1 Heat and mass transfer on MHD flow ofNon-Newtonian fluid over an infiniteverticalporous plate with Hall effects D.Dastagiri Babu 1 , S. Venkateswarlu 2 , E.Keshava Reddy 3 1 Research Scholar of JNTUA Anantapuramu,A.P. India 2 Dept. of Mathematics, RGM College of Engg. and Technology, Nandyal,Kurnool, A.P, India 3 Dept.of Mathematics,JNTUA College of Engg.Anantapuramu, A.P. India 1 corresponding author:[email protected]2 [email protected]3 [email protected]Abstract We have considered the boundary-layer flow of a heat-absorbing MHD non-Newtonian fluid along a semi-infinite vertical porous moving plate in the presence of thermal buoyancy effect and taking hall current effects into account. The dimensionless governing equations are solved analytically using two-term harmonic and non-harmonic functions. Computational analysis of the results is presented with a view to revealfor the velocity, temperature and concentration profiles within the boundary layer. The Skin friction, Nusselt number and Sherwood number are also examined with the reference to governing parameters. Keywords:MHD flows,Non-Newtonian flow, Porous mediumandunsteady flows Nomenclature: In this paperx, z and t are the dimensional distance along and perpendicular to the plate and dimensional time, respectively u and v are the components of dimensional velocities along x and z directions respectively is the fluid density, is the kinematic velocity, p C is the specific heat at constant pressure, is the fluid electrical conductivity, 0 B is the magnetic induction, k is the permeability of the porous medium. T is the dimensional temperature, 0 Q is the dimensional heat observation co-efficient, is the thermal diffusivity, g is the gravitational acceleration and T is the coefficient of volumetric thermal expansion, C isvolumetric expansion coefficient for concentration and 1 is the kinematic visco-elasticity. p U , w T are the wall dimensional velocity temperature, respectively , U T are the free steam dimensional velocity, temperature respectively, 0 and u are constants. 0 2 kw K v is the permeability of the porous medium, Pr p C k is the prandtl number, 2 2 0 2 0 B M w is the magnetic field International Journal of Pure and Applied Mathematics Volume 119 No. 15 2018, 87-103 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ Special Issue http://www.acadpubl.eu/hub/ 87
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1
Heat and mass transfer on MHD flow ofNon-Newtonian
fluid over an infiniteverticalporous plate with Hall effects
z
D.Dastagiri Babu1, S. Venkateswarlu2, E.Keshava Reddy3 1Research Scholar of JNTUA Anantapuramu,A.P. India
2Dept. of Mathematics, RGM College of Engg. and Technology, Nandyal,Kurnool, A.P, India 3Dept.of Mathematics,JNTUA College of Engg.Anantapuramu, A.P. India
In this paperx, z and t are the dimensional distance along and
perpendicular to the plate and dimensional time, respectively u and v are the
components of dimensional velocities along x and z directions respectively is
the fluid density, is the kinematic velocity, pC is the specific heat at constant
pressure, is the fluid electrical conductivity, 0B is the magnetic induction, k is
the permeability of the porous medium. T is the dimensional temperature, 0Q is
the dimensional heat observation co-efficient, is the thermal diffusivity, g is
the gravitational acceleration and T is the coefficient of volumetric thermal
expansion, C isvolumetric expansion coefficient for concentration and 1 is the
kinematic visco-elasticity. pU , wT are the wall dimensional velocity temperature,
respectively ,U T are the free steam dimensional velocity, temperature
respectively, 0 andu are constants. 0
2
kwK
v is the permeability of the porous
medium, PrpC
k
is the prandtl number,
22 0
2
0
BM
w
is the magnetic field
International Journal of Pure and Applied MathematicsVolume 119 No. 15 2018, 87-103ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/
87
2
parameter,
3
0
GrT g T T
w
is the grashof number, Sc
v
D is the Schmidth
number, 2
1 0
2
wRm
is the dimensionless form visco-elasticity parameter of the
Rivlin-Ericksen Fluid.
