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International Journal of Applied Environmental Sciences
ISSN 0973-6077 Volume 12, Number 9 (2017), pp. 1693-1706
© Research India Publications
http://www.ripublication.com
Natural Convection Magnetohydrodynamic Flow of a
Micro Polar Fluid past a Semi infinite Vertical
Porous Flat Moving Plate
S. Panda1, S. S. Das2 and N. C. Bera3
1Department of Physics, KISS, KIIT Campus-10, Patia, Bhubaneswar-751 024 (Odisha), India.
*2Department of Physics, KBDAV College, Nirakarpur, Khordha-752 019 (Odisha), India.
3Department of Physics, KIIT University, Patia, Bhubaneswar-751 024 (Odisha), India.
Abstract
The objective of this paper is to analyze an unsteady natural convection flow
of a viscous incompressible electrically conducting micro polar fluid past a
semi-infinite vertical porous flat moving plate in presence of transverse
magnetic field. The vertical porous plate is subjected to move with a uniform
velocity in the upward direction in its own plane and the free stream velocity
follows an exponentially increasing or decreasing small perturbation law. The
porous plate absorbs the polar fluid with a suction velocity varying with time.
The governing equations for velocity, angular velocity and temperature of
flow field have been solved using perturbation technique. The effects of the
important flow parameters such as Grashof number for heat transfer Gr, magnetic parameter M, viscosity ratio , plate velocity Up and Prandtl number Pr etc. on the velocity, angular velocity and temperature profiles of the flow field are discussed with the help of figures. This paper has some relevance in
geothermal and oceanic circulation.
Keywords: Unsteady, magnetohydrodynamic, free convection, micro polar,
moving plate
mailto:[email protected]
1694 S. Panda, S.S. Das and N.C. Bera
NOMENCLATURE
A suction velocity parameter B0 magnetic flux density Cp specific heat at constant pressure Gr Grashof number for heat transfer g acceleration due to gravity k thermal conductivity M magnetic field parameter Nu Nusselt number n dimensionless exponential index Pr Prandtl number T temperature t dimensionless time U0 scale of free stream velocity u,v components of velocities along and perpendicular to the plate, respectively V0 scale of suction velocity x, y distances along and perpendicular to the plate, respectively
Greek symbols
Fluid thermal diffusivity Dimensionless viscosity ratio f Coefficient of volumetric thermal expansion of the working fluid Spin-gradient viscosity Scalar constant (1) Electrical conductivity Fluid density Λ Coefficient of gyro-viscosity Fluid dynamic viscosity Fluid kinematic viscosity r Fluid kinematic rotational viscosity Dimensionless temperature Angular velocity vector
Superscripts
differentiation with respect to y
* dimensional properties
Subscripts
p plate w wall condition free stream condition
Natural Convection Magnetohydrodynamic Flow of a Micro Polar Fluid past… 1695
1. INTRODUCTION
The phenomenon of flow and heat transfer in an electrically conducting polar fluid
past a vertical porous plate under the influence of a magnetic field has been a subject
of interest of a good number of researchers because of its applications in several fields
of science and technology for example in the boundary layer control in the field of
aerodynamics (Aero et al. [1]), geothermal energy extractions, oil exploration and in plasma studies. Physically, micro polar fluids represent fluids consisting of randomly
oriented particles suspended in a viscous medium (Dep [2] and Lukaszewicz [3]).
Several Darcian porous MHD studies have been carried out considering the effects of
magnetic field on electrically conducting fluid with or without heat transfer in various
configurations. Chamkha [4] investigated the unsteady convective heat and mass
transfer flow of an electrically conducting fluid in presence of transverse magnetic field
past a semi-infinite vertical permeable moving plate with heat absorption. Mbeledogu
and Ogulu [5] analyzed the effect of heat and mass transfer on unsteady natural
convection flow of a rotating electrically conducting fluid past a vertical porous flat
plate in the presence of radiative heat transfer and transverse magnetic field. Das and
his co-workers [6] estimated the effect of mass transfer on magnetohydrodynamic flow
and heat transfer past a vertical porous plate through a porous medium under oscillatory
suction and heat source and observed that a growing magnetic parameter reduces the
velocity of the flow field at all points of the flow field.
