effect of heat and mass transfer on mhd … vol1… · · 2016-03-18the jeffery fluid flow with...

JORIND 13(2) December, 2015. ISSN 1596-8303. www.transcampus.org/journal; www.ajol.info/journals/jorind

EFFECT OF HEAT AND MASS TRANSFER ON MHD OSCILLATORY FLOW OF JEFFERY

FLUID IN A POROUS CHANNEL WITH THERMAL CONDUCTIVITY AND SORET

A.S .Idowu and A. Jimoh

Department of Mathematics University of Ilorin, Ilorin, Nigeria, and

L.O. Ahmed

Hassan Usman Katsina Polytechnic Katsina, Katsina, Nigeria.

E-mail: [email protected]+234-803-582-0415

Abstract

In this paper, an investigative analysis has been carried out to study the effect of heat and mass transfer

on magnetohydrodynamic(MHD) oscillatory flow of Jeffery fluid through a porous medium in a channel

in the presence of thermal radiation and soret. The partial differential equations governing the flow are

transformed in to a system of non-linear ordinary differential equations by perturbation technique and

are solved numerically by using the Shooting technique with fourth order Runge-Kutta method with the

aid of MAPLE software. The results obtained are displayed graphically and in tabular form to illustrate

the effect of various parameters on the dimensionless velocity, temperature and concentration profiles.

The convergence of fourth order Runge-Kutta method solution is also discussed and a good agreement is

found in numerical solution.

Keywords: MHD flow, Jeffery fluid, shooting technique, thermal radiation, Soret

Introduction

The Jeffery fluid flow with heat and mass transfer

problem have many practical applications in

sciences and technology. It is used in agricultural

engineering to study the underground water

resources, seepage of water in river-beds as well as

in food processing, magneto hydrodynamics

(MHD) generators, geothermal energy extraction,

electromagnetic propulsion, polymer solution and

the boundary layer control in the field of

aerodynamics and so on. The combined heat and

mass transfer has many applications in difference

branches of science and technology. The influence

of heat transfer on MHD oscillatory flow of Jeffery

fluid in a channel has been done by Kavita et .al.

(2013). The problem was solved analytically using

perturbation technique and the solution obtained

shows that a rise in the chemical reaction

parameter leads to reduction of the velocity as well

as species concentration.

Vidyasadar et al.(2013), investigated the radiation

effect on MHD free convection flow of

Kuvshinshiki fluid with mass transfer past a

vertical porous plate through porous medium. The

solution to the problem were obtained by using the

Perturbation technique, it is observed that the

velocity decreases as visco-elastic increases.

Idowu et al. (2013) also investigated the effect of

heat and mass transfer on unsteady MHD

oscillatory flow of Jeffery fluid in a horizontal

channel with chemical reaction. The significant of

their study revealed that the velocity is more of

Jeffery fluid than that of Newtonian fluid.

Srinivasa et al. (2013), worked on effect of heat

transfer on MHD oscillatory flow of Jeffery fluid

through a porous medium in a tube. The governing

partial differential equation have been solved by

using perturbation technique, it is found that, the

velocity is more for Jeffery fluid than that of

Newtonian fluid. Finite element solution of heat

and mass transfer in MHD flow of a viscous fluid

past vertical porous plate under oscillatory suction

velocity has been discussed by Rao et al. (2012),

The problem has being solved by Galerking finite

element method. It was found that the increase in

magnetic field contributes to the decrease in the

velocity field.

The importance of Soret (thermal -diffusion) and

Dufor (diffusion thermo) as it influenced the

fluids with very light molecular weight as well as

medium molecular weight have received

attention by researchers and useful results have

been reported. Sharma, Yadav, Nidhish and

Mishra et al.(2012), considered the Soret and

Dufor effect on unsteady MHD mixed convection

flow past a radiative vetical porous plate

embedded in a porous medium with a chemical

mailto:[email protected]


reaction. The problems were solved numerically

applying explicit finite difference method, the

solution obtained shows that the fluids with

medium molecular weight (H2, air), Dufor and

Soret effect should not be neglected.