1. Introduction: The phenomenon of flows through porous medium has been a subject of
interest of many researchers because of its wide range of application in different
fields such as petroleum engineering, chemical engineering etc. In petroleum
engineering, it is dealt with the movement of natural gas and oil through
reservoirs. Further, the study on underground water resources, seepage of water
in river bed is also related to the flow through porous medium. The free
convection flow past vertical plate in the presence of viscous dissipates heat
using the perturbation method was discussed by Gupta et al. [1]. Kafousias and
Raptis [2] extended the work with mass transfer effects. Chambre and Young [3]
analysed the problem of the first-order chemical reactions over a horizontal
plate. Soundalgekar et al. [4] discussed the mass transfer effects on the flow past
a vertical plate impulsively started with variable temperature and constant heat
flux. Elbashbeshy [5] discussed the effects of a magnetic field in mass transfer
along a vertical plate. The thermal, mass diffusion, magnetic field and hall
current effects were discussed by Takhar et.al [6] . Investigation of the
significance of step change in wall temperature is meaningful and it was done in
the fabrication of thin-film photovoltaic devices by Chandran et.al[7]
.Muthucumaraswamy and Muralidharan [8] discussed the effects of thermal
radiation on a linearly accelerated vertical plate with variable temperature and
uniform mass flux. Double dispersion convection effects on the combined heat
and mass transfer in a non-Newtonian fluid-saturated porous medium has been
discussed by Kari and Murthy [9]. Free convective power-law fluid flow past a
vertical plate in a non-Darcian porous medium in the presence of a homogeneous
chemical reaction was studied by Chamkha et. al [10]. The internal heat
generation or absorption is important in problems involving chemical reactions
where heat may be generated or absorbed in the course of such reactions and it
was investigated by Rao et. al [11]. Rajput and Kumar [12] investigated
radiation effects on magnetic field flow of an impulsively started vertical plate
with non-uniform heat and mass transfer. Siva raj and Rushi Kumar [13]
visualized the flow of viscoelastic fluid over a moving vertical plate and flat plate
with variable electric conductivity. Fluid flows past an infinite plate are of much
importance due to its large practical applications, such as motions due to wall
shear stress and it was studied by Seth et. al [14]. Several authors have carried
out their research works for the importance of the chemical reaction effects, on
some mass transfer flow problems. Poornima et. al [15] studied the thermal
radiation and chemical reaction effects on free Convective flow past a Semi-
infinite vertical porous moving plate in the presence of magnetic field. Okedoye
[16] reported the non-uniform heat source/sink in controlling the heat transfer in
the boundary layer region. Rashad et al. [17] have studied the viscous dissipation
effects on the free convective heat transfer of nanofluids. Singh and Makinde [18]
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analysed the combined convection slip flow with temperature variation along a
moving plate in free stream. Sheikholeslami and Ganji [19] discussed heat
Transfer effects in Nano fluid flow between parallel plates.Recently, Krishna et
al. [20-23] discussed the MHD flows of an incompressible and electrically
conducting fluid in planar channel.
Motivated by the above studies, in this paper we have consideredMHD
flow of non-Newtonian (Rivlin-Ericksen) fluid past a semi-infinite moving porous
plate.
