NUMERICAL INVESTIGATION OF AN UNSTEADY MIXED...

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International Journal of Mechanical Engineering and Technology (IJMET)

Volume 7, Issue 3, May–June 2016, pp.170–193, Article ID: IJMET_07_03_016

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ISSN Print: 0976-6340 and ISSN Online: 0976-6359

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NUMERICAL INVESTIGATION OF AN

UNSTEADY MIXED CONVECTIVE MASS

AND HEAT TRANSFER MHD FLOW WITH

SORET EFFECT AND VISCOUS

DISSIPATION IN THE PRESENCE OF

THERMAL RADIATION AND HEAT

SOURCE/SINK

Suresh

Dept. of Mathematics, Gulbarga University,

Kalaburagi-585106, Karnataka, India

P.H. Veena

Dept. of Mathematics, Smt. V. G. Womens College,

Kalaburagi-585102, Karnataka, India

V. K. Pravin

Dept. of Mechanical Engineering,

P. D. A. College of Engineering, Kalaburagi-585102,

Karnataka, India

ABSTRACT

In the present paper an analysis of mixed convection of an unsteady

magneto hydrodynamic (MHD) flow of an incompressible viscous fluid

through porous media due to a vertical porous stretching sheet in the presence

of viscous dissipation, thermal radiation and heat source /sink has been

carried out. The fluid considered is viscous and incompressible. The

governing partial differential equations of the flow, mass and heat transfer are

highly non linear hence are converted into a system of ordinary differential

equations using suitable similarity transformations. These ordinary

differential equations are further converted into 7 first order ordinary

differential equations and are solved numerically by Matlab ode-45 solver via

shooting method. The effects of various physical parameters such as magnetic

parameter, porous parameter, mixed convection parameter, Eckert number,

heat source/sink parameter, unsteady parameter, Soret number, the Prandtl

Numerical Investigation of an Unsteady Mixed Convective Mass and Heat transfer MHD

flow with Soret effect and Viscous Dissipation in the presence of Thermal Radiation and Heat

Source/sink


number of the flow, mass and heat transfer characteristics are analyzed and

illustrated through graphs.

Cite this Article: Suresh, P.H. Veena and V. K. Pravin, Numerical

Investigation of an Unsteady Mixed Convective Mass and Heat transfer MHD

flow with Soret effect and Viscous Dissipation in the presence of Thermal

Radiation and Heat Source/sink. International Journal of Mechanical

Engineering and Technology, 7(3), 2016, pp. 170–193.

http://www.iaeme.com/currentissue.asp?JType=IJMET&VType=7&IType=3

NOMENCLATURE

A, b, c constants

Externally imposed transverse magnetic field strength

C concentration of the species

Free stream concentration

Local skin-friction coefficient

Specific heat at constant pressure

Coefficient of mass diffusivity

Thermal diffusivity

Ec Eckert number

Dimensionless suction velocity

g acceleration due to gravity

k thermal conductivity

k* mass absorption co-efficient

Porous parameter

R thermal radiation parameter

Nusselt number

Pr Prandtl number

radiative heat flux in the y- direction

Local Reynolds number

Sc Schmidt number

Sr Soret number

Sherwood number

T fluid temperature

Stretching sheet temperature

Temperature far away from the stretching sheet

u,v velocity components in the x- and y-directions

t time variable

Heat source/sink parameter

x,y flow directional coordinate and normal to the stretching sheet

http://www.iaeme.com/currentissue.asp?JType=IJMET&VType=7&IType=2

Suresh, P.H. Veena and V. K. Pravin


Greek Symbols

Coefficient of thermal expansion

Coefficient of expansion with concentration

Similarity variable

Dynamic viscosity

Kinematic viscosity

Density of the fluid

Stream function

Electrical conductivity of the fluid

Stephan–Boltzmann constant

1. INTRODUCTION

The analysis of viscous incompressible mass and heat transfer of magneto

hydrodynamic mixed convection flow through porous medium has received

considerable attention with numerous industrial applications in hydrodynamics viz

chromatography, crystal magnetic damping control, chemical catalytic reactors,

geophysics, energy related engineering problems including polymer sheets and metal

sheets. It also includes in the aerodynamic extrusion of polymer sheets, nano fluid

power plants, heat exchange between soil and atmosphere, packed sphere beds

migration of moisture through air contained in fibrous insulation, granular insulation

materials of high performance insulation buildings, transpiration cooling, packed bed

chemical reactors and continuous filament extrusion from a dye. These industrial

observations explain, how existence of practical applications of flow, mass and heat

transfer has drawn in many areas in the world.