Bhavana et al. (2013), presented the soret effect

on free convective unsteady MHD flow over a

vertical plate with heat source. The governing

equation for this problem are solved analytically

by using perturbation method, the result obtained

shows that the velocity profile increase with an

increasing of soret number, i.e the fluid velocity

rises due to greater thermal diffusion. Anuradha et

al (2014), in their work, studied the Heat and

mass transfer on unsteady MHD free convective

flow past a semi-infinite vertical plate with soret

effect. The problem was investigated for both

aiding and opposing flow using perturbation

technique and their result shows that the soret

number increases as the velocity profile increases.

Uwanta and Omokhuale (2014), considered the

effects of variable thermal conductivity on heat

and mass transfer with Jeffery fluid.

Hence, in this present study, special attention is

given to heat and mass transfer on magnetohydro-

dynamics(MHD) oscillatory flow of Jeffery fluid

through a porous medium in a channel in the

presence of thermal radiation and Soret. The

expressions are obtained for velocity, temperature

and concentration numerically. The influence of

various parameters on the velocity, temperature

and concentration are discussed in detail through

several graphs.

Mathematical formulation of the problem

We considered the two-dimensional free

convective and mass transfer flow of an

incompressible Jeffrey fluid in a channel of width

h under the influence of electrically applied

magnetic field and radiative heat transfer as

shown in the figure below. It is assumed that the

fluid has very small electrical conductivity and

electromagnetic force respectively. We choose the

Cartesian coordinate system (x, y), where x- is

taken along center of the channel and y- axis is

taken normal to the flow direction.

y =h 1TT

u

h

y

y=0 0TT

x

Flow Configuration and coordinate system of the Problem.

Hence,

2*2*

11 ttS

where , is the dynamic viscousity,

1 is the ratio of

relaxation to retardation times, 2 is the retardation time and

*t

is the shear rate.

The basic equations of momentum and energy governing of such a flow, subject to the

Boussinesq approximation, are:

Continuity equation

0

y

u


(1)

Momentum equation:

'

'

'2

0''''

2'

'2

'

''

'

'

)()(11

uk

uB

CCgTTgy

u

x

P

y

u

t

u

(2)

Temperature equation:

''

'

'

0

'

''

'

' 1)(

y

q

Cpy

TTK

yCp

k

y

T

t

T r

(3)

Concentration equation

)( ''

2'

'2

2'

'2

'

''

'

'

CCK

y

T

T

KD

y

CDm

y

C

t

Cr

m

Tm (4)

The boundary conditions are

,0u 0T , 0C at y=0 (5)

,0u1TT ,

1CC at y=h (6)

Where u is the axial velocity, T is the

fluid temperature, P is the pressure,

is the fluid density, 0B and is the

magnetic field strength, is the

conductivity of the fluid, g is the

acceleration due to gravity, is the

coefficient of volume expansion due to

temperature, Cp is the specific heat at

constant pressure. k is the thermal

conductivity and q is the radiative heat

flux, Dm is the coefficient of mass

diffusivity, TK is the thermal

diffusion ratio, m is a constant, mT is

the mean fluid temperature and rK is

the chemical reaction, qr is the

radiation heat flux, is the kinematic

viscosity and subscript denotes the

conditions in the free stream.

For optically thick fluid we have

modest (1993) as follows radiative

heat flux term by using the Rosseland

approximation given by;

'

4'

'

''

3

4

y

T

kq

r

r

(7)

where is the Stelfan-Boltzmann

constant. It should be noted that by

using the Rosseland approximation,

the present analysis is limited to

optically thick fluids. If the

temperature differences within the

flow are sufficient small, then equation

(7) can be literalized by expanding 4'T into the Taylor series about

'

T ,

which after neglecting higher order

terms take the form

4''3'4' 34 TTTT (8)


'

''

3

16

r

rk

Tq

(9)

Substituting (8) and (9), into equation (3) we have

2'

'23'

'

'

'

0

'

''

'

'

3

16)(

y

TT

Cpke

s

y

TTK

yCp

k

y

T

t

T

(10)

The thermal conductivity depends on temperature. This was given by Molla et al. (2005), as:

''

0 1)( TTmkTK

where 0k is the thermal conductivity of the ambient fluid and is defined by

T

TK

TK

)(

)(

1

0 > 0 is the suction parameter and 0 < 0 is the injection parameter. On introducing the

following non-dimensional variables

,0

'

U

uu ,

'

0

yUy ,

4

'2

0

tUt ,

''

''

TT

TT

w

,''

''