2.Formulation and Solution of the problem:
We consider the MHD flow of anelectrically conducting and heat absorbing Non-
Newtonian fluid (Rivlin-Ericksen type)over a semi-infinite vertical permeable
moving plate embedded in a porous medium with a uniform transverse magnetic
field and taking hall currents into account(Fig.1).It is assumed that there is no
applied voltage which implies the absence of an electrical field the transversely
applied magnetic field and magnetic Reynolds number are assumed to be very
small to that the induced magnetic field and the hall effects are negligible.A
consequence of the small magnetic Reynolds number is a the uncoupling of the
Navier-stokes equations from Maxwell’s equations the governing equations for
this investigation are based on the balances of mass,linear momentum made
above,these equations can be written in Cartesian frame of reference as follows:
x g
Momentum boundary layer
Thermal boundary layer
Porous medium
0 1 ntw w Ae Concentration boundary layer
0
z
Fig. 1 Physical configuration of the Problem
0w
z
(1)
2 3 3
1 02 2 3
1y T C
u u p u u uw v B J u T T C C
t z x z t z z k
(2)
2 3 3
1 02 2 3
1x
v v v v v+w = B J v
t z y z z t z k
(3)
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2
0
2
p
QT T T+w T T
t z z C
(4)
2
2
C CD
t z
(5)
The magnetic and viscous dissipations are neglected in this study. It is
assumed that the permeable plate moves with aconstant velocity in the direction
of fluid flow and the free steam velocity follows the exponentially increasing
small perturbation law. Inaddition, it is assumed that the temperature at the
wall as well as the suction velocity is exponentially varying with time. Under
then assumptions the appropriate conditions for the velocity, temperature fields
are
0 at 0nt nt
p w w w wu U , v , T T T T e ,C C C C e z ,
, 0 , ,0 1 , asntu U v u e T T C C z (6) It is clear
from equation(1) that the suction velocity at the plate surface is a function of
time only assuming that it takes the following exponential form
0 1 ntw w Ae (7)
When the strength of the magnetic field is very large, the generalized
Ohm’s law is modified to include the hall current so that
1e e
e
O e
J J B E V B PB e
(8)
The ion-slip and thermo electric effects are not included. Further it is
assumed that ee ~ 0 (1) and ,1ii where i and i are the cyclotron
frequency and collision time for ions respectively. In the equation (8) the electron
pressure gradient, the ion-slip and thermo-electric effects are neglected. We also
assume that the electric field E=0 under assumptions reduces to
x y 0J m J σB v
(9)
y x 0J m J σB u
(10)
Where e em is the Hall parameter.
On solving equations (9) and (10) we obtain,
( )1
0x 2
σBJ v mu
m
(11)
( )1
0y 2
σBJ mv u
m
(12)
Substituting the equations (11) and (12) in (3) and (2) respectively, we obtain
22 3 3
012 2 3 2
1( )
1
T C
σBu u p u u uw v mv u u
t z x z t z z m k
T T C C
(13)
22 3 3
01 22 2 3
1( )
1
σBv v v v vv mu+w = v
mt z y z z t z k
(14)
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Combining equations (13) and (14), let andq u iv x iy , weobtain,
22 3 3
01 22 2 3
1
1( )
C
Bq q p q q q vw q
mt z z z t z k
g g C C
(15)
Where A is a real positive constants, and are small less than unity, and 0w is
a scale of suction velocity which has non-zero positive constant outside the
boundary layer equation(2) gives
2
0
1 Up= U B U
t k
(16)
We introduced the following non-dimensional variables 2
* * * * * 10 0
0 0 0 0
, , , , , , ,p
p
w w
uw z twU C Cu vu v z U U t
U w v U U C C
Making use of non-dimensional variables the governing equations reduces
to (Droppingaskerisks) .
2 2 3
2 2 2
3
3
11 Gr Gm
1
1
nt
nt
Uq q q M ue U q Rm
t z t z m K t z
qe
z
(17)
2
2
11
Pr
nte = Qt z z
(18)
2
2
11
Sc
ntet z z
(19)
The dimensionless form boundary conditions equations become
1 1 1 at 0
0 0 0 at
nt nt nt
pq U , e , e ,U e z
q U , , ,U z
(20)
Equation (10) and (11) represent a set of partial differential equations that
cannot be solved in closed form however if can be reduced to a setoff ordinary