The transient free convection MHD flow past an infinite vertical porous flat plate in
presence of mass transfer has been studied by Panda and his team [7]. Das et al. [8] reported the free convection effects on unsteady viscous flow of an electrically
conducting fluid past an infinite vertical porous plate with heat source/sink in presence
of magnetic field. Patil and Kulkarni [9] observed the effect of chemical reaction on
free convective flow of a polar fluid through a porous medium in the presence of
internal heat generation. Das and his group [10] analyzed unsteady mixed convective
flow of a polar fluid past a semi-infinite vertical porous moving plate in presence of
transverse magnetic field and noticed that an increase in magnetic parameter enhances
the velocity as well as angular velocity of the flow field at all points of the flow field.
Recently, Das and his associates [11] discussed the magnetohydrodynamic convective
mass transfer flow of a polar fluid past a semi infinite vertical porous flat moving plate
embedded in a porous medium and reported that their results are in good agreement
with those of the previous group [10].
The present study investigates an unsteady natural convection flow of a viscous
incompressible electrically conducting micro polar fluid past a semi-infinite vertical
porous flat moving plate which is subjected to move with a uniform velocity in
upward direction in its own plane in presence of a transverse magnetic field. The
plate absorbs the polar fluid with a suction velocity varying with time and the free
1696 S. Panda, S.S. Das and N.C. Bera
stream velocity follows an exponentially increasing or decreasing small perturbation
law. The effects of the flow parameters on the flow field across the boundary layer
have been discussed and analyzed with the aid of figures. This paper has some
relevance in geothermal and oceanic circulation.
2. MATHEMATICAL FORMULATION OF THE PROBLEM
Consider an unsteady viscous incompressible flow of an electrically conducting micro
polar fluid past a semi-infinite vertical porous moving plate embedded in a porous
medium and subjected to a transverse magnetic field in the presence of a pressure
gradient. The transversely applied magnetic field and magnetic Reynolds number are
very small and hence the induced magnetic field is negligible (Cowling [12]). Viscous
and Darcy’s resistance terms are taken into account with constant permeability of the
porous medium. The hole size of the porous plate is assumed to be significantly larger
than a characteristic microscopic length scale of the porous medium. The geometry of
the problem is shown in Figure A. Following Yamamoto and Iwamura [13], we
assume the porous medium as an assembly of small identical spherical particles fixed
in space. The flow variables are functions of y* and t* only.
FigureA. Geometry of the problem
Natural Convection Magnetohydrodynamic Flow of a Micro Polar Fluid past… 1697
Under the above conditions, the governing equations for mass, momentum and energy
conservation in Cartesian coordinates are given by:
Continuity:
,yv
0
(1)
Linear momentum:
*r20f2
2
r y2uBTTg
yu
xp1
yuv
tu
(2)
Angular momentum:
2
2
*
*
*
**
*
**
yyv
tj
, (3)
Energy:
2
2
***
* yT
yTv
tT
, (4)
where , , r , g , f , , B0, *j , * , , T and are respectively the density,
kinematic viscosity, kinematic rotational viscosity, acceleration due to gravity,
coefficient of volumetric thermal expansion of the fluid, electrical conductivity of the
fluid, magnetic induction, micro-inertia density, component of the angular velocity
vector normal to xy-plane, spin-gradient viscosity, temperature and effective fluid
thermal diffusivity; x and
y are the dimensional distances longitudinal and perpendicular to the plate respectively and
u , v are the components of dimensional
velocities along x and
y directions respectively.
It is assumed that the porous plate moves with a constant velocity (*pu ) in the
longitudinal direction and the free stream velocity (*U ) follows an exponentially
increasing or decreasing small perturbation law. We, further assume that the suction
velocity ( *v ) and the plate temperature (T) vary exponentially with time. For small velocities, the heat due to viscous dissipation is neglected in the energy equation (4).
With the above assumptions, the boundary conditions for velocity and temperature
fields are
*t*n
ww*p
* eTTTT,uu , 22
*
*
*
*
yu
y
at
*y 0
1698 S. Panda, S.S. Das and N.C. Bera
*t*n** eUUu 10 , 0,TT* as
*y , (5)
where 0
U is a scale of free stream velocity and *n is a scalar constant.
It is clearly seen from the continuity equation (1) that the suction velocity normal to
the plate is a function of time only and therefore, we shall take it in the form:
*t*nAeV*v 10 , (6)
where A is a real positive constant, and A are small and less than unity and 0
V is a scale of suction velocity which is a non-zero positive constant. Outside the boundary
layer, equation (2) gives
*20*
*
*
*UB
dtdU
dxdp1
(7)
We now introduce the following non-dimensional variables:
TTTT
,Vt
t,VU
,Uu
U,UU
U,yV
y,Vvv,
Uuu
w
**
*p
p
**** 20
0000
0
00
, ,Vnn
20
*
*2
20 j
Vj
,
kC
P pr is the Prandtl number, 20
20
VB
M
is the magnetic
parameter,2
00VU
)TT(gG wfr
is the Grashof number for heat transfer. (8)
Again, the spin-gradient viscosity which gives the relationship between the
coefficients of viscosity and micro-inertia is defined as
)(jj)A( ** 2
11
2, (9)
where denotes the dimensionless viscosity ratio which is defined as
(10)
Here, Λ is the coefficient of gyro-viscosity or vortex viscosity.
