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