MIXED CONVECTION OVER AN ISOTHERMAL VERTICAL FLAT
PLATE EMBEDDED IN A POROUS MEDIUM WITH MAGNETIC
FIELD, RADIATION AND VARIABLE VISCOSITY WITH HEAT
GENERATION.
T.RajaRani1 , C.N.B.Rao2
Higher College of Technology, Muscat, Oman1
S.R.K.R. Engineering College, Bhimavaram, India2
[email protected], [email protected]
ABSTRACT
This paper focuses on the numerical solutions of
the effects of magnetic field, radiation, variable
viscosity and heat generation on similarity solutions
of mixed convection adjacent to an isothermal
vertical plate which is embedded in a porous
medium. A similarity transformation is used to
reduce the partial differential equations governing
the problem into ordinary differential equations and
the equations are solved numerically subject to
appropriate boundary conditions by the use of
Runge-Kutta-Gill method together with a shooting
technique. The flow and heat transfer quantities of
similarity equations are found to be the functions of
QRdC ,,, and RP where C is the magnetic
interaction parameter, Rd is the radiation parameter,
is viscosity variation coefficient, Q is the heat
generation/absorption parameter and RP is the
mixed convection parameter which is the ratio of
Rayleigh to cletep numbers. In the present work
the cases of assisting and opposing flows are
discussed. It has been found that in opposing flow
case, dual solutions exist for negative values of RP
and boundary layer separation occurs. It is observed
that depending on the values of RP there exists no
solution, a unique solution or dual solutions and
also the temperature decreases significantly with
increase in Q and C. Skin friction, heat transfer
coefficient, velocity and temperature fields are
studied and discussed with the help of a table and
graphs.
KEY WORDS Mixed Convection , Porous medium, Darcy model,
Heat generation, Variable viscosity.
1. 1.INTRODUCTION 2. 3. During last five decades much insightful
work has been done on mixed convection
boundary-layer flows in porous media. The
analogous problems have important
applications in the fields such as geothermal
systems, food processing and grain storage,
solar power collectors, oil reservoir modeling
and the dispersion of chemical contaminants in
different industrial processes in the
environment. References [10] and [14] stand
evident to the fact that convection flows in
porous media are of vital importance to such
processes.
Reference [12] discussed the Internal heat
generation and Radiation effects on a Certain
Free convection Flow and reference [1]
discussed the Numerical study of the combined
free-forced convection and mass transfer flow
past a vertical porous plate in a porous medium
with heat generation and thermal diffusion.
Reference [8] discussed the effect of variable
viscosity on convective heat transfer in the
three different cases of natural convection,
mixed convection and forced convection taking
fluid viscosity to vary inversely with
temperature. The authors have discussed the
effect of the appropriate parameters on the flow
and heat transfer quantities. The authors
however did not discuss hot plate and cold
plate cases separately in free convection and
did not discuss opposing flow case in mixed
convection.
International Journal of Digital Information and Wireless Communications (IJDIWC) 2(1): 7-17 The Society of Digital Information and Wireless Communications, 2012(ISSN 2225-658X)
7
Reference [11] discussed laminar natural
convection flow (nonporous) and heat transfer
of fluids with and without heat sources in
channels with constant wall temperatures.
Reference [5] discussed mixed convection
boundary layer flow over a vertical surface for
the Darcy model when viscosity varies
inversely as a linear variation of temperature.
Results of both assisting flow and opposing
flow were discussed as function of mixed
convection parameter and variable viscosity parameter c . In the opposing flow case, the
existence of dual solutions and boundary layer
separation were noticed. Mixed convection
boundary layer flow on a vertical surface in a
saturated porous medium is studied in reference
[9]. In that paper the flow of a uniform stream
past an impermeable vertical surface embedded
in a saturated porous medium and which is
supplying heat to porous medium at a constant
rate is considered. Reference [6] discussed
radiation effects on natural convection in an
inclined porous surface with internal heat
generation.