Das et al.[1] investigated the Numerical solution of mass transfer effects on

unsteady flow past accelerated vertical porous plate with suction and solved

numerically by finite difference scheme. Prasad et al.[2] have studied the Radiation

and mass transfer effects of two-dimensional flow past an impulsively started infinite

vertical plate and solved governing equations by finite difference method. Mohamed

Abd El-Aziz [3] has interpreted the results of Radiation effect on the flow and heat

transfer over an unsteady stretching sheet using fifth order Runge-Kutta Fehlberg

integration scheme to solve differential equations via shooting technique.

Hari Rani and Chang Kim [4] worked on the numerical study of Dufour and Soret

effects on unsteady natural convection flow past an isothermal vertical cylinder by

applying a Crank-Nicolson type of implicit finite difference method with a tri-

diagonal matrix manipulation. Hayat et al [5] investigated an Unsteady flow with heat

and mass transfer of a third grade fluid over a stretching surface in the presence of

chemical reaction, and solved the system of equations by means of homotopy analysis

method (HAM). They discussed the Convergence of derived series solutions

explicitly. Vempati and Gari[6] discussed about the Soret and Dufour effects on

unsteady MHD flow past an infinite vertical porous plate with thermal radiation and

solved governing PDEs by Finite Element Method . Zaman and Ayub [7] investigated

the Series solution of an unsteady free convective flow with mass transfer along an

accelerated vertical porous plate with suction. Dulal Pal and Mondal [8] have studied

the numerical effects of Soret, Dufour, chemical reaction and thermal radiation on

MHD non-Darcy unsteady mixed convective heat and mass transfer flow over a

stretching sheet via shooting algorithm with Runge–Kutta Fehlberg integration



Source/sink


scheme. Chamkha and Ahmed [9] made an investigation on the Unsteady MHD Heat

and Mass Transfer by Mixed Convection Flow in the Forward Stagnation Region of a

Rotating Sphere in the Presence of Chemical Reaction and Heat Source. Singh and

Kumar [10] have interpreted the results of Fluctuating Heat and Mass transfer on

unsteady MHD free convection flow of radiating and reacting fluid past a vertical

porous plate in slip-flow regime. Chamkha et al [11] investigated an Unsteady MHD

natural convection flow from a heated vertical porous plate in a micropolar fluid with

Joule heating, chemical reaction and radiation effects. Hayat et al [12] discussed about

Mass transfer effects on the unsteady flow of UCM fluid over a stretching sheet and

Homotopy analysis method is used for the development of series solution of the

arising nonlinear problem. Husnain et al [13] discussed about the Heat and Mass

transfer analysis in unsteady boundary layer flow through porous media with variable

viscosity and thermal diffusivity solved analytically and numerically by using the

homotopy analysis method and the Runge–Kutta method via shooting technique.

Alamgir et al [14] studied the Effects of Thermophorosis on Unsteady MHD free

convective heat and mass Transfer along an Inclined porous plate with heat generation

in presence of Magnetic Field and by employing Nachtsheim-Swigert shooting

iteration technique along with sixth order Runge-Kutta integration scheme. AL-

ODAT, GHAMDI[15] studied Numerical investigation of Dufour and Soret effects on

unsteady MHD natural convection flow past a vertical plate embedded in non-Darcy

porous medium and used implicit finite difference scheme of the Crank-Nicolson type

with tri diagonal matrix manipulation method to solve governing non linear

dimensionless equations. Mustafa et al.[16] presented his work On heat and mass

transfer in the unsteady squeezing flow between parallel plates and used Homotopy

Analysis Method(HAM) to construct the series solution of the problem. Husnain et

al.[17] discussed heat and mass transfer analysis in unsteady boundary layer flow

through porous media with variable viscosity and thermal diffusivity and solved

analytically by Homotopy analysis Method and Numerically by Runge-Kutta shooting

method. Vajravelu et al.[18] interpreted Unsteady convective boundary layer flow of

a viscous fluid at a vertical surface with variable fluid properties and solved non linear

PDEs by Second order finite difference scheme known as Kellar Box Method.