Using equations (6)-(10), the governing equations (2)-(4) are reduced to the following
non-dimensional form:
y
2uUMGy
u1dt
dUyuAe1
tu
r2
2nt
, (11)
Natural Convection Magnetohydrodynamic Flow of a Micro Polar Fluid past… 1699
2
2nt
y1
yAe1
t
, (12)
2
211
yPyAe
t rnt
, (13)
where
22j (14)
The corresponding boundary conditions now take the form
2
2
1y
uy
,e,Uu ntp
at y=0,
0,0,Uu as y (15)
3. METHOD OF SOLUTION
In order to solve the equations (11)-(13) by reducing these partial differential
equations to a
system of ordinary differential equations in non-dimensional form, we assume the
linear
velocity, angular velocity and temperature as
210 Oyueyuu nt (16)
210 Oyey nt (17)
210 Oyey nt (18)
Substituting equations (16)-(18) in equations (11)-(13) and equating the harmonic and
non-harmonic term, neglecting the coefficient of O(2 ), we get the following pairs of
equations for 000 ,,u and 111 ,,u .
00r00"0 2GMMuuu1 (19)
11r011"1 2GuAnMunMuu1 (20)
000 (21)
0111 An (22)
000 rP (23)
0111 rrr APnPP (24)
1700 S. Panda, S.S. Das and N.C. Bera
Here the primes denote differentiation with respect to y.
The corresponding boundary conditions now reduce to:
1,1,u,u,0u,Uu 1011001p0 at y=0
0,0,0,0,1u,1u 101010 as y (25)
The solutions of equations (19)-(24) satisfying boundary conditions (25) are given by
y5m4y
53yrP
210 eAeAAeAAyu (26)
y7m14
y5m12
y3m1311
y1m10
y9
yrP871 eAeAeAAeAeAeAA)y(u
(27)
y50 eAy (28)
y3m14y
61 eAeAy (29)
yrPey 0 (30)
y1m0yrP
01 eA1eAy
(31)
where,
r2
rr1 P4PP21m ,
r2
rr2 P4PP21m ,
n421m 23 ,
n421m 24 , 1M4112
1m5 ,
1M41121m6 ,
1nM41121m7 , 1nM4112
1m8 ,
r2r1
2r
0 PmPmAPA
,
651 mm
MA , r6r5
r2 PmPm
GA
,
653 mm
2A ,
p53214 UAAAAA ,
2
532
3
p2125
2r2
5 mAA
UAAmPAA ,
43
52
6 mmAA
A ,
877 mm
nMA , 8r7r
r28 mPmP
PAAA
,
87
6539 mm
A2AAAA ,
8171
0r10 mmmm
A1GA
,
83313
11 mmmmm2
A
, 8575
24
12 mmmmAA
A
,
2723113
2512
2110
29
2r86
13 mmAmmAmAAPAA
A
2723113
121098727
mmAmAAAAAm
,
Natural Convection Magnetohydrodynamic Flow of a Micro Polar Fluid past… 1701
1213111098714 AAAAAAAA . (32)
3.1. Skin friction
The skin friction at the wall of the plate is given by
000
y
*w
w yu
VU (33)
Using Equations (16), (26) and (27) in Equation (33), the skin friction becomes
1471251311310198rnt
45532rw AmAmAAmAmAAPeAmAAAP (34)
3.2. Rate of heat transfer
The rate of heat transfer or the heat flux at the wall in terms of Nusselt number is
given by
0
yu y
N (35)
Using Equations (18), (30) and (31) in Equation (35), the heat flux becomes
01r0nt
ru A1mPAePN (36)
4. DISCUSSIONS AND RESULTS
The study reported herein investigates an unsteady natural convection flow of a
viscous incompressible electrically conducting micro polar fluid past a semi-infinite
vertical porous flat moving plate which is subjected to move with a uniform velocity
in upward direction in its own plane in presence of a transverse magnetic field. The
governing equations for linear velocity, angular velocity and temperature of the flow
field have been solved employing perturbation technique and the effects of the flow
parameters on the linear velocity, angular velocity and temperature of flow field
across the boundary layer have been discussed and analyzed with the aid of velocity
profiles (1-4), angular velocity profiles (5-6) and temperature profile (7) as detailed
below.
