Heat and Mass transfer on MHD Rotating flow of a …Heat and Mass transfer on MHD Rotating flow of a...
Transcript of Heat and Mass transfer on MHD Rotating flow of a …Heat and Mass transfer on MHD Rotating flow of a...
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 9 (2019) pp. 2107-2120
© Research India Publications. http://www.ripublication.com
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Heat and Mass transfer on MHD Rotating flow of a Visco-elastic fluid
through porous medium
Gorakhnath Waghmode 1* and S.V.Suneetha2
1Research Scholar, Department of Mathematics, Rayalaseema University, Kurnool, Andhra Pradesh – 518007, India.
2Department of Mathematics, Rayalaseema University, Kurnool, Andhra Pradesh – 518007, India.
Abstract
We discussed the heat and mass transfer on MHD free
convective rotating flow of a visco-elastic incompressible
electrically conducting fluid past a vertical porous plate
through a porous medium with suction and heat source. We
analyze the effect of time dependant fluctuative suction on a
visco-elastic fluid flow The coupled nonlinear partial
differential equation are turned to ordinary by super imposing
a solution with steady and time dependant transient part.
Finally, the set of ordinary differential equations is solved
with a perturbation scheme to meet the inadequacy of
boundary condition. Elasticity of the fluid and the Lorentz
force reduce the velocity and it is more pronounced in case of
heavier species. Most interesting observation is the fluctuation
of velocity appears near the plate due to the presence of sink
and presence of elastic element as well as heat source reduces
the skin friction.
Keywords: Convective flows, heat and mass transfer, porous
medium, visco-elastic fluids, oscillatory permeability.
1. INTRODUCTION
An important class of two dimensional time dependent flow
problem dealing with the response of boundary layer to
external unsteady fluctuations of the free stream velocity
about a mean value attracted the attention of many
researchers. MHD flow with heat and mass transfer has been a
subject of interest of many researchers because of its varied
application in science and technology. Such phenomena are
observed in buoyancy induced motions in the atmosphere, in
water bodies, quasi-solid bodies such as earth, etc. In natural
processes and industrial applications many transportation
processes exist where transfer of heat and masstakes place
simultaneously as a result of thermal diffusion and diffusion
of chemical species. Several researchers have analyzed the
free convective and mass transfer flow of a viscous fluid
through porous medium. The permeability of the porous
medium is assumed to be constant while the porosity of the
medium may not be necessarily being constant. Kim [1]
studied the unsteady MHD convective heat past a semiinfinite
vertical porous moving plate with variable suction. The
problem of three dimensional free convective flow and heat
transfer through porous medium with periodic permeability
has been discussed by Singh and Sharma [2]. Singh and Singh
[3] have analyzed the heat and mass transfer in MHD flow of
a viscous fluid past a vertical plate under oscillatory suction
velocity. Bathul [4] discussed the heat transfer in a three
dimensional viscous flow over a porous plate moving with
harmonic disturbance. Postelnicu [5] studied numerically the
influence of a magnetic field on heatand mass transfer by
natural convection from vertical surfaces in porous media
considering Soret and Dufour effects. Singh and Gupta [6]
have investigated the MHD free convective flow of a viscous
fluid through a porous medium bounded by oscillatory porous
plate in slip flow regime with mass transfer. Ogulu and
Prakash [7] considered heat transfer to unsteady magneto
hydrodynamic flow past an infinite vertical moving plate with
variable suction. Das et al. [8] analyzed the mass transfer
effect on unsteady flow past an accelerated vertical porous
plate with suction employing numerical methods. The basic
equations of incompressible MHD flow are non-linear. But
there are many interesting cases where the equations became
linear in terms of the unknown quantities and may be solved
easily. Linear MHD problems are accessible to exact solutions
and adopt the approximations that the density and transport
properties be constant. No fluid is incompressible but all may
be treated as such whenever the pressure changes are small in
comparison with the bulk modulus. Ferdows et al. [9]
analyzed free convection flow with variable suction in
presence of thermal radiation. Alam et al. [10] studied Dufour
and Soret effect with variable suction on unsteady MHD free
convection flow along a porous plate. Mishra et al. [16] have
studied fluctuating flow of a non-Newtonian fluid past a
porous flat plate with time varying suction. Majumdar and
Deka [11] gave an exact solution for MHD flow past an
impulsively started infinite vertical plate in the presence of
thermal radiation. Muthucumaraswamy et al. [12] studied
unsteady flow past an accelerated infinite vertical plate with
variable temperature and uniform mass diffusion. Asghar et
al. [13] have reported the flow of a non-Newtonian fluid
induced due to oscillations of a porous plate. Moreover, Dash
et al. [14] have studied free convective MHD flow of a visco-
elastic fluid past an infinite vertical porous plate in a rotating
frame of reference in the presence of chemical reaction. The
specific area of application of heat and mass transfer
phenomena is common in chemical process industries such as
food processing and polymer production. Recently, Gireesh
Kumar and Satyanarayana [17], Sivraj and Rushi Kumar [18]
have considered MHD flow of visco-elastic fluid with short
memory (Walters model). Many authors [20-24] discussed
MHD flows through porous medium in planar channels.