Turkyilmazoglu and Pop [19] investigated heat and mass transfer of unsteady natural

convection flow of some nanofluids past a vertical infinite flat plate with radiation

effect by exact analytical method. Madhusudhan et al.[20] studied unsteady MHD

free convective heat and mass transfer flow past a semi-infinite vertical permeable

moving plate with heat absorption, radiation, chemical reaction and Soret effects by

analytical methods. Zheng et al.[21] presented the Unsteady heat and mass transfer in

MHD flow over an oscillatory stretching surface with Soret and Dufour effects by

HAM. Shankar and Yirga[22] discussed unsteady heat and mass transfer in MHD

flow of nanofluids over stretching sheet with a non-uniform heat source/sink and solved boundary layer equations by Kellar box method. Mohamed et al.[23] presented

the results on unsteady MHD double-diffusive convection boundary-layer flow past a

radiate hot vertical surface in porous media of chemical reaction and heat sink. The

governing equations are solved in closed form by Laplace-transform technique in his

study.

Reddy et al.[24] made a note on the thermal radiation and magnetic field effects

on unsteady mixed convection flow and mass transfer over a porous stretching surface

with heat generation and Numerical solution of governing equations are obtained by

Runge-Kutta Forth order scheme along with shooting technique.



Idowu et al. [25] presented the effect of heat and mass Transfer on Unsteady

MHD Oscillatory Flow of Jeffrey fluid in a horizontal channel with chemical reaction

are evaluated using perturbation technique. Mohammed Ibrahim et al.[26] studied

radiation effect on unsteady MHD free convective heat and mass transfer flow of past

a vertical porous plate embedded in a porous medium with viscous dissipation by

employing shooting method with 4th

order RK integration scheme. Haroun et al.[27]

studied On unsteady MHD mixed convection in a nanofluid due to a

stretching/shrinking surface with suction/injection using the spectral relaxation

method. Nayak et al.[28] investigated Soret and Dufour effects on mixed convection

unsteady MHD boundary layer flow over stretching sheet in porous medium with

chemically reactive species and solved governing non linear differential equation by

RK4 method. Hunegnaw and Kishan[29] studied unsteady MHD heat and mass

transfer flow over stretching sheet in porous medium with variable properties

considering viscous dissipation and chemical reaction. Agarwal and Bhadauria[30]

studied the Unsteady heat and mass transfer in a rotating nano fluid Layer. Ferdows et

al.[31] interpreted the results on boundary layer flow and heat transfer of a nanofluid

over a permeable unsteady stretching sheet with viscous dissipation and solved the

flow equations numerically using the Nactsheim–Swigert shooting technique together

with Runge–Kutta six-order iteration scheme. Sengupta[32] made an analysis of

unsteady heat and mass transfer flow of radiative chemically reactive fluid past an

oscillating plate embedded in porous medium in presesnce of Soret effect. Das et

al.[33] presented an unsteady free convection flow past a vertical plate with heat and

mass fluxes in the presence of thermal radiation by analytical method. Ahmad and

Khan [34] investigated Unsteady heat and mass transfer magneto hydrodynamic

(MHD) nanofluid flow over a stretching sheet with heat source–sink using quasi-

linearization technique. Ravindran and Samyuktha [35] have investigated the

unsteady mixed convection flow over stretching sheet in presence of chemical

reaction and heat generation or absorption with non-uniform slot suction or injection

using the quasi linearization technique in combination with an implicit finite

difference scheme.