The aim of the present paper is to
investigate the effects of variable viscosity or
temperature dependent viscosity, magnetic
field, radiation and heat generation/absorption
on mixed convection boundary layer flow over
a vertical surface embedded in a porous
medium. It is assumed that viscosity of the
fluid varies as a linear function of temperature
as in [4] & [13]. Both cases of assisting and
opposing flows are considered. The physical
coordinate system of the problem is presented
in Fig-1. The governing partial differential
equations are transformed into ordinary
differential equations by similarity variable and
the equations are solved numerically using
Runge-Kutta-Gill method with a shooting
technique for some values of the governing
parameters. Variations in flow and heat transfer
characteristics are presented in a table and in
some figures and discussed. Quantitative
comparisons with the existing results for the
case of constant viscosity, without magnetic
field and radiation effects as reported by
Merkin [9], Chin and Nazar [5] are presented.
2. FORMULATION AND SOLUTION
Let an isothermal flat plate be embedded
vertically in a porous medium saturated with
a viscous incompressible electrically
conducting, gray, emitting, absorbing and
non-scattering fluid. It is also assumed that
internal heat generation/absorption is
present. The plate is maintained at a constant
temperature, x-axis is taken vertically along
the plate and y-axis perpendicular to it so
that the plate can be described by y = 0.
Orientation of the plate can be seen in Fig-1.
Using the boundary layer and Boussinesq
approximations the equations governing the
problem, i.e., the Continuity equation, the
Darcys law and the energy equations are presented as:
0
y
v
x
u (2.1)
020
UuBUuK
gx
pp
(2.2)
0
v
Ky
p (2.3)
International Journal of Digital Information and Wireless Communications (IJDIWC) 2(1): 7-17 The Society of Digital Information and Wireless Communications, 2012(ISSN 2225-658X)
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TT
c
Q
y
q
y
T
c
k
y
Tv
x
Tu
p
r
p
m
0
2
2
(2.4)
Here u, v are fluid velocity components, T is
fluid temperature, K is permeability, mk is
effective thermal conductivity of the fluid
saturated porous medium, 0B is the magnetic
flux, is the electric conductivity and rq is
the radiative heat flux transverse to the
vertical plate. The Rosseland approximation
is used in the energy equation to describe the
thermal radiative heat transfer. It may be
noted that by the use of Rosseland
approximation, the applicability of the
present analysis is limited to optically thick
fluids only and 0Q is the heat
generation/absorption constant.
The appropriate boundary conditions are:
0,,,
0,0,0,0
xyasUuTT
xyatvATT
(2.5)
where U is free stream velocity. Taking
U as BU where B is a constant,
introducing a stream function , cleteP
number xPe , non-dimensional functions
,f ; a similarity variable and radiative
flux rq through the relations
x
UcQQ
y
T
k
Tq
Pex
y
TT
TT
Pe
f
xUPe
p
c
sr
x
xm
m
x
0
3
2
1
0
2
1
3
16
)(
)(
(2.6)
where s is the Stefan-Boltzmanns
constant and ck is the mean absorption
coefficient and eliminating fluid pressure
from (2.2), (2.3) the governing equations are
obtained as:
)(1
2
11 RPCfCfC
(2.7)
fQ
Rd2
3
41
(2.8)
Here 2*
*
MK
KC
is the magnetic field
parameter, K
LK
LBM
f
2*
22
02 &
e
s
kk
TRd
34
is the radiation parameter, and
x
x
Pe
RaRP is Mixed convection parameter,
f
x
xTTgKRa
0 is the Rayleigh
number,
2
11 f ( Ref. [4] &
[13]) where viscosity variation
coefficient.
The boundary conditions (2.5) in terms of f
and are
0,0,
,0,1,0
fas
fat
(2.9)
Equation (2.7) can be integrated once using
the condition on f at to get
( )1
11
2
C RPf
C
(2.10)
which gives the slip velocity.