Recently, Krishna and Gangadhar Reddy [25] discussed the
unsteady MHD free convection in a boundary layer flow of an
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electrically conducting fluid through porous medium subject
to uniform transverse magnetic field over a moving infinite
vertical plate in the presence of heat source and chemical
reaction. Krishna and Subba Reddy [26] have investigated the
simulation on the MHD forced convective flow through
stumpy permeable porous medium (oil sands, sand) using
Lattice Boltzmann method. Krishna and Jyothi [27] discussed
the Hall effects on MHD Rotating flow of a visco-elastic fluid
through a porous medium over an infinite oscillating porous
plate with heat source and chemical reaction. Reddy et al.[28]
investigated MHD flow of viscous incompressible nano-fluid
through a saturating porous medium. Recently, Krishna et al.
[29-32] discussed the MHD flows of an incompressible and
electrically conducting fluid in planar channel. Veera Krishna
et al. [33] discussed heat and mass transfer on unsteady MHD
oscillatory flow of blood through porous arteriole. The effects
of radiation and Hall current on an unsteady MHD free
convective flow in a vertical channel filled with a porous
medium have been studied by Veera Krishna et al. [34]. The
heat generation/absorption and thermo-diffusion on an
unsteady free convective MHD flow of radiating and
chemically reactive second grade fluid near an infinite vertical
plate through a porous medium and taking the Hall current
into account have been studied by Veera Krishna and
Chamkha [35]. Veera Krishna et al. [36] discussed the heat
and mass transfer on unsteady, MHD oscillatory flow of
second-grade fluid through a porous medium between two
vertical plates under the influence of fluctuating heat
source/sink, and chemical reaction. Veera Krishna et al. [37]
investigated the heat and mass transfer on MHD free
convective flow over an infinite non-conducting vertical flat
porous plate. Veera Krishna and Jyothi [38] discussed the
effect of heat and mass transfer on free convective rotating
flow of a visco-elastic incompressible electrically conducting
fluid past a vertical porous plate with time dependent
oscillatory permeability and suction in presence of a uniform
transverse magnetic field and heat source. Veera Krishna and
Subba Reddy [39] investigated the transient MHD flow of a
reactive second grade fluid through a porous medium between
two infinitely long horizontal parallel plates. Veera Krishna et
al. [40] discussed heat and mass-transfer effects on an
unsteady flow of a chemically reacting micropolar fluid over
an infinite vertical porous plate in the presence of an inclined
magnetic field, Hall current effect, and thermal radiation taken
into account. Veera Krishna et al.[41] discussed Hall effects
on steady hydromagnetic flow of a couple stress fluid through
a composite medium in a rotating parallel plate channel with
porous bed on the lower half. Veera Krishna et al. [42]
discussed Hall effects on unsteady hydromagnetic natural
convective rotating flow of second grade fluid past an
impulsively moving vertical plate entrenched in a fluid
inundated porous medium, while temperature of the plate has
a temporarily ramped profile. Veera Krishna and Chamkha
[43] discussed the MHD squeezing flow of a water-based
nanofluid through a saturated porous medium between two
parallel disks, taking the Hall current into account. Veera
Krishna et al. [44] discussed Hall effects on MHD peristaltic
flow of Jeffrey fluid through porous medium in a vertical
stratum.
In the above studies mentioned, therefore, the objective of the
present study is to analyze the effect of permeability variation
and oscillatory suction velocity in free convective mass
transfer rotating flow of a visco-elastic fluid (Walters
B’fluid model) past an infinite vertical porous plate through
a porous medium in presence of a uniform transverse
magnetic field and heat source.
2. FORMULATION AND SOLUTION OF THE
PROBLEM
The unsteady free convective rotating flow of a visco-elastic
(Walters B’) fluid past an infinite vertical porous plate in a
porous medium with time dependent oscillatory suction in
presence of a transverse magnetic field is considered. Let x-
axis be along the plate in the direction of the flow and y-axis
normal to it.
Fig. 1: The Physical model
Let us consider the magnetic Reynolds number is much less
than unity so that the induced magnetic field is neglected in
comparison with the applied transverse magnetic field. The
basic flow in the medium is, therefore, entirely due to the
buoyancy force caused by the temperature difference between
the wall and the medium. It is assumed that initially, at t > 0,
the plate as well as fluids is at the same temperature and also
concentration of the species is very low so that the Soret and
Dofour effects are neglected. When t > 0, the temperature of
the plate is instantaneously raised to Tw and the concentration
of the species is set to Cw. The physical configuration of the
problem as shown in Fig. 1.