Thus motivated by the above analyses, the main objectives of present paper is to

study the Numerical investigation of unsteady mass and heat transfer of MHD viscous

flow with Soret and viscous dissipation in presence of thermal radiation and heat

source/sink. The problem addresses here is fundamental one that arises in many

practical situations. The non- linearity and mathematical difficulties associated in

basic equations encourages to use numerical techniques. The governing equations of

the given problem after similarity transformations are solved numerically by ode45 of

MATLAB differential equation of first order solver via shooting method. The effect

of various governing parameters on velocity, temperature and concentration are

analyzed and exhibited through graphs.

2. PROBLEM FORMULATION

Considered a typical laminar mixed convection boundary layer flow with heat and

mass transfer of a viscous incompressible electrically conducting fluid over a vertical

porous medium placed in the plane y = 0 and moving with the velocity

( , )1

w

cxU x t

t

with temperature distribution

0

22 1

w

T cxT T

t

where c and

are constants, 0T is the reference temperature. X-axis is taken along the stretching

direction of the sheet and y-axis is normal to the surface of sheet. A magnetic field of



Source/sink


uniform strength 0B is applied in the negative direction of the y-axis. The fluid is

considered to be gray, absorbing and emitting radiation but non scattering medium

and to describe the radiative heat flux, the Rosseland approximation is considered in

Heat equation and is considered negligible along x-axis compared to the other axis.

Under the above considerations the governing boundary layer equations of the

model areas follows

0u v

x y

(1) 22

* 0

2 '( ) ( )

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

t x y y k

(2)

22

0

2

1( )r

p p p

QqT T T k T uu v T T

t x y c y c y c y

(3)

2 2

2 2M T

C C C C Tu v D D

t x y y y

(4)

The boundary conditions for the flow, heat and mass distribution fields are

, , , 0w w w wu U v V T T C C at y

0 , ,u T T C C as y (5)

Where t is the time variable, u and v are the velocity components along x- and y-

axis respectively. 1

w

cV

t

is the velocity of suction parameter( wV > 0),

is the kinematic viscosity, is the volumetric coefficient of thermal expansion, g is

the acceleration due to gravity , 0B is the uniform magnetic field , is the Electrical

conductivity, pC is the Specific heat at constant pressure, is the Density, T the

Temperature, T is the temperature for away from the stretching surface, k is the

coefficient of the thermal conductivity of the fluid, 0Q is the Heat generation constant,

rq is the Radiation heat flux, is the Effective dynamic viscosity, mD is the mass

diffusivity, mD is the Thermal diffusivity, By Rosseland approximation, the radiative

heat flux is given by 4

*

4

3s

r

Tq

k y

where s is the Stefan Boltzman Constant and

*k is the absorption coefficient. Temperature difference in the flow is assumed to be

sufficiently small so that 4T can be expressed as linear function of T using truncated

Talyer’s series about the free stream temperature T and neglecting higher order

terms we get 4 3 44 3T T T T then equation (3) becomes

23 2

0

* 2

16( )s

p p p

T QT T T k T uu v T T

t x y c c k y c y

(6)



Introducing now the following non dimensional variables, by choosing the stream

function

( , )x y Such that ,u vy x

. Thus continuity equation (1) satisfies Identically

with this of ( , )x y where

( , )1

cx y xf

t

, 1

cy

t

,

( )w

T T

T T

,

0 2

( )2 1

cxT T T

t

( )w

C C

C C

,

0 2

( )2 1

cxC C C

t

0

p w

Q xQ

C U

By consequence of the above similarity variables in (2), (4)and (6), following

ordinary differential equations are obtained ,

2

''' '' ' ' '' ' '

1 2 2

10

2f ff f S f f Mf K f

(7)

'' ' ' ' ''Pr

01 2

Sf f Q Ecf

R

(8)

'' ' ' ' '' 02

SSc f f Sr

(9)

Boundary conditions corresponding to the above non-linear differential equations

converts to

'(0) , (0) 1, (0) 1f fw f

'( ) 0, ( ) 0, ( ) 0f (10)

Where A is the unsteady parameter, 1 and 2 are mixed convection parameters,

M is the Magnetic parameter, 2K is the Permeability parameter, Pr is the Prandtl

number, R is the thermal radiation parameter, Q is the heat source/sink parameter,

is the thermal diffusivity, Ec is the Eckert number, Sc is the Schimidt number, Sr is

the Soret number, where 1

w w

tf V

c

is the dimensionless velocity which

determines the transpiration rate at the surface with wf > 0 for suction and wf < 0 for



Source/sink


injection and wf = 0 represents impermeable sheet, prime denotes the partial

differentiation with respect to similarity variable . Analysis of Flow (velocity) '( )f

, Temperature ( ) and Concentration ( ) allows us to determine the important

characteristics of engineering design problems.

