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can

be observed that temperature profiles are decreases as the increase.

Figure 8 illustrates the dimensionless temperature distribution ( ) for different values of T where

M r r c r cM 2.0, 2.0, 2.5,Q 1.0, 0.5,N 1.0,P 2.0,S 0.6, S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can

be observed that temperature profiles are decreases as theT

increase.

Figure 9 displays the dimensionless temperature distribution ( ) for different values of Q where

M T r r c r cM 2.0, 2.0, 2.5, 2.0, 0.5,N 1.0,P 2.0,S 0.6, S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can

be observed that temperature profiles are increases as the Q increase.

Figure 10 exhibits the dimensionless temperature distribution ( ) for different values of rN where

M T r c r cM 2.0, 2.0, 2.5, 2.0, 0.5,Q 0.5,P 2.0,S 0.6 , S 2.0, 0.5,E 0.01 = = = = = = = = = = = Then for above case it can be

observed that radiation leads to a significant change in temperature profiles, increases as ther

N increase.

Figure 11 represents the dimensionless temperature distribution ( ) for different values of rS where

M T r c r cM 2.0, 2.0, 2.5, 2.0, 0.5,Q 0.5,P 2.0,S 0.6 , N 1.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can be

observed that temperature profiles are increases as ther

S increase.

Figure 12 depicts the dimensionless concentration distribution ( ) for different values of where

T M r r c r cM 2.0, 4.0, 2.0,Q 1.0, 2.0,N 1.0,P 2.0,S 0.6,S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can

be observed that concentration profiles are increases as the increase.

Figure 13 shows the dimensionless concentration distribution ( ) for different values of where

T M r r c r cM 2.0, 4.0, 2.0,Q 1.0, 2.0,N 1.0,P 2.0,S 0.6, S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can

be observed that concentration profiles are decreases as the increase.

Figure 14 portrays the dimensionless concentration distribution ( ) for different values of M where

T r r c r cM 2.0, 2.0, 2.0,Q 1.0, 0.5,N 1.0,P 2.0,S 0.6, S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can be

observed that concentration profiles are decreases as the M increase.

Figure 15 displays the dimensionless concentration distribution ( ) for different values of cS where

T r r M r cM 2.0, 2.0, 2.0,Q 1.0, 0.5,N 1.0,P 2.0, 2.5, S 2.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can

be observed that concentration profiles are decreases as thec

S increase.

Figure 16 exhibits the dimensionless concentration distribution ( ) for different values of rS where

M T r c r cM 2.0, 2.0, 2.5, 2.0, 0.5,Q 0.5,P 2.0,S 0.6 , N 1.0, 0.5,E 0.01. = = = = = = = = = = = Then for above case it can be

observed that concentration profiles are increases as ther

S increase.


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Since the physical interest of the problem, the skin-friction coefficient ( )1

2

xf eC R

, the Nusselt number ( )1

2

xu eN R

at the sheet and the Sherwood number ( )1

2

xh eS R

at the sheet are plotted against Unsteadiness parameter ( ) and

illustrated in Figures 17-25.

Figure 17 represents the skin-friction coefficient ( )1

2

xf eC R

for different values ofT

where M 2.0, 2.0,= =

M r r c r c2.5,Q 1.0,N 1.0,P 2.0,S 0.6, S 2.0, 0.5,E 0.01. = = = = = = = = Then for above case it can be observed that skin-

friction coefficient are increases as theT

increase.

Figure 18 depicts the skin-friction coefficient ( )1

2

xf eC R

for different values ofM

whereT

M 2.0, 2.0, 2.0, = = =

r r c r cQ 1.0,N 1.0,P 2.0,S 0.6 , S 2.0, 0.5,E 0.01= = = = = = = Then for above case it can be observed that skin-friction

coefficient are increases as theM

increase.

Figure19 shows the skin-friction coefficient ( )

1

2

xf e

C R

for different values of

Mwhere M T

2.5, 2.0, 2.0, = = =

r r c r cQ 1.0,N 1.0,P 2.0,S 0.6 , S 2.0, 0.5,E 0.01= = = = = = = Then for above case it can be observed that skin-friction

coefficient are decreases as the Mincrease.

Figure 20 displays the rate of heat transfer ( )1

2

xu eN R

for different values ofT

whereM

2.5,Q 1.0, = =

r r c r cN 1.0,P 2.0,S 0.6, S 2.0, 0.5,E 0.01.= = = = = = Then for above case it can be observed that heat transfer rate are

increases as theT

increase.

Figure 21 represents the rate of heat transfer ( )1

2

xu eN R

for different values ofr

N where M 2.0, 2.0,= =

M T r c r c2.5, 2.0,Q 0.5,P 2.0,S 0.6, S 2.0, 0.5,E 0.01 = = = = = = = = Then for above case it can be observed that heat

transfer rate are increases as ther

N increase.