CC

CC

w

,Pr

k

C

,)(

2

0U

TTgGr

,)(

2

0U

CCgGc

,2

0

2

0

U

BM

,

4 3'

ekk

TR

,

DmSc

,)(

)(''

''

CCT

TTDmKSr

m

T

and ,

2

0U

kK r ,

2

0

0

U

),( ''

TTm ,2

0

'UkK

(11)

where U is the mean flow velocity, into the equations (2), (3) and (4) with equation (1) identically

satisfied the following set of differential equations, (after dropping bars)

uK

MGcGry

ue

y

u

t

u ti

1

11

1

4

12

2

(12)

Pr

1Pr

1

Pr4

12

22

R

yyyt

(13)

Kry

SoyScyt

2

2

2

21

4

1 (14)

where M is the Hartman number, Gr is the Grashof number, Pr is Prandlt number and R is the radiation

parameter.

The corresponding non-dimensional boundary conditions are

,0u 0 , 0 at y=0 (15)

,0u 1 , 1 at y=1 (16)

Method of solution


In order to solve equations (12)-(14) for purely oscillatory flow, we assumed

tiex

p

(17)

tieyutyu )(),( 0 (18)

tieyty )(),( 0 (19)

tieyty )(),( 0 (20)

where is the real constant and is the frequency of the oscillation.

Substituting equations (17)-(20) in to equations (12)-(16), we get

)(11)(11

)(1

1111)()(

11)(114

00

02

0

2

00

yGcyGr

yuK

My

yu

y

yuyu

i

(21)

(22))(

Pr

)(

Pr

)()(1

Pr

)()(

4

2

00

2

0

2

0

2

00

0

y

yuEcy

R

y

y

y

yy

y

yy

i

(23) )()()(

)(4

02

0

2

2

0

2

00 yScKr

y

yScSo

y

y

y

yScy

Sci

with the boundary conditions

,00 u 00 , 00 at y=0 (24)

,00 u 10 , 10 at y=1 (25)

Results and discussion


Effect of Hartman number M on velocity for Pr=0.71,Gr=1, Gc=1, =1, R=0.3, K=1,Kr=1, w=1,i=

, =0.1,Ec=1,Sc=0.26,So=5, =1, 1 =1, is shown in Figure 1. It is found that the velocity

decreases as M increases.


Fig.2.

Effect of Grashof number Gr on velocity u for Pr=0.71,M=0.5, Gc=1, =1, R=0.3, K=1,Kr=1, w=1,i=

, =0.1,Ec=1,Sc=0.26,So=5, =1, 1 =1, is shown in Figure 2. It is observed that the velocity

increases as Gr increases.

Fig.3


Figure.3. This depicts the influence of Grashof number Gc on velocity u for Pr=0.71, =1,M=0.5,

Gr=1, =1, R=0.3, K=1,Kr=1, w=1,i= , =0.1,Ec=1,Sc=0.26,So=5, 1 =1, . It is noted from the

graph that the velocity increases as Gc increases.

Figure.4. Show the effect of permeability porous parameter K on velocity u for Pr=0.71, M=0.5, Gc=1,

=1, R=0.3, Kr=1, w=1, i= , =0.1, Ec=1, =1, Sc=0.26, So=5, 1 =1. It is evident that, the

velocity increases as K increases.


Fig.5. show the effect of on velocity u for Pr=0.71,M=0.5, Gc=1, =1, R=0.3, Kr=1, w=1,i= ,

=0.1,Ec=1,Sc=0.26,So=5, 1 =1, is shown in fig.3. It is observed that, the velocity increases as

increases.

Figure.6. Illustrated the effect of 1 on velocity u for Pr=0.71, M=0.5, Gc=1, =1, R=0.3, Kr=1,

=1, i= , =0.1, Ec=1, Sc=0.26, So=5, =1 is shown in fig.3. It is observed that, the velocity

increases as 1 increases.


Figure.7. Illustrated the effect of on velocity u for Pr=0.71,M=0.5, Gc=1, =1, R=0.3, Kr=1, i=

, =0.1,Ec=1,Sc=0.26,So=5, 1 , =1, =1 is shown in fig.3. It is noted that, as increases

velocity decreases.