III i) SKIN FRICTION

The wall shear stress may be expressed in terms of local skin friction coefficient as

0 ''

2

21

(0)2

y

f p e

w

u

yC C R f

U

III ii) NUSSELT NUMBER

The Local Nuselt number of the rate of heat transfer coefficient in the stretching

surface may be expressed as

0 '(0)y u

u

w e

Tx

y NN

T T R

III iii) SHREWOOD NUMBER

The local Sherwood number signifies the rate of mass transfer which is expressed as

Where we

U xR

Is the local Reynolds Number

0 '(0)y

w e

Cx

y ShSh

C C R

4. SOLUTION METHODOLOGY

The solution of the set of ordinary differential equations (7) to (9) corresponding to

laminar boundary conditions (10) are obtained Numerically by MATLAB ode45

solver with shooting technique. The values of ', f and are known at end points i.e

and these end conditions are utilized to generate three unknown initial

conditions at 0 by using shooting method. The most difficulty of this method is to

choose suitable initial finite values. Starting with some initial guess values and solve

the boundary value equations(2),(5) and (6) to obtain ''(0)f ,

'(0) and '(0) by

MATLAB software. The velocity, temperature and concentration field can easily to

be obtained for a particular set of physical parameters. The results are discussed in the

next section.

5. RESULTS AND DISCUSSION

To discuss the clear insight of the physical problem, we have been carried out

Numerical computation of the model by MATLAB ode45 solver along with shooting

technique for various values of governing parameters A, M, B, Pr, Q, Ec, Sc, K2, Sr



and fw. Many graphs are drawn for velocity field, temperature field and concentration

fields. Fig1(a). depicts the velocity profiles for different values Magnetic parameter M

and it is observed from the figure that as M increases the rate of velocity distribution

decreases slightly near the plate and outside there is no significant effect on velocity

profile. This is because of the fact that magnetic field has tendency to give resistive

type force known as Lorentz force which has a property to slow down the motion of

the fluid. This result agrees qualitating with expectation with the published result but

in case of temperature and concentration profiles it is different as shown in fig1.(b)(c).

Fig.2 (a)(b)(c) represents the graphs of velocity, temperature and concentration

profiles for various values of permeability parameter 2K . By observing the graphs it

is shown that velocity profile decreases with increasing values of 2K where the

temperature and concentration distribution increases with increasing values of 2K .

Physically it reveals the fact that increasing the tightness of porous medium results in

increasing the resistance against flow and thus fluid velocity decreases.

Fig3 (a) (b) (c). Displays the graphs to show the influence of suction parameter fw

on flow, temperature and concentration. One can depicts easily from the figures that

increasing in suction parameter results in decreasing concentration, temperature and

flow profiles.

Fig4 (a)(b)(c)explain the influence of unsteady parameter S on flow, heat and

mass transfer distribution respectively. One can observe from the figures that the

profile for temperature, concentration and flow decreases with increasing values of

Unsteady parameter S. Physically it means that increasing the values of unsteady

parameter S is to reduce the thickness in the boundary layer near the wall and fluid

velocity increases away from the wall and it is same in heat and concentration

profiles.

Figs 5(a)(b)(c) presents the graphs to show the effect of Schimidt number on flow,

temperature and concentration .It is observed from the figures that temperature and

concentration profiles decrease with increasing values of Sc. Here it is noted that

Schimidt number is inversely proportional to the diffusion coefficient.

Fig.6 (a)(b)(c) Shows the effect of Soret number on ', f and . For increasing

values of Soret number, the Velocity and Concentration increases and temperature

decreases. The effects of Prandtl number are drawn in the figures7(a)(b)(c)which

represents respectively velocity, temperature and concentration profiles and it can be

seen from the figures that distribution near the boundary layer flow are decreasing by

increasing values of Prandtl number i.e boundary layer thickness decreases or slew

rate of thermal diffusion.