Let the suction velocity be of the form
0( ) (1 )i tw t w e (1)
where 0 0w and 1 are positive constants.
Under the above assumption with usual Boussinesq’s
approximation, the governing equations with respect to the
rotating frame are given by
0u vx y
(2)
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2 3 3
0
2 2 3
12
ku u p u u uw vt z x z z t z
2
*0 ( ) ( )p
B u u g T T g C CK
(3)
2
2
23 3
0 0
2 3
2
1
p
v vw ut z
p vy z
k Bv vv v vz t z K
(4)
2
12( )
T T Tw k S T Tt z z
(5)
2
2
C C Cw Dt z z
(6)
The boundary conditions are
0, ,
at 0
i tw
i tw
u T T T T e
C C C C e y
(7)
0, , asu T T C C y (8)
Combining (1) & (2) we obtain, let
andq u iv x iy
3 3 3
0
2 2 3
2
1
q qw i qt z
kp q q qz z t z
2
0 ( ) *( )p
B vq q g T T g C CK
(9)
Introducing the non-dimensional quantities,
2
0 0
2
0 0
4* , * , * , * ,
4
, ,
w w
w z w t qz t qw w
T T C pPT T C
Making use of non-dimensional variables, the governing
equations reduces to
22
2
3 3
2 3
1 1(1 ) 2
4
(1 )
i t
i t
q q qe iEq P M qt z z K
q qRc et z z
Gr Gc (11)
2
2
1 1(1 )
4 Pr
i te St z z
(12)
2
2
1 1(1 )
4 Sc
i tet z z
(13)
The boundary conditions are,
0, 1 , 1 at 0
0, 0, 0 as
i t i tq e e zq z
(14)
Where,
1
2
0
SSv
is the Heat source parameter,
2
0
2
pw KK
is the
Permeability parameter, 2
2 0
2
0
BMw
is the Hartmann
number, 2
0
Ew
is the rotation parameter, Prk
is the
Prandtl number, 3
0
*( )Gr wg T T
w
is the Thermal
Grashof number, 3
0
( )Gc wg C C
w
is the mass
Grashof number, ScD
is the Schmidt number,
0 0
24
k wRc
is the visco-elastic parameter.
In view of periodic suction, temperature and concentration at
the plate let us assume the velocity, temperature,
concentration in the neighbourhood of the plate be
0 1, i tq y t u y u y e (15)
0 1, i ty t T y T y e (16)
0 1, i ty t C y C y e (17)
The above method of solution has been adopted by Mishra et
al. [16], Das et al. [19] and Gireesh Kumar and Satyanarayana
[17] to solve the periodically fluctuating flow problems.
Substituting Eqs. (15) – (17) into (11) – (13) and equating the
non-harmonic and harmonic terms, we get
'" '" ' 2
0 0 0 0
0 0
1Rc 2
Gr Gc
u u u M iE uK
T C (18)
'" '" ' 2
1 1 1 1
3
0 0 01 13
1Rc 1 Rc 2
Rc Gr Gc
u i u u M iE uK
d u du uT Cdy dy K
(19)
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" '
0 0 0Pr Pr 0T T ST (20)
" ' '
1 1 1 0Pr Pr4
iT T P S T T
(21)
" '
0 0Sc 0C C (22)
" ' '
1 1 1 0Sc Sc Sc 4
iC C T C (23)
The boundary conditions reduce to
0 1 0 1 0 1
0 1 0 0 1
0, 1, 0 at 0,
0, 0, 0, as
u u T T C C yu u T T C C y
(24)
Eqs. (18) and (19) are of third order but two boundary
conditions are available. Therefore the perturbation method
has been applied using 1Rc Rc , the elastic parameter
as the perturbation parameter.