Figure.8. Illustrated the effect of R on temperature for Pr=0.71,M=0.5, Gc=1, =1, 1 =1, Kr=1, i=

, =0.1,Ec=1,Sc=0.26,So=5, =1, =1 . It is noted that, as R increases temperature axis

decreases.

Figure.9. Show the effect of on temperature for Pr=0.71,M=0.5, Gc=1, 1 =1, R=0.3, Kr=1, i=

, =0.1,Ec=1,Sc=0.26,So=5, =1, =1is shown in fig.9. It is observed that, as increas

temperature axis decreases.


Figure.10. Show the effect of Pr on temperature for Pr=0.71,M=0.5, Gc=1, =1, 1 =1, R=0.3,

Kr=1, i= , =0.1,Ec=1,Sc=0.26,So=5, =1, =1. It is found that, as Pr increases temperature axis

also increases.

Figure.11. Show the impact of Ec on temperature for Pr=0.71,M=0.5, Gc=1, =1, 1 =1, R=0.3,

Kr=1, i= , =0.1,Ec=1,Sc=0.26,So=5, =1, =1. It is found that, as Ec increases temperature axis

also increases.


Figure.12. Depicts the effect of Sc on concentration for Pr=0.71,M=0.5, Gc=1, =1, 1 =1, R=0.3,

Kr=1, i= , =0.1,Ec=1, So=5, =1, =1. It is noted that, as Sc increases concentration axis

decreases.

Figure.13. Depicts the effect of So on concentration for Pr=0.71,M=0.5, Gc=1, =1, 1 =1, R=0.3,

Kr=1, i= , =0.1,Ec=1, Sc=0.26, =1, =1. It is noted that, as Sc increases concentration axis

decreases.


Fig.14. illustrates the effect of Kr on concentration for Pr=0.71,M=0.5, Gc=1, =1, 1 =1, R=0.3,

So=5, i= , =0.1,Ec=1, Sc=0.26, =1, =1. It is noted that, as Kr increases concentration axis

also increases.