Figure 22 illustrates the rate of heat transfer ( )1

2

xu eN R


S where M 2.0, 2.0,= =

M T r c r c2.5, 2.0,Q 0.5,P 2.0,S 0.6,N 1.0, 0.5,E 0.01 = = = = = = = = Then for above case it can be observed that heat

transfer rate are decreases as ther

S increase.

Figure 23 depicts the rate of mass transfer ( )1

2

xh eS R

for different values of whereT

M 2.0, 4.0,= =

M r r c r c2.0,Q 1.0, 2.0,N 1.0,P 2.0,S 0.6 , S 2.0,E 0.01 = = = = = = = = Then for above case it can be observed that mass

transfer rate are decreases as the increase.

Figure 24 displays the rate of mass transfer ( )1

2

xh eS R

for different values ofc

S where M 2.0, 2.0,= =

T r r M r c2.0,Q 1.0,N 1.0,P 2.0, 2.5, S 2.0, 0.5,E 0.01 = = = = = = = = Then for above case it can be observed that mass

transfer rate are decreases as thec

S increase.

Figure 25 shows the rate of mass transfer ( )1

2

xh eS R


S where M 2.0, 2.0,= =

M T r c r c2.5, 2.0,Q 0.5,P 2.0,S 0.6,N 1.0, 0.5,E 0.01 = = = = = = = = Then for above case it can be observed that mass

transfer rate are increases as ther

S increase.


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Fig. 2: Velocity profiles for different values of

Unsteadiness parameter ( ) . Fig. 3: Velocity profiles for different valuesof Suction parameter ( ) .

Fig. 4: Velocity profiles for different values of

Thermal convective parameter ( )T .Fig. 5: Velocity profiles for different values

of Mass convective parameter ( ).M

Fig. 6: Velocity profiles for different values ofMagnetic parameter ( ).M Fig. 7: Temperature profiles for differentvalues of Unsteadiness parameter ( ) .


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Fig. 8: Temperature profiles for different values

of Thermal convective parameter ( )T .Fig. 9: Temperature profiles for different

values of Heat source parameter ( )Q .

Fig. 10: Temperature profiles for different

values of Radiation parameter ( )rN .Fig. 11: Temperature profiles for different

values of Soret number ( ).rS

Fig. 12: Concentration profiles for different

values of Unsteadiness parameter ( ) . Fig. 13: Concentration profiles for differentvalues of Chemical reaction parameter ( ).


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values of Mass convective parameter ( ).M Fig. 15: Concentration profiles for different

values of Schmidt number ( ).cS


values of Soret number ( ).rS Fig. 17: Skin-friction coefficient for different

values ofThermal convective parameter ( ).T

Fig. 18: Skin-friction coefficient for differentvalues of Mass convective parameter ( ).M Fig. 19: Skin-friction coefficient for differentvalues of Magnetic parameter ( ).M


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Fig. 20: Heat transfer rate for different values of

Thermal convective parameter ( )T .Fig. 21: Heat transfer rate for different

values of Radiation parameter ( )rN .

Fig. 22: Heat transfer rate for different values of

Soret number ( ).rS Fig. 23: Mass transfer rate for different

values of Chemical reaction parameter ( ).

Fig. 24: Mass transfer rate for different values ofSchmidt number ( ).cS Fig. 25: Mass transfer rate for differentvalues of Soret number ( ).rS


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5 Conclusion

The objective of the present work is studying the effect of thermal radiation, heat generation and chemicalreaction on MHD mixed convective flow of heat and mass transfer over an unsteady stretching permeable

surface. The governing equations are approximated to a system of non-linear ordinary coupled differential

equations by similarity transformation. Numerical calculations has been carried out for various values of

the dimensionless parameters of the problem. The conclusions of present study given below:

a. The momentum and concentration boundary layer thickness increases with increase in the unsteadinessparameter whereas the thermal boundary layer thickness decreases.

b. As Suction parameter increases the momentum boundary layer thickness decreases gradually.c. The momentum boundary layer thickness and skin-friction coefficient reduces as magnetic parameter is

increased.

d. For increasing Thermal convective parameter the momentum boundary layer thickness rises steeplywhereas the thermal boundary layer reduces. Also the skin-friction coefficient and heat transfer rate are

rises.

e. The thermal boundary layer thickness and heat transfer rate increases with increase in the radiationparameter.

f. Concentration boundary layer reduces whereas momentum boundary layer rises as the Mass convectiveparameter is increased. Also the rising effect found for skin-friction coefficient.

g. As heat source parameter increases the thermal boundary layer thickness increases gradually.h. The thermal and concentration boundary layer thickness and surface mass transfer rates raise where as

heat transfer rate decreases as Soret number increase.

i. The concentration boundary layer thickness and surface mass transfer rates reduces as Schmidt numberincrease.

j. The concentration boundary layer thickness and surface mass transfer rates reduces as Chemicalreaction parameter increase.

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Documents

Heat Generation, Thermal Radiation and Chemical Reaction Effects on MHD Mixed Convection Flow over an Unsteady Stretching Permeable Surface