2
0 00 01
2
1 10 11
( ) Rc ( ) Rc ,
( ) Rc ( ) Rc
u u y u y O
u u y u y O
(25)
Inserting Eq. (25) into (18) and (19) and equating the
coefficients of Rc0 and Rc to zero we have following sets of
ordinary differential equations zeroth order and first or der;
" ' 2
00 00 00 0 0
12 Gr Gc u u M iE u T C
K
(26)
" ' 2 '"
01 01 01 00
12u u M iE u u
K
(27)
" ' 2
10 10 10
' 0000 1 1
12
Gr Gc
u u M iE uK
uu T CK
(28)
" ' 2
11 11 11
'" " '" 0110 10 00
12
u u M iE uK
uu i u uK
(29)
The corresponding boundary conditions are:
00 01
10 11
0, as 0,
0, as
u u yu u y
(30)
Solving these differential equations with the help of boundary
conditions we get,
4
1
1 22
Sc
1 22
( , )1
2
12
m yo
m y y o
Pq y t a a eM iE
KPa e a e
M iEK
4 4 1 Sc
3 4 5 3 4 5( ) )m y m y m y ycR a a a e a e a e a e
5 34
1 2
5 34
2 1
118 12 13
6
Sc 114 15 16 17
6
25 31 26 27
Sc
25 29 30
( )
m y m ym y
m y m yy
m y m ym y
m y m y y i t
Pa e a e a ea
Pa e a e a e aa
Rc a a e a e a e
a e a e a e e
(31)
3 31 114( , )
m y m ym y m y i timy t e e e e e
(32)
24 4
( , ) 1c cS y S ym y i tc ciS iSy t e e e e
(33)
The skin friction at the plate in terms of amplitude and phase
angle is given by
0
0 010
0
cosy
i ty
y
u uue N ty y y
(34)
Where, , tan ir i
r
NN N iNN
The rate of heat transfer, i.e. heat flux at the (Nu) in terms
of amplitude and phase is given by,
0
0 10
0
0 cos
y
i tu y
y
T TN ey y
T R ty
(35)
, tan ir i
r
RR R iRR
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The mass transfer coefficient, i.e, the Sherwood number (Sh)
at the plate in terms of amplitude and phase is given by
0
0 10
0
0 cos
y
i th y
y
C CS ey y
C Q ty
(36)
Where, , tan ir i
r
QQ Q iQQ
4. RESULTS AND DISCUSSION
The present study considers the effect of mass transfer on free
convective rotating flow and heat transfer of a visco-elastic
incompressible electrically conducting fluid past a vertical
plate through a porous medium with time dependent suction in
presence of a constant transverse magnetic field and heat
source. The significance of the present study is to bring out
the effect of oscillatory characteristics of suction velocity,
plate temperature and concentration. The common feature of
the velocity profile is parabolic with picks near the plate. It is
interesting to note that due to dominating effect of heat source
no oscillatory behaviour of the profile is exhibited but in the
presence of sink an oscillation in the profile is well marked.
The result of the present study is verified by comparing with
that of Das et al. [19] for viscous case Rc = 0. The effect of Rc
is to thinning of velocity boundary layer which may be
attributed due to elastic property of the fluid under
consideration. Our result is in conformity with the result of
Das et al. [19].
The flow governed by the non-dimensional parameters
Hartmann number M, permeability parameter K, rotation
parameter E, thermal Grashof number Gr, mass Grashof
number Gc, Elastic parameter Rc, the frequency of
oscillation, Schmidt number Sc, Prandtl number Pr, Heat
source parameter S, time t. The velocity, temperature and
concentration profiles are shown on Figures (2-12), Figures
(13-16) and Figures (17-18) respectively. Tables (1-3)
represent the skin friction, Nusselt number and Sherwood
number for different variations in the governing parameters.
We noticed that from the Fig. 2, the velocity components u
and v reduce with increasing the Hartmann number M.
Further, due to interaction of applied magnetic field with
conducting fluid the current is generated. Consequently, it
gives rise to an induced magnetic field and ponders motive
force. That force acts opposite to the direction of the flow and
unless an additional electric field is applied to counter act. In
the present study the Lorentz force decelerates the fluid flow
consequently resulted in thinning of boundary layer. The
resultant velocity is also reduces with increasing the intensity
of the magnetic field. Both the velocity components u and v
enhance with increasing permeability parameter K. i.e., both
the real and imaginary parts of the velocity enhance with
increasing the permeability parameter K throughout the fluid
region. We also observe that lower the permeability lesser the
fluid speed in the entire fluid medium (Fig. 3). The similar
behaviour is observed with increasing the thermal Grashof
number Gr, mass Grashof number Gc or rotation parameter E
(Figs. 4-6). It is also observed that, from the Figures (7-8)
both the velocity components continuously retardation in its
magnitude with increasing elastic parameter Rc or the
frequency of oscillation . The figure (9) reveals that, the
primary velocity component u reduces and secondary velocity
component v increases for 2z and the gradually reduces
with increasing Schmidt number Sc. Likewise, from the
figures (10 & 11) both the velocity components u and v
reduces with increasing Heat source parameter S or Prandtl
number Pr. The resultant velocity is also reduces with
increasing S or Pr. The effect of source parameter is well
marked from Fig. (10). Presence of source reduces the
velocity, significantly in all the layers. Most interesting case is
due to the presence of sink which gives rise to a span wise
oscillatory flow after a few layers from the plate. This is due
to depletion of thermal energy causing a sudden reduction of
velocity. Again it is compensated due to buoyancy effect.
Finally with increasing time the velocity components u, v and
resultant velocity enhance throughout the fluid region Figs.