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3. SOLUTION OF THE PROBLEM
3.1 Parameters of the Problem and Their Effect on the Flow and Heat Transfer
The flow and heat transfer depend on the
parameter C, , RP, Rd and Q where C is
the magnetic field parameter, is the
viscosity variation coefficient, RP is the
mixed convection parameter which is the
ratio of the Rayleigh number to cleteP
number, Rd is the radiation parameter and
Q is the heat generation parameter. Positive
and negative values of A correspond to
TT0 , TT0 and in turn to assisting
flow and opposing flow respectively. The
parameter takes positive as well as
negative values, the limiting values being
-2 and +2. Irrespective of the values of
0T and T , zero value of corresponds to
constant viscosity. In this paper solutions are
found for the values of -1, 0, 0.5 and 1 of .
The mixed convection parameter RP
takes positive values for assisting flow and
negative values for opposing flow. When RP
is zero, the results correspond to the forced
convection case. Enhanced flow can
correspond to an increase in the positive
value of RP, as an increase in its value can
be due to an increase in the temperature
difference )( 0 TT . Calculations are done
for a wide range of positive and negative
values of RP.
When there is no magnetic field, the
parameter C takes the value unity and for
increasing intensity of the magnetic field,
the parameter takes values smaller than
unity. In the present study, solutions are
found for the values 0.5 and 1 of C.
Reduced flow can be expected for smaller
values of C or for increased intensity of the
magnetic field as the Lorentz force (due to
the magnetic field) obstructs the flow.
When transfer of heat energy through
radiation is neglected, the parameter Rd
takes zero value and for increasing intensity
of thermal radiation, the parameter takes
larger values. Solutions are found for the
values 0, 0.5, 10 of the parameter Rd.
Thermal radiation causes thickening of the
thermal boundary layer and hence increasing
values of the parameter Rd can increase
thermal boundary layer thickness.
When transfer of heat energy through
generation of heat is neglected the parameter
Q takes zero value and for heat generation,
the parameter Q takes positive values like
0.5 and 0.7. However for heat absorption,
the parameter Q takes negative values, say,
-0.2. When heat absorption occurs (Q is negative) the fluid exerts a dragging force
on the surface and opposite is the case for
heat generation (Q is positive). Effects of
Simultaneous variation of the values of the
parameters on the flow and heat transfer are
presented in the discussion.
3.2 NUMERICAL SOLUTION
The equations for f and , i.e.,
equations (2.7), (2.8) are integrated
numerically subject to boundary conditions
(2.9) by Ruge-Kutta-Gill method together
with a shooting technique. The accuracy of
the method is tested by comparing
appropriate results of the present analysis
with available results. Present work agreed
well with Merkin [9], Chin etal.[5], which is
shown in Fig-2 for skin friction )0(f and in
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Table-1 for heat transfer coefficient ' (0) '
respectively for no magnetic field, no
radiation, constant viscosity and no heat
generation i.e., 1, 0, 0 & 0C Rd Q
Table -1
Variation of heat transfer )'0(' with RP
)'0(' for Upper Solution RP Chin[5] Present work
0.0 0.79790 0.7978845
-0.1 --------- 0.7720349
-0.3 0.71730 0.7172853
-0.5 0.65750 0.6574822
-0.8 0.55396 0.5539434
-1.0 0.46962 0.4695999
-1.1 0.41915 0.4191351
-1.2 0.35848 0.3584588
-1.3 0.27448 0.2743369
-1.35 0.19388 0.1908006
RPc=-1.354 0.15589 0.1556460
)'0(' for Lower Solution -1.1 0.00176 0.0017597
-1.2 0.01849 0.0184873
-1.3 0.06528 0.0652853
-1.35 0.13223 0.1322655
RPc=-1.354 0.15589 0.1561500
3.3 Discussion of the Results
In the following, discussion of both
assisting and opposing flows are presented.