differential equations in dimensionless form that can be solved analytically this
can be done representing the velocity and temperature as,
2
0 1
ntq q z e q z O (21) 2
0 1
ntz e z O (22)
2
0 1
ntz e z O (23)
Substituting the equations (21), (22) and (23) into equation (17), (18) and
(19),equating the harmonic and non-harmonic terms, the neglecting and higher
order terms of 2O ,one obtains the following pairs of equations
0 0 0 1 1 1, , and , ,q q ,
3 2 2 2
0 0 00 03 2 2 2
1 1Gr
1 1
d q d q dq M MRm q
dz dz dz m K m K
(24)
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3 2 2
1 1 11 13 2 2
32
0 01 2 3
11
1
1
1
d q d q dq MRm nRm q nq
dz dz dz m K
d q dqMn Gr RmA A
m K dz dz
(25)
2
0 002
Pr Pr 0d d
Qd z dz
(26)
2
01 11 12
Pr Pr Pr Prdd d
n Q Ad z dz dz
(27)
2
0 0
2Sc 0
d d
d z dz
(28)
2
01 112
Sc Sc Scdd d
n Adz dz dz
(29)
The corresponding boundary conditions can be written as
0 1 0 1 0 10 1 1 1 1 at 0pq U ,q , , , , z
0 1 0 1 0 11 1 0 0 0 0 atq , q , , , , z (30)
Without going into detail,the solution of equation (24)-(29) subject to
conditions(30) can be show to be 1
0
m zq e
(31)
3 1
1 2 1
m z m zq a e a e
(32)
Equations (24) and (25) are third degree order differential equations when
0Rm and we have two boundary conditions so we follows bears and Walter’s as
2
0 01 02q q Rmq O Rm (33)
2
1 11 12q q Rmq O Rm (34)
Substituting equations (33) and (34) into (24) and (25), equating different powers
of Rm and neglecting 2O Rm are,
1 1
2 2 2
01 01012 2 2
1 1Gr Gm
1 1
m z n zd q dq M Mq e e
dz dz m K m K
(35)
3 2 2
01 02 02023 2 2
10
1
d q d q dq Mq
z dz dz m K
(36)
31
31
2 2
11 1111 1 22 2
2
011 2 2
1Gr
1
1Gm
1
m zm z
n zn z
d q dq Mn q n a e a e
dz dz m K
dqMb e b e A
m K dz
(37)
33 2 2 2
01 0211 12 11 12123 2 2 2 3
1
1
d q dqd q d q d q dq Mn n q
dz dz dz dz m K dz dz
(38)
The corresponding boundary conditions are
01 02 11 120 0 0 at 0pq u ,q ,q ,q z
01 02 11 121 0 1 0 asq ,q ,q ,q , z (39)
We get zeroth order and first order solutions of Rm
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6 61 1
01 4 3 4 3 1m z m zm z n z
q a e a e b e b e
(40)
8 6 8 6 1 1
02 7 5 7 5 6 6
m z m z m z m z m z n zq a e a e b e b e a e b e
(41)
10 3 3 61 1
11 16 9 15 5 10 12 14
m z m z n z m zm z n zq a e a e a e b e b e a e a
(42)
6 8 10 312 1
12 27 22 23 24 25 26
m z m z m z m zm z m zq a e a e a e a e a e a e
(43)
In view of the above solutions, the velocity and temperature
distributions in the boundary layer become
0 1, ntq z t q z e q z
6 61 1
8 6 8 6 1 1
10 3 3 61 1
6 812 1 1
4 3 4 3
7 5 7 5 6 6
16 9 15 8 12 12 14
27 22 23 16 24
1m z m zm z n z
m z m z m z m z m z n z
m z m z n z m zm z n znt
m z m zm z m z n z
a e a e b e b e
Rm a e a e b e b e a e b e
e a e a e a e b e b e a e a
Rm a e a e a e b e a e
10 3 3
25 26 17
m z m z n za e a e b e
0 1, ntz t z e z
31 1
2 1
m zm z m znte e a e a e (44)
0 1, ntz t z e z
31 1
2 1
n zn z n znte e b e be (45)
The skin friction co-efficient, Nusselt number and Sherwood number are
important physical parameters for this type of boundarylayer flow.These
parameters can be defined and determined as follows.
28 29 30 31
0
nt
z
qa Rma e a Rma
z
(46)
32 33
0
nt
z
Nu a e az
(47)
33 35
0
nt
z
Sh a e az
(48)
3.Results and Discussion:
We noticed that, the velocity component u and v reduces with increasing
the intensity of the magnetic field or Hartmann number M. The similar
behaviour is observed for the resultant velocity Figs.(2). It is obvious that the
effect of increasing values of the magnetic field parameter M results in a
decreasing velocity components u and v across the boundary layer. Figs.(3)
depicts the effect of permeability of the porous medium parameter (K) on velocity
distribution profiles for u and v and it is obvious that as permeability parameter
(K) increases, the velocity components for u and v increases along the boundary
layer thickness which is expected since when the holes of porous medium become
larger, the resistive of the medium may be neglected. Similar behaviour is
observed with increasing permeability parameter K for the resultant velocity.