12). Hence we observed that the free convection due to
thermal buoyancy, mass buoyancy and porosity of the
medium enhances the fluid velocity whereas Schmidt number
and Prandtl number reduce it. Kumar et al. [17] observed that
the velocity profile increases with increase of thermal Grashof
number and Solutal Grashof number and they have also
observed that the velocity decreases with increase in magnetic
parameter. Hence our observation in respect of these
parameters agree with [17] in the absence of porous medium.
Thus, the above result indicates that heavier species with
lower thermal conductivity reduces the fluid flow.
Figs. (13-16) exhibits the effects of Prandtl number (Pr),
source/sink parameter S, the frequency of oscillation and time
on temperature field. The variation is asymptotic in nature.
Sharp fall of temperature is noticed in case of water (Pr = 7.0)
for both source/sink. Absence of source contributes to
increase the thickness of thermal boundary layer. Higher
Prandtl number fluid causes lower thermal diffusivity and
hence reduces the temperature at all points. It is interesting to
note that the effect of source is to lower down the temperature
than the effect of sink. The temperature increases with
increasing frequency of oscillation and time throughout the
fluid region. From Figs. (17-19), it is observed that the
concentration distribution asymptotically decreases with the
heavier species. This result is in good agreement with Das et
al. [19]. Likewise the concentration enhances with increasing
frequency of oscillation and time throughout the fluid region.
We noticed from the Table (1) we present the values of skin
friction. The increase of visco-elastic parameter (Rc) leads to
decrease the skin friction. Further, it is to note that an increase
in Gr, Gc, Pr, and K leads to exert greater skin friction on
the boundary whereas Sc, M, S, t and Rc reduce it.
From the Table 2 it is seen that an increase in Pr or S leads to
an increase in Nusselt number. Nusselt number reduces with
increasing the frequency of oscillation. There is no effect of
time on Nusselt number.
It is noticed from the table (3) that the Schmidt number i.e.
mass transfer coefficient affects Sherwood number in a
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similar manner as that of Prandtl number, the frequency of oscillation and time in heat transfer.
Fig. 2 The velocity profiles for the components u and v against M
0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1K E S t
Fig. 3 The velocity profiles for the components u and v against K
1, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1M E S t
Fig. 4 The velocity profiles for the components u and v against E
1, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1M K S t
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Fig. 5 The velocity profiles for the components u and v against Gr
1, 0.5, 0.5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1M K E S t
Fig. 6 The velocity profiles for the components u and v against Gc
1, 0.5, 0.5,Gr 5,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1,M K E S t
Fig. 7 The velocity profiles for the components u and v against Rc
1, 0.5, 0.5,Gr 5,Gc = 3, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1M K E S t
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Fig. 8 The velocity profiles for the components u and v against
1, 0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, 0.5,Sc 0.22,Pr 0.71, 0.1M K E S t
Fig. 9 The velocity profiles for the components u and v against Sc
1, 0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, 0.5,Sc 0.22,Pr 0.71, 0.1M K E S t
Fig. 10 The velocity profiles for the components u and v against S
1, 0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6,Sc 0.22,Pr 0.71, 0.1,M K E t
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Fig. 11 The velocity profiles for the components u and v against Pr
1, 0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22, 0.1M K E S t
Fig. 12 The velocity profiles for the components u and v against t
1, 0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71M K E S
Fig. 13 The temperature profiles for against Pr with 0.01, / 6, 0.5, 0.1S t
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Fig. 14 The temperature profiles for against S with 0.01, / 6,Pr 0.71, 0.1t
Fig. 15 The temperature profiles for against with 0.01, 0.5,Pr 0.71, 0.1S t
Fig. 16 The temperature profiles for against t with 0.01, / 6, 0.5,Pr 0.71S
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Fig. 17 The Concentration profiles for against t with 0.01, / 6,Sc 0.22, 0.1t
Fig. 18 The Concentration profiles for against t with 0.01,Sc 0.22, 0.1t
Fig. 19 The Concentration profiles for against t with 0.01, / 6,Sc 0.22
Table 1: Shear stresses
M K S Gr Gc Sc ω Rc Pr t
1 0.5 0.5 5 3 0.22 / 6 0.001 0.71 0.1 0.647996
1.5 0.509526
2 0.400457
1 0.714341
1.5 0.749688
1 0.601219
1.5 0.579154
6 0.651212
7 0.653632
4 0.666612
5 0.682836
0.3 0.621506
0.6 0.544473
/ 4 0.913730
/ 3 1.135170
0.01 0.647996
0.05 0.647996
3 2.666330
7 -0.11972
0.5 0.438557
0.8 0.281477
Table 2: Nusselt number
Pr ω t S Nu
0.71 / 6 0.1 0.5 1.049300
3 3.443800
7 7.491330
/ 4 1.049020
/ 3 1.048870
0.5 1.049300
0.8 1.049300
0.8 1.189220
1 1.270730
Table 3: Sherwood number
Sc ω t Sh
0.22 / 6 0.1 0.220035
0.3 0.300029
0.6 0.600013
/ 4 0.220062
/ 3 0.220087
0.5 0.220035
0.8 0.220035
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© Research India Publications. http://www.ripublication.com
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4. CONCLUSIONS
We have considered the effect of heat and mass transfer on
free convective rotating flow of a visco-elastic incompressible
electrically conducting fluid past a vertical porous plate
through a porous medium with time dependent suction in
presence of a uniform transverse magnetic field and heat
source. The conclusions are made as the following.