However more attention is paid to the
solutions of the opposing flow case. When
0T T the opposing flow case arises in
which RP takes negative values and dual
solutions exist. The solution corresponding
to relatively larger values of (0)f and
' (0) ' is referred to as the upper solution
and the one corresponding to smaller values
of (0)f and ' (0) ' as the lower solution.
It can be understood that at the critical
values of RP ( )cRP , the first solution (upper
solution) and the second solution (lower
solution) meet. When 0T T the assisting
flow case arises in which RP takes positive
values and single solution exist. The local
drag coefficient is proportional to the skin
friction, (0)f and the local Nusselt number
is directly proportional to the heat transfer
coefficient or wall transfer rate, ' (0) ' .
Qualitatively interesting results related to
the skin friction, heat transfer coefficient,
velocity and temperature profiles are
presented; some of them are in the form of
table-1 and others in the form of figures 2 to
16. The changes in skin friction with
negative values of RP (opposing flow) are
shown in the Fig-2, 3, 4 and with positive
values of RP(assisting flow) in the Fig-5.
The corresponding changes in heat transfer
rate are shown in Fig-6, 7 & 8.
From figures 3 and 4, it can be observed
that when Q is positive (heat generation) the
local drag coefficient (the values of (0)f )
takes negative values and dual solution
exist. However for Q negative (heat
absorption) the local drag coefficient takes
positive values and single solution exists.
Physically, positive sign of (0)f implies
that the fluid exerts a dragging force on the
surface and negative sign implies the
opposite. It is also observed as Q increases
from 0.5 to 0.7 the range of admissible
absolute values of RP also decrease. From
Fig-3 it is observed that local drag
coefficient takes larger values in the
presence of magnetic field (C=0.5) than in
its absence (C=1). The coefficient of drag is
also observed to be larger in the absence of
radiation than in its presence. The range
(critical value of RP) over which solution
exist can be seen to be considerably larger
for negative variation of viscosity
( 0.5) than for constant variation of
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11
viscosity ( 0 ). However for positive
variation of viscosity ( 0.5) the range is
less than for constant variation of viscosity
( 0) .
From Fig-5 (assisting flow), it is
observed that as RP increases the magnitude
of (0)f increases for all values of
, , &C Rd Q . This is because the fluid
velocity increases when the buoyancy force
increases and hence increases the skin
friction.
It is observed from Fig-6 that ' (0) '
decreases when RP becomes more negative.
This is because the buoyancy force works
against the fluid flow and therefore heat
transfer process is retarded. Like skin
friction ( (0))f , heat transfer coefficient
(' (0) ') is negative for heat generation and
is positive for heat absorption. From Fig-6
and 7 it is observed that, as the intensity of
magnetic field increases i.e., (C decreases
from 1 to 0.5) heat transfer coefficient
' (0) ' increases in case of opposing flow
and opposite is the behavior in case of
assisting flow. Also it can be observed that
for a fixed value of RP, the value of
' (0) ' for negative variation of viscosity
)5.0( is always higher than the value
of ' (0) ' for constant viscosity ( 0)
and the value of ' (0) ' for positive
variation of viscosity ( 0.5) is always
less than for constant viscosity ( 0) . It is
observed that with variable viscosity, the
separation of boundary layer is delayed for
positive variation of viscosity than for
negative variation of viscosity. In case of
assisting flow, from Fig-8, it can be viewed
that heat transfer coefficient ' (0) '
increases as RP increases positively. This is
because the buoyancy force works along
with the fluid flow, therefore heat transfer
process is accelerated.
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Variations in temperature are shown in
Fig-9 to Fig-13. From Fig-9, it may be
observed that thermal boundary layer
thickness is more in case of heat generation
(Q is positive) than in case of heat absorption
(Q is negative). Boundary layer thickness is
more for lower solution than for upper
solution. As magnitude of RP increases
thermal boundary layer thickness increases.