4.1. Velocity field
The flow parameters affecting the velocity of the flow field are Grashof number for
heat transfer Gr, magnetic parameter M, viscosity ratio and the plate velocity Up. The velocity of the flow field is found to change more or less with the variation of the
1702 S. Panda, S.S. Das and N.C. Bera
above parameters. The effects of these parameters on the velocity field are discussed
with the help of Figures 1-4.
0
1
2
3
4
5
0 1 2 3 4 5
y
u
Gr= -5
Gr= 0
Gr=2
Gr=5
Figure1. Effect of Grashof number for heat transfer Gr on Velocity profiles against y
The effect of Grashof number for heat transfer Gr on the velocity field is presented in Figure 1. The Grashof number for heat transfer is found to accelerate the velocity of
the flow field near the plate. Figure 2 depicts the effect of magnetic parameter M on the velocity field. A growing magnetic parameter is noticed to enhance the velocity of
the flow field at all points.
0
2
4
6
8
10
12
14
0 1 2 3 4 5y
u
M=0
M=0.5
M=2
M=10
Figure2. Effect of magnetic parameter M on Velocity profiles against y
Natural Convection Magnetohydrodynamic Flow of a Micro Polar Fluid past… 1703
0
5
10
15
20
25
30
0 1 2 3 4 5y
u
Figure3. Effect of viscosity ratio on Velocity profiles against y
The effect of viscosity ratio on the velocity of the flow is shown in Figure 3. The
viscosity ratio is observed to decelerate the velocity of the flow field at all points.
Figure 4, elucidates the effect of the plate velocity Up on the velocity field. From the curves of the said figure, it is clearly observed that an increase in plate velocity has an
accelerating effect on the velocity of the flow field near the plate. The results shown
in velocity profiles 1, 2 and 3 are in good agreement with those of Das et al. [11] and Figure 4 closely matches with those of Das and his group [10].
0
1
2
3
4
5
6
7
0 1 2 3 4 5y
u Up=0 Up=0.2Up=2 Up=4
Figure4. Effect of plate velocity Up on Velocity profiles against y
1704 S. Panda, S.S. Das and N.C. Bera
4.2. Angular velocity field
The effect of magnetic parameter M and the viscosity ratio on the angular velocity of the flow field has been shown in Figures 5-6. The profiles of Figure 5 clearly show
that a growing magnetic parameter M leads to enhance the angular velocity of the flow field at all points. Figure 6 elucidates the effect of plate velocity Up on the angular velocity of the flow field. It is observed that an increase in of plate velocity
leads to decrease the angular velocity of the flow field. The angular velocity profiles
are in good agreement with those of Das et al. [10, 11].
0
5
10
15
20
25
30
0 1 2 3 4 5y
M=0 M=0.5
M=2 M=10
Figure5. Effect of magnetic parameter M on Angular velocity profiles against y
0
2
4
6
8
10
12
14
0 1 2 3 4 5y
Up=0 Up=0.2
Up=2 Up=4
Figure6. Effect of plate velocity Up on Angular velocity profiles against y
Natural Convection Magnetohydrodynamic Flow of a Micro Polar Fluid past… 1705
4.3. Temperature field
The plot of temperature of the flow field against y for different values of the Prandtl number Pr is shown in Figure 7. Comparing the curves of the said figure, it is observed that an increase in Prandtl number reduces the temperature of the flow field
at all points. The temperature profiles shown in Figure 7 closely matches with those
of Das et al. [11].
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5y
Pr=0.71 Pr=1 Pr=5
Figure7. Effect of Prandtl number Pr on Temperature profiles against y
5. CONCLUSIONS
We summarize the conclusions drawn from the above analysis on the linear velocity,
angular velocity and temperature of the flow field.
1. The Grashof number for heat transfer Gr accelerates the velocity of the flow field near the plate.
2. A growing magnetic parameter M enhances the linear velocity as well as angular velocity of the flow field at all points.
3. The effect of increasing viscosity ratio is to retard the linear velocity of the
flow field at all points.
4. The plate velocity Up is observed to enhance the linear velocity of the flow field near the plate while it shows reverse effect in case of angular velocity.
5. An increase in Prandtl number Pr decreases the temperature of the flow field at all points.
1706 S. Panda, S.S. Das and N.C. Bera
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