1. Presence of heat source prevents the oscillatory flow
whereas the sink favours it. Elasticity of the fluid is
responsible for thinning of the boundary layer.
2. Application of magnetic field decelerates the fluid
flow.
3. The heavier species with low conductivity reduces
the flow within the boundary layer.
4. An increase in elasticity of the fluid leads to
decrease the velocity which is an established result.
5. The concentration distribution asymptotically
decreases with the heavier species.
6. Elasticity’s of the fluid and heat source reduce the
skin friction providing a favourable condition for
stretching.
Appendix:
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REFERENCES
[1]. Y.J. Kim, Unsteady MHD convective heat transfer past
a semi-infinite vertical porous moving plate with
variable suction, Int. J. Eng. Sci. 38 (8) (2000) 833–
845.
[2]. K.D. Singh, R. Sharma, Three dimensional free
convective flow and heat transfer through a porous
medium with periodic permeability, Indian J. Pure
Appl. Math. 33 (6) (2002) 941–949.
[3]. A.K. Singh, N.P. Singh, Heat and mass transfer in
MHD flow of a viscous fluid past a vertical plate under
oscillatory suction velocity, Indian J. Pure Appl.Math.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 9 (2019) pp. 2107-2120
© Research India Publications. http://www.ripublication.com
2119
34 (3) (2003) 429–442.
[4]. S. Bathul, Heat transfer in three dimensional viscous
flow over a porous plate moving with harmonic
disturbances, Bull. Pure Appl. Sci. E 24 (1) (2005) 69–
97.
[5]. A. Postelnicu, Influence of a magnetic field on heat
and mass transfer by natural convection from vertical
surfaces in porous media considering Soret and Dufour
effects, Int. J. Heat Mass Transfer 47 (2004) 1467–
1472.
[6]. P. Singh, C.B. Gupta, MHD free convective flow of a
viscous fluid through a porous medium bounded by an
oscillating porous plate in slip flow regime with mass
transfer, Indian J. Theor. Phys. 53 (2) (2005) 111–120.
[7]. A. Ogulu, J. Prakash, Heat transfer to unsteady
magneto hydrodynamics flow past an infinite moving
vertical plate with variable suction, Phys. Scr. 74
(2006) 232–239.
[8]. S.S. Das, S.K. Sahoo, G.C. Dash, Numerical solution
of mass transfer effects on unsteady flow past an
accelerated vertical porous plate with suction, Bull.
Malays. Math. Sci. Soc. 29 (1) (2006) 33–42.
[9]. M. Ferdows, M.A. Sattar, M.N.A. Siddiqui, Numerical
approach on parameters of the thermal radiation
interaction with convection in boundary layer flow at a
vertical plate with variable suction, Thammasat Int. J.
Sci. Technol. 9 (2004) 19–28.
[10]. M.S. Alam, M.M. Rahman, M.A. Samad, Dufour and
Soret effects on unsteady MHD free convection and
mass transfer flow past a vertical plate in a porous
medium, Nonlinear Analysis: Modeling and Control 11
(2006) 211–226.
[11]. M.K. Majumder, R.K. Deka, MHD flow past an
impulsively started infinite vertical plate in the
presence of thermal radiation, Rom. J. Phys. 52 (5–7)
(2007) 565–573.
[12]. R. Muthucumaraswamy, M. Sundarraj, V.
Subhramanian, Unsteady flow past an accelerated
infinite vertical plate with variable temperature and
uniform mass diffusion, Int. J. Appl. Math. Mech. 5 (6)
(2009) 51–56.
[13]. S. Asghar, M.R. Mohyuddin, T. Hayat, A.M. Siddiqui,
The flow of a non-Newtonian fluid induced due to the
oscillation of a porous plate, Math. Probl. Eng. 2
(2004) 133–143.
[14]. G.C. Dash, P.K. Rath, N. Mohapatra, P.K. Dash, Free
convective MHD flow through porous media of a
rotating visco-elastic fluid past an infinite vertical
porous plate with heat and mass transfer in the
presence of a chemical reaction, AMSE Model. B 78
(4) (2009) 21–37.