Table Parameters Skin friction Nusselt number Sherwood number

M=0.5 0.786509234063952 1.53417341847432 0.343328859491213

M=1.0 0.753412715685220 1.52632752660818 0.353627882766775

M=1.5 0.723175569984999 1.51948502413997 0.362609789139015

Gr=1 0.753412715685220 1.52632752660818 0.353627882766775

Gr=3 1.10138054040062 1.63608868436843 0.209455496844945

Gr=5 1.47433688633081 1.80399640907544 0.0110898039159554

Gc=1 0.753412715685220 1.52632752660818 0.353627882766775

Gc=3 0.976986598624846 1.59867708062249 0.258533981342738

Gc=5 1.18043796373808 1.68806016817170 0.141006044399500

K=0.1 0.440412497174142 1.47057307039731 0.426813641022650

K=0.3 0.631641082488556 1.50066746343869 0.387310847746046

K=0.5 0.695445926632879 1.51348331452627 0.370488009409741


Pr=1 0.753115799401325 1.56043807588558 0.308968246728234

Pr=3 0.751166948675777 1.79349990912435 0.00391702204178215

Pr=5 0.749374736416404 2.02308188110744 -0.29643958012034

=0.5 0.757042573385750 1.28136728192595 0.676013700330644

=1 0.753412715685220 1.52632752660818 0.353627882766775

=1.5 0.750680481264154 1.772522182666212 0.0306441171977730

R=1 0.756306233408737 1.40113583317533 0.519592075897442

R=2 0.760072895474194 1.24351611540520 0.728689492977149

R=3 0.763453971250303 1.10740524208311 0.909403774148898

Kr=5 0.771026079591195 1.53059236679131 0.513879174009839

Kr=10 0.800377157340369 1.53789467323389 0.789998520896985

Kr=15 0.842761929709430 1.54889862708871 1.20206599361492

Sc=0.22 0.757938140270703 1.52726259043143 0.453043584132732

Sc=0.60 0.714469455097467 1.51850417726917 -0.491697567932285

Sc=0.97 0.671026357641268 1.51025267009304 -1.41405749145483

So=5 0.753412715685220 1.52632752660818 0.353627882766775

So=10 0.720321813286112 1.51954256306286 -0.321688581931267

So=15 0.687795517071836 1.51317525526134 -0.980847623545117

1=0.1 0.753412715685220 1.52632752660818 0.353627882766775

1=0.3 0.863687592726171 1.54992090521622 0.322665856146396

1=0.5 0.967799743890190 1.57464962526463 0.290216102372919

=0.1 0.333909883406734 1.46898187259426 0.428853660609605

=0.3 0.427164430282350 1.47850627820395 0.416355917190486


=0.5 0.520400821408634 1.48987108303624 0.401446008674616

Ec=5 0.749854280325509 1.83852939621731 -0.056107616710659

Ec=10 0.745573091163929 2.22114674848053 -0.558172018644775

Ec=15 0.741461223475744 2.59579951585727 -1.04970017176913

=1 0.753412715685220 1.52632752660818 0.353627882766775

=3 0.722687967786060 1.45397973095791 0.430676412890135

=5 0.694374167170062 1.38603254505062 0.500752356264130

The table above depict the various parameters , A

strong flow circulation is observed when the

following parameters were high Gr,Gc, K, 1,

, . It was discovered that increase in M and Skin friction reduces, as the temperature gradient

increase the parameters of Pr, Ec, also increases

while R is dropping, the table result of contradict the graph result: the table shows

increase while graph depict decreases velocity . It

was noted that Sherwood number Increases in

concentration profile, the Sc and So decreases,

while Kr is getting high.

The results obtained could be used for design and

development of MHD generator, food processing,

heat exchanger as in the nuclear reactor and some

metallurgical processes. It is also use in

agricultural engineering to study the underground

water resources.

Conclusion

The effect of heat and mass transfer on MHD

oscillatory flow of Jeffery fluid through a porous

medium in a channel in the presence of thermal

radiation and Soret is investigated in this paper.

The expressions for the velocity, temperature and

concentration are solved numerically by using the

shooting technique with fourth order Runge-kutta-

method, it is observed that, velocity increseases

with increasing , K, Gr, Gc, 1 , while it

decreases with increasing M, . Also it was found

that the temperature profile increases with

increasing Pr, Ec, while it decreases with

increasing and R, It is discovered that

concentration profile increases with increasing

Kr, while it decreases with increasing Sc and So.

The illustrations depicts that, the velocity is more

of Jeffery fluid than that of Newtonian fluid.

References

Anand, J.Rao, R. Srinivasa R. and Sivaiah,

S.(2012): Finite element solution of heat and mass

transfer inMHD flow of a viscous fluid past

vertical porous plate under oscillatory suction

velocity. Journal ofapplied fluid mechanics, Vol

5, No 3, pp. 1-10,2012

Anuradha, S.(2104): Heat and mass transfer on

unsteady MHD free convective flow past a

semi-infinite vertical plate with soret effect.

International journal of science and technoledge,

Appliedfluid mechanics, Vol 5, No 3, pp. 1

10,2012

Bhupendra K. Sharma : Soret and Dufor effect on

unsteady MHD mixed convection flow past a

radiative vetical porous plate embedded in a

porous medium with a chemical reaction.


(http://www, SciRP.org/journal/am). Applied

Mathematics, 2012, 3,717-723

Idowu, A.S, Joseph, K.M and Danniel (2013):

Effect of heat and mass transfer on unsteady

MHDoscillatory flow of Jeffery fluid in a

horizontal channel with chemical reaction. IOSR:

Journal of Mathematics (IOSR-Jm), e-ISSN: 2278-

5728, p-ISSN: 2319-765X. Volume8, Issue 5(Nov-

Dec,2013),pp74-87

Kavita K.(2012) : Influence of heat transfer on

MHD oscillatory flow of Jeffery fluid in a

channel’’. Adv.Appl.Sci.Res.3 (4):2312-2325.

Srinivasa, K. R. and Koteswara, P. R.(2012):

Effect of heat transfer on MHD oscillatory flow of

Jeffery

fluid through a porous medium in a tube.

International journal of mathematical Archive

3(11) Uwanta,I. J. and Omokhuale, E.(2014): Effects

variable thermal conductivity on heat and mass

transfer with Jeffery fluid. International journal of

mathematical Archive-5(3),2014,135-149.

Vidyasadar, G.(2013):Radiation effect on MHD

free convection flow of kuvshinshiki fluid with

mass transfer past a vertical porous plate through

porous medium. Asian Journal of Current

Engineering andMaths. 2:3 May-June (2013)

170-174.

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