Figs 8(a) (b) displays the influence of viscous dissipation parameter Ec or Eckert

number on velocity and temperature profiles. The fluid velocity and thermal boundary

layer increases with increasing values of Ec. Physically by definition of Eckert

number it is the ratio of Kinetic energy of the flow to the boundary layer Enthalpy

difference. It corporate the conversion of the kinetic energy into thermal energy by

work done against the viscous fluid stresses. Plate is cooled by the influence of

positive Eckert number. There fore for maximum values of Eckert number, thermal

boundary layer and velocity increases.

The distribution of radiation parameter R on flow, heat and concentration are

shown in figs 9(a)(b)(c). For various values of R, the temperature and concentration

profiles causing to increase of increasing values of R while velocity profile decreases.



Source/sink


Figs 10(a)(b)(c) show the influence of Heat source/sink parameter on three

distributions, it is clearly observed from the figures that velocity decreases with

increasing values of Q and temperature and concentration profile due increasing by

increasing values of Q.

In figs 11(a)(b)(c) and 12(a)(b) , for increasing mixed convection parameter t he

variations of fluid flow and thermal boundary layers are decreasing and increasing in

case of concentration profile are shown.

Fig1 (a). Longitudinal Velocity distribution for different values of Magnetic parameter M, for

fixed fw = 0.8; S = 0.3; = 0.2; = 0.2; Pr=1; Sc = 0.4;R =0.4;Q =0.3;k2= 01; Ec =0.1; Sr =

1;

Fig1 (b).Temperature profiles for different values of Magnetic parameter M, with fixed values

of A =0.5 ; = 0.2; = 0.2; fw =0.5 ;Pr=2; Sc = 0.4; R =0.3; Q =0.6;k2=1; Ec =0.1; Sr = 2;

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f ' (

)

M = 0, 0.2, 0.4

S = 0.3; 1 = 0.2; 2 = 0.2; Pr=1; Sc = 0.4;

R =0.4; Q =0.3;k2= 01; Ec =0.1; Sr = 1;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

M = 0, 0.5, 1.0



Fig1(c).Concentration profiles for different values of Magnetic parameter M, with fixed

values of S =0.5 ;fw=0.5; =0.3 ; =0.2 ; Pr=2; Sc = 0.4; R =0.3; Q =0.6;k2 =1;Ec=0.1; Sr

= 2;

Fig2 (a). Longitudinal Velocity distribution for different values of Porous parameter k2, for

fixed fw = 0.8;S = 0.3; = 0.2; = 0.2; Pr=1; Sc = 0.4; R =0.4; Q =0.3;M= 0.2; Ec =0.1; Sr =

1;

Fig2 (b).Temperature profiles for different values of Porous parameter K2, with fixed values

of A =0.5 ; = 0.2; = 0.2; fw =0.5 ;Pr=2; Sc = 0.4; R =0.3; Q =0.6;M=0.5; Ec =0.1; Sr =

2;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

M = 0.2, 1.0, 2.0

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f '(

)

K2 = 1.0, 2.0, 3.0

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

K2 = 1.0, 2.0, 3.0



Source/sink


Fig2 (c).Concentration profiles for different values of porous parameter k2, with fixed values

of S =0.5 ; fw=0.5; =0.3 ; =0.2 ;M =0.1 ;Pr=2; Sc = 0.4; R =0.3; Q =0.6;Ec=0.1; Sr =

2;

Fig3 (a). Longitudinal Velocity distribution for different values of Suction Parameter fw, for

fixed S = 0.3; = 0.2; = 0.2; M= 0.2; Sc = 0.4; Pr =1; Q =-2; R =0.3; k2=1; Ec = 0.1; Sr

= 5;