Figs. (4)illustrate the variation in velocity components u and v with span wise
coordinate n for several values of Rm. We observed that both u reduces and v
increases with increasing visco-elastic fluid parameter of the Rivlin-Ericksen
fluid Rm. It was found that an increase in Rm leads to a decrease in the
resultant velocity distribution across the boundary layer. Figs.(5) illustrate the
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velocity profiles for u and v for different values of the Grashof number Gr. It can
be seen that an increase in Gr leads to a rise in velocity u and v profiles. The
resultant velocity is also enhances throughout the fluid region with increasing
the thermal Grashof number Gr. We noticed that from the Figs. (6), the
magnitude of the velocity component u reduces and v enhances with increasing
Suction parameter A throughout the fluid region. The resultant velocity is
reduces with increasing Suction parameter A. Figs. (7) presents the velocity
distribution profiles for different values of the Prandtl number (Pr). The results
show that the effect of increasing values of the prandtl number results in
anincrease in u and in a decrease in the velocity componentv. The resultant
velocity is also reduces with Prandtl number Pr.Figs. (8) shows the velocity
profiles for u and v for different values of dimensionless heat absorption
coefficient Q. Clearly as Q increase the pack values of velocity components u
increase and v tends to decrease. The resultant velocity reduces with increasing
heat absorption coefficient Q, physically the presence of heat absorption
coefficient has the tendency to reduce the fluid temperature. This causes the
thermal buoyancy effects to decrease resulting in a net reduction in the fluid
velocity.The Figs. (9) shows the velocity profiles for u and v against span wise
direction for different values of the scalar constant . It was found that an
increase in the value of leads to an increase in the resultant velocity
distribution across the boundary layer. The velocity component u diminishes first
and then experiences enhancement andwhereasv increases with increasing the
scalar constant . Figs. (10) Depicts the effect of the frequency of oscillation n on
the velocity distribution. The primary velocity component u and secondary
velocity v reducewith increasing the frequency of oscillation n . The resultant
velocity reduces with increasing the frequency of oscillation throughout the fluid
medium. Finally, the
Figs. (11) Shows the velocity profiles for u and v against hall parameter m. It
was found that an increase in the value of m leads to an increase in the
resultant velocity distribution across the boundary layer. The velocity component
u and v enhances with increasing the hall parameter m.
We noticed that from the Figs. (12), the temperature reduces with increasing
Prandtl number Pr or the frequency of oscillation n or suction velocity A or heat
absorption coefficient Q.The result display that an increase in the value of Q
results in decrease in the temperature profiles as expected. The temperature
profiles with span wise coordinate n for various scalar constant . The numerical
results show that the effect of increase value of e results in an increase thermal
boundary layer thickness and more uniform temperature distribution across the
boundary layer. Similar behaviour is observed in entire fluid region with
increasing time.
We noticed from the Figs.(13), the temperature reduces with increasing
Schmidt number Sc or the frequency of oscillation n or suction velocity A. The
Concentration profiles with span wise coordinate n for various scalar constant .
The numerical results show that the effect of increase value of results in an
increase concentration boundary layer thickness and more uniform concentration
distribution across the boundary layer.
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We have also shown Table (1) of the surface skin friction coefficient
against some parameters. The stress components x and the magnitude of y are
reduces with increasing the intensity of the magnetic field M, Schmidt number
Sc or the suction velocity A. The reversal behaviour is observed throughout the
region with increasing hall parameter mGrashof number Gr or heat absorption
coefficient Q or scalar constant , because an increase in Gr influences the
buoyancy that results in skin friction. The stress component x enhances and the
magnitude of y reduces with increasing permeability parameter K, whereas x
decreases and the magnitude of y boost up with increasing visco-elastic fluid
parameter of the Rivlin-Ericksen fluid Rm or Prandtl number Pr or frequency of
oscillationn. The rate of heat transfer (Nu) is shown in the Table (2) with reference to
all governing parameters. The magnitude of the Nusselt number rise up
throughout the fluid region with increasing scalar constant , suction velocity A,
Prandtl number Pr, heat absorption parameter Q and the frequency of oscillation
n.
The rate of mass transfer (Sh) is shown in the Table (3). The magnitude of
the Sherwood number enhances throughout the fluid region with increasing
scalar constant , suction velocity A, Schmidth number Sc, and the frequency of
oscillation n.
Figures 2. The velocity Profiles for andu v against M