[15]. S.K. Khan, E. Sanjayanand, Visco-elastic boundary
layer flow and heat transfer over an exponential
stretching sheet, Int. J. Heat Mass Transfer 48 (2005)
1534–1542.
[16]. S.P. Mishra, G.C. Dash, J.S. Roy, Fluctuating flow of a
non-Newtonian fluid past a porous flat plate with time
varying suction, Indian J. Phys. 48 (1974) 23–36.
[17]. J. Gireesh Kumar, P.V. Satyanarayana, Mass transfer
effect on MHD unsteady free convective Walters
memory flow with constant suction and heat sink, Int.
J. Appl. Math Mech. 7 (19) (2011) 97–109.
[18]. R. Sivraj, B. Rushi Kumar, Unsteady MHD dusty
visco-elastic fluid Couette flow in an irregular channel
with varying mass diffusion, Int. J. Heat Mass Transfer
55 (2012) 3076–3089.
[19]. S.S. Das, A. Satapathy, J.K. Das, J.P. Panda, Mass
transfer effects on MHD flow and heat transfer past a
vertical porous plate through a porous medium under
oscillatory suction and heat source, Int. J. Heat Mass
Transfer 52 (2009) 5962–5969.
[20]. A. Postelnicu, Influence of a magnetic field on heat
and mass transfer by natural convection from vertical
surfaces in porous media considering Soret and Dufour
effects, Int. J. Heat Mass Transfer 47 (2004) 1467–
1472.
[21]. P. Singh, C.B. Gupta, MHD free convective flow of a
viscous fluid through a porous medium bounded by an
oscillating porous plate in slip flow regime with mass
transfer, Indian J. Theor. Phys. 53 (2) (2005) 111–120.
[22]. A. Ogulu, J. Prakash, Heat transfer to unsteady
magneto hydrodynamics flow past an infinite moving
vertical plate with variable suction, Phys. Scr. 74
(2006) 232–239.
[23]. S.S. Das, S.K. Sahoo, G.C. Dash, Numerical solution
of mass transfer effects on unsteady flow past an
accelerated vertical porous plate with suction, Bull.
Malays. Math. Sci. Soc. 29 (1) (2006) 33–42.
[24]. M. Ferdows, M.A. Sattar, M.N.A. Siddiqui, Numerical
approach on parameters of the thermal radiation
interaction with convection in boundary layer flow at a
vertical plate with variable suction, Thammasat Int. J.
Sci. Technol. 9 (2004) 19–28.
[25]. Veera Krishna.M., M.Gangadhar Reddy, MHD Free
Convective Boundary Layer Flow through Porous
medium Past a Moving Vertical Plate with Heat Source
and Chemical Reaction, Materials Today: Proceedings,
vol. 5, pp. 91–98, 2018.
https://doi.org/10.1016/j.matpr.2017.11.058.
[26]. Veera Krishna.M., G.Subba Reddy, MHD Forced
Convective flow of Non-Newtonian fluid through
Stumpy Permeable Porous medium, Materials Today:
Proceedings, vol. 5, pp. 175–183, 2018.
https://doi.org/10.1016/j.matpr.2017.11.069.
[27]. Veera Krishna.M., Kamboji Jyothi, Hall effects on
MHD Rotating flow of a Visco-elastic Fluid through a
Porous medium Over an Infinite Oscillating Porous
Plate with Heat source and Chemical reaction,
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 9 (2019) pp. 2107-2120
© Research India Publications. http://www.ripublication.com
2120
Materials Today: Proceedings, vol. 5, pp. 367–380,
2018. https://doi.org/10.1016/j.matpr.2017.11.094.
[28]. Reddy.B.S.K, M. Veera Krishna , K.V.S.N. Rao, R.
Bhuvana Vijaya, HAM Solutions on MHD flow of
nano-fluid through saturated porous medium with Hall
effects, Materials Today: Proceedings, vol. 5, pp. 120–
131, 2018.
https://doi.org/10.1016/j.matpr.2017.11.062.
[29]. VeeraKrishna.M., B.V.Swarnalathamma, Convective
Heat and Mass Transfer on MHD Peristaltic Flow of
Williamson Fluid with the Effect of Inclined Magnetic
Field,” AIP Conference Proceedings, vol. 1728, p.
020461, 2016. DOI: 10.1063/1.4946512
[30]. Swarnalathamma. B. V., M. Veera Krishna, Peristaltic
hemodynamic flow of couple stress fluid through a
porous medium under the influence of magnetic field
with slip effect AIP Conference Proceedings, vol.
1728, p. 020603, 2016. DOI: 10.1063/1.4946654.
[31]. VeeraKrishna.M., M.Gangadhar Reddy MHD free
convective rotating flow of Visco-elastic fluid past an
infinite vertical oscillating porous plate with chemical
reaction, IOP Conf. Series: Materials Science and
Engineering, vol. 149, p. 012217, 2016 DOI:
10.1088/1757-899X/149/1/012217.