Fig3 (b).Temperature profiles for different values of Suction parameter fw, with fixed values

of A =0.5 ; = 0.2; = 0.2; M =0.5 ;Pr=2; Sc = 0.4; R =0.3; Q =0.6;k2=1; Ec =0.1; Sr = 2;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

k2 = 2, 4, 6

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f '(

)

fw = 0, 0.5, 1.0

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

fw = 0.0, 0.5, 1.0



Fig3 (c). Concentration profiles for different values of Suction parameter fw, with fixed

values of S =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=2; Sc = 0.4; R =0.3; Q =0.6; k2= 01; Ec

=0.1; Sr = 2;

Fig4 (a). Longitudinal Velocity distribution for different values of Unsteady parameter S, for

fixed fw = 0.8; = 0.2; = 0.2; Pr=1; Sc = 0.4; R =0.4; Q =0.3;M= 0.2; k2=1;Ec =0.1; Sr =

1;

Fig4 (b). Temperature profiles for different values of Unsteady parameter S, with fixed values

of fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=1; Sc = 0.4; R =0.3; Q =0.6; k2= 01; Ec =0.1; Sr =

2;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

fw = 0.5, 1.0, 1.5

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f '(

)

S = 0, 0.5, 1.0

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

S = 0.2, 0.4, 0.6



Source/sink


Fig4 (c ). Concentration profiles for different values of Unsteady parameter S, with fixed

values of fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=2; Sc = 0.4; R =0.3; Q =0.6; k2= 01; Ec

=0.1; Sr = 2;

Fig5(a). Longitudinal Velocity profiles for different values of Schmidt Number Sc, for fixed

fw = 0.8; A = 0.3; = 0.1; = -0.3; Pr=1; R =0.4; Q =0.3;M= 0.2; k2=1;Ec =0.1; Sr = 1;

Fig5(b). Temperature profiles for different values of Schmidt Number Sc, with fixed values of

S =0.3 ; fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=2; R =0.3; Q =0.6 ; k2= 01; Ec =0.1; Sr = 2;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

S = 0.2, 0.6, 1.0

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f ' (

)

Sc = 0. 0.5, 1.0

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Sc = 0.1, 0.5, 1.0



Fig 5 (c ). Concentration profiles for different values of Schmidt Number Sc, with fixed

values of S =0.3 ; fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=2; R =0.3; Q =0.6;k2= 01;Ec =0.1;

Sr = 2;

Fig 6 (a). Longitudinal Velocity distribution for different values of Soret number Sr, for fixed

fw = 0.8;S = 0.3; = 0.2; = 0.2; M= 0.2; Sc = 0.4; Pr =1; Q =0.3; R =0.3; k2=1; Ec = 0.1;

Fig 6 (b). Temperature profiles for different values of Soret number Sr, with fixed values of S

=0.3 ; fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=1; Sc = 0.4; R =0.3; Q =0.6;k2= 01; Ec =0.1;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Sc = 0.3, 0.4, 0.5

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f ' (

)

Sr = 1.0, 3.0, 5.0

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Sr = 1, 3, 5



Source/sink


Fig 6 (c ). Concentration profiles for different values of Soret Number Sr, with fixed values of

S =0.3 ; fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=2; Sc = 0.4; R =0.3; Q =0.6;k2= 01; Ec =0.1;

Fig 7 (a). Longitudinal Velocity profiles for different values of Prandtl number Pr, for fixed

fw = 0.8;S = 0.3; = 0.2; = 0.2; M= 0.2; Sc = 0.4; R =0.4; Q =0.3; k2=1;Ec =0.1; Sr = 1;

Fig 7 (b). Temperature profiles for different values of Prandtl Number Pr, with fixed values of

S =0.3 ; fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ; Sc = 0.4; R =0.3; Q =0.6; k2= 01; Ec =0.1; Sr

= 2;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Sr = 1, 1.5, 2

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f '(

)

Pr = 1.0, 2.0, 5.0

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Pr = 2, 3, 4



Fig 7 (c). Concentration profiles for different values of Prandlt number Pr, with fixed values

of S =0.3 ; fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ; Sc = 0.4; R =0.3; Q =0.6; k2= 01; Ec =0.1;

Sr = 2;

Fig 8 (a). Longitudinal Velocity distribution for different values of Eckart number Ec, for

fixed fw = 0.8;S = 0.3; = 0.2; = 0.2; M= 0.2; Sc = 0.4;Pr =1; Q =0.3; R =0.3; k2=1; Sr =