From Fig-10 it may be viewed that
boundary layer thickness is more in presence
of radiation than in its absence. Also it may
be observed that increase in thermal radiation
parameter (Rd) produces significant increase
in the thickness of the thermal boundary layer
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of the fluid and so the temperature profile
( ) increases and tends to zero at the edge of
the boundary layer. This is due to the fact that
the presence of thermal radiation causes
thickening of the thermal boundary layer. As
C decreases (magnetic effect increases) there
is no significant change in upper solution but
there is an increase in the lower solution. This
may be due to the resistance offered by the
Lorentz force due to the flow, and as a result
an increase in temperature.
In Fig-11 are shown variations of
temperature with variations of viscosity for
fixed values of C, Rd, Q and RP. It may be
noticed that the nature of the lower solution is
just opposite to the nature of the upper
solution.
From Fig-12 it may be observed that
temperature and thermal boundary layer
thickness are more for opposing flow than for
assisting flow. It can be seen from Fig-13 that
in case of assisting flow, thermal boundary
thickness for positive variation of viscosity
)5.0( is more than for constant viscosity
)0( . But, for negative variation viscosity
)5.0( it is less than for constant viscosity.
As RP (mixed convection parameter) increases
thermal boundary layer thickness decreases and
also it is observed for RP=50, after certain
stage, temperature is becoming negative.
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From Fig-14, slip velocity 9.0)0( f is
same in case of upper and lower solutions for
heat generation (Q= 0.5, 0.7) as well as for heat
absorption )2.0( Q when 0,1.0 RP .
It can be observed that as RP becomes more
negative slip velocity decreases for both heat
generation and absorption. The velocity of the
fluid is more for heat absorption than for heat
generation. Hydrodynamic boundary layer
thickness is more for lower solution than for
upper solution.
From Fig-15, it can be observed that slip
velocity and fluid velocity for negative
variation of viscosity ( 5.0 ) as well as for
positive variation of viscosity )5.0( is
more than for constant viscosity ( 0
).Velocities of aiding and opposing flows are
shown in Fig-16. As RP increases, velocity and
slip velocity increase and slip velocity of the
aiding flow is found to be more than unity
whereas for opposing flow they are less than
unity. It may be noted that slip velocity can be
directly calculated from the equation 2.10.
From Fig-9 and Fig-14 it can be observed
that thermal boundary layer thickness is more
than hydrodynamic boundary layer thickness.
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4. Conclusions
This paper analyzes momentum and heat
transfer as affected by magnetic field,
radiation, variable viscosity and heat
generation in similarity solutions of mixed
convection adjacent to an isothermal vertical
plate embedded in a porous medium.
Numerical solutions for momentum and heat
transfer are obtained by employing Runge-
Kutta Gill method together with a shooting
technique. The following conclusions are
drawn from the numerical results.
1. In case of opposing flow in the presence of heat generation (Q is positive), the local
drag coefficient (skin friction) takes
negative values and dual solutions exist.
However in case of heat absorption (Q is
negative), skin friction takes positive values
and single solution exists. Physically
positive sign of skin friction implies that the
fluid exerts a dragging force on the surface
and negative sign implies opposite.
2. Local drag coefficient takes larger values in the presence of magnetic field than in its
absence. The coefficient of drag is also
observed to be larger in the absence of
radiation than in its presence.
3. The separation of boundary layer is delayed for positive variation of viscosity than for
negative variation of viscosity.
4. As mixed convection parameter (RP) is more negative (opposing flow), heat transfer
coefficient decreases because the buoyancy
force works against the fluid flow, therefore
heat transfer process is retarded. In case of
assisting flow heat transfer coefficient
increases as RP increases positively. This is
because the buoyancy force works along
with the fluid flow, therefore heat transfer
process is accelerated.
5. Thermal boundary layer thickness is more than hydrodynamic boundary layer
thickness. The velocity of the fluid is more
for heat absorption than for heat generation.
Slip velocity in aiding flow is more than in
opposing flow.
6. Thermal boundary layer increases as Rd(radiation) increases, due to the fact that
the presence of thermal radiation causes
thickening of the thermal boundary layer.
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