[32]. VeeraKrishna.M., G.Subba Reddy Unsteady MHD
convective flow of Second grade fluid through a porous
medium in a Rotating parallel plate channel with
temperature dependent source, IOP Conf. Series:
Materials Science and Engineering, vol. 149, p.
012216, 2016. DOI: 10.1088/1757-
899X/149/1/012216.
[33]. Veera Krishna.M., B.V.Swarnalathamma and J.
Prakash, “Heat and mass transfer on unsteady MHD
Oscillatory flow of blood through porous arteriole,
Applications of Fluid Dynamics, Lecture Notes in
Mechanical Engineering, vol. XXII, pp. 207-224, 2018.
Doi: 10.1007/978-981-10-5329-0_14.
[34]. Veera Krishna.M, G.Subba Reddy, A.J.Chamkha,
“Hall effects on unsteady MHD oscillatory free
convective flow of second grade fluid through porous
medium between two vertical plates,” Physics of
Fluids, vol. 30, 023106 (2018); doi: 10.1063/1.5010863
[35]. Veera Krishna.M, A.J.Chamkha, Hall effects on
unsteady MHD flow of second grade fluid through
porous medium with ramped wall temperature and
ramped surface concentration, Physics of Fluids 30,
053101 (2018), doi:
https://doi.org/10.1063/1.5025542
[36]. Veera Krishna.M., K.Jyothi, A.J.Chamkha, Heat and
mass transfer on unsteady, magnetohydrodynamic,
oscillatory flow of second-grade fluid through a porous
medium between two vertical plates, under the
influence of fluctuating heat source/sink, and chemical
reaction, Int. Jour. of Fluid Mech. Res., vol. 45, no. 5,
pp. 459-477, 2018b. DOI:
10.1615/InterJFluidMechRes.2018024591.
[37]. Veera Krishna.M., M.Gangadhara Reddy,
A.J.Chamkha, Heat and mass transfer on MHD free
convective flow over an infinite non-conducting
vertical flat porous plate, Int. Jour. of Fluid Mech. Res.,
vol. 46, no. 1, pp. 1-25, 2019. DOI:
10.1615/InterJFluidMechRes.2018025004.
[38]. Veera Krishna.M., K.Jyothi, Heat and mass transfer on
MHD rotating flow of a visco-elastic fluid through
porous medium with time dependent oscillatory
permeability, J. Anal, vol. 25, no. 2, pp. 1-19, 2018.
https://doi.org/10.1007/s41478-018-0099-0.
[39]. Veera Krishna.M., Subba Reddy.G., Unsteady MHD
reactive flow of second grade fluid through porous
medium in a rotating parallel plate channel, J. Anal., vol. 25, no. 2, pp. 1-19, 2018.
https://doi.org/10.1007/s41478-018-0108-3.
[40]. Veera Krishna.M., B.V.Swarnalathamma,
Ali.J.Chamkha, Heat and mass transfer on
magnetohydrodynamic chemically reacting flow of
micropolar fluid through a porous medium with Hall
effects, Special Topics & Reviews in Porous media – An International Journal, vol. 9, no. 4, pp. 347-364,
2018.
DOI: 10.1615/SpecialTopicsRevPorousMedia.2018024
579.
[41]. Krishna, M. V., Prasad, R. S., & Krishna, D. V., Hall
effects on steady hydromagnetic flow of a couple stress
fluid through a composite medium in a rotating
parallelplate channel with porous bed on the lower
half. Advanced Studies in Contemporary Mathematics (Kyungshang), vol. 20, no. 3, pp. 457–468, 2010.
[42]. Veera Krishna.M., M.Gangadhara Reddy,
A.J.Chamkha, Heat and Mass Transfer on MHD
Rotating Flow of Second Grade Fluid Past an Infinite
Vertical Plate Embedded in Uniform Porous Medium
with Hall Effects, Applied Mathematics and Scientific Computing, Trends in Mathematics, pp. 417-427. DOI:
https://doi.org/10.1007/978-3-030-01123-9_41
[43]. Veera Krishna.M, Ali J. Chamkha, Hall effects on
MHD Squeezing flow of a water based nano fluid
between two parallel disks, Journal of porous media,
vol. 22, no. 2, pp. 209-223, 2019. DOI:
10.1615/JPorMedia.2018028721
[44]. M. Veera Krishna, K.Bharathi, Ali.J.Chamkha, Hall
effects on MHD peristaltic flow of Jeffrey fluid
through porous medium in a vertical stratum,
Interfacial Phenomena and Heat transfer, vol. 6, no. 3,
pp. 253-268, 2019. DOI: 10.1615/InterfacP
henomHeatTransfer.2019030215