1;

Fig 8 (b). Temperature profiles for different values of Eckart number Ec, with fixed values of

S =0.3 ; fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=1; Sc = 0.4; R =0.3; Q =0.6;k2= 01; Sr = 2;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Pr = 3, 4, 5

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f '(

)

Ec = -1.0, 0, 1.0

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Ec = 0.1, 0.3, 0.5



Source/sink


Fig 9 (a). Longitudinal Velocity distribution for different values of Radiation parameter R, for

fixed fw = 0.8;S = 0.3; = 0.2; = 0.2; M= 0.2; Sc = 0.4; Pr =1; Q =0.3; k2=1;Ec =0.1; Sr =

1;

Fig 9 (b). Temperature profiles for different values of Radiation parameter R, with fixed

values of S =0.3 ; fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=2; Sc = 0.4; Q =0.6; k2= 01; Ec

=0.1; Sr = 2;

Fig 9 (c). Concentration profiles for different values of Radiation parameter R, with fixed

values of S =0.3 ; fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=2; Sc = 0.4; Q =0.6; k2= 01; Ec

=0.1; Sr = 2;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f '(

)

R = 0, 0.4, 0.8

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

R = 0.2, 0.4, 0.6

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

R = 0.2, 0.4, 0.6



Fig 10 (a). Longitudinal Velocity profiles for different values of parameter Q, for fixed fw =

0.8;S = 0.3; = 0.2; = 0.2; M= 0.2; Sc = 0.4; Pr =1; R =0.3; k2=1;Ec =0.1; Sr = 1;

Fig 10 (b). Temperature profiles for different values of Heat source parameter Q, with fixed

values of S =0.3 ; fw =0.3 ; =0.3 ; =0.2 ;M =0.1 ;Pr=1; Sc = 0.4; R =0.3; k2= 01; Ec

=0.1; Sr = 2;

Fig10 (c).Concentration profiles for different values of Heat Source/sink parameter Q, with

fixed values of A = 0.3; = 0.2; = 0.2; M= 0.2; Pr =1;Sc = 0.4; R =0.3; fw=0.8; k2=1;Ec

=0.1; Sr = 1;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f ' (

)

data1

data2

data3Q = -1, 0, 0.5

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Q = 0.2, 0.4, 0.6

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

Q = -1, 0, 0.5



Source/sink


Fig 11 (a). Longitudinal Velocity profiles for different values of Convection parameter , for

fixed fw = 0.8;S = 0.3; = 0.2; Pr=1; Sc = 0.4; R =0.4; Q =0.3;M= 0.2; k2=1;Ec =0.1; Sr =

1;

Fig 11 (b). Temperature profiles for different values of convection parameter , with fixed

values of S =0.3 ; fw =0.3 ; =0.2 ;M =0.1 ;Pr=2; Sc = 0.4; R =0.3; Q =0.6;k2= 01; Ec =0.1;

Sr = 2;

Fig 11 (c).Concentration profiles for different values of Magnetic parameter M, with fixed

values of S =0.5 ;fw=0.5; =0.2 ; M =0.1;Pr=2; Sc = 0.4; R =0.3; Q =0.6;k2 =1;Ec=0.1; Sr

= 2;

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f ' (

)

1= -0.1, 0.0, 0.1

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

1 = 0.1, 0.4, 0.7

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(

)

1= 0.5, 1.0, 1.5



Fig12(a). Longitudinal Velocity distribution profiles for different values of Convection

parameter , for fixed fw = 0.8;S = 0.3; = 0.1; Pr=1; Sc = 0.4; R =0.4; Q =0.3;M= 0.2;

k2=1;Ec =0.1; Sr = 1;

Fig12 (b). Concentration profiles for different values of convection parameter , with fixed

values of S =0.3 ; fw =0.3 ; =0.3 ; M =0.1 ;Pr=2; Sc = 0.4; R =0.3; Q =0.6; k2= 01; Ec

=0.1; Sr = 2;

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“Authors are thankful to University Grants Commission, New Delhi for supporting

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