-
Study the Micropolar Fluid Flow near the Stagnation on a
Vertical Plate
with Prescribed Wall Heat Flux
Gitima Patowary1, Dusmanta Kumar Sut
2
1Assistant Professor, Dept. of Mathematics, Barama College,
Barama, Assam, email:
[email protected] 2Assistant Professor, Dept. of
Mathematics, Jorhat Institute of
Science and Technology, Jorhat, Assam, email:
[email protected]
Abstract
This paper investigates the
influence of the material parameter,
buoyancy parameter and Prandtl number
on the skin friction coefficient and the heat
transfer rate at the surface on a steady,
two dimensional flow of an incompressible
micropolar fluid near the stagnation point
on a vertical plate with prescribed surface
heat flux in presence of a magnetic filed.
The free stream velocity and the surface
heat flux are assumed to be proportional to
the distance from the stagnation point.
Similarly transformation is employed to
transform the governing partial differential
equations to a set of ordinary differential
equations. The effects of the material
parameter, buoyancy parameter and
Prandtl number on the skin friction
coefficient and the heat transfer rate at the
surface are discussed and the
corresponding velocity, temperature and
microrotation profiles are shown
graphically. Both assisting and opposing
flows are considered and it is found that
dual solutions exist for both cases.
Key words: micropolar fluid, vertical
plate, mixed convection, stagnation point
flow, magnetic parameter, boundary layer,
fluid mechanics, shooting method.
1. Introduction The theory of micropolar fluid, first
introduced and formulated by Eringen
(1966, 1972) and derived the constitutive
laws of fluid with micro-structure has been
a field of very active research for the last
decades. This theory is capable to explain
the complex fluids behaviour such as
liquid crystals, polymeric suspensions,
animal blood etc. by taking into account
the effect arising from local structure and
micro-motions of the fluid elements. An
extensive review of micropolar fluids and
their applications has been done by Ariman
et al. (1973).
Many researchers have been
studied the free or mixed convection flow
of a micropolar fluid towards a vertical
surface. Bhargava et al. (2003) have
analyzed the effect of temperature
dependent heat sources on the fully
developed free convection electrically
conducting micropolar fluid between two
parallel porous vertical plates in a strong
magnetic field by solving the governing
differential equations using the quasi
linearization method. Jena and Mathur
(1984) considered the laminar mixed
convection boundary layer flow of a
micropolar fluid from an isothermal
vertical flat plate. Gorla (1995) studied the
unsteady laminar mixed convection
boundary layer flow of a micropolar fluid
over a vertical flat plate by assuming the
free stream velocity undergoes arbitrary
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variation with time. Lok et al. (2006)
studied the unsteady mixed convection
boundary layer flow of a micropolar fluid
near the stagnation point flow towards a
horizontal permeable plate immersed in a
micropolar fluid. They obtained the
solutions using homotopy analysis method
and found that their solutions are in
favourable agreement with the numerical
solutions previously reported by Attia
(2006). Azizah et al. (2010) studied
considering a steady mixed convection
flow of micropolar fluid near the
stagnation point on a vertical plate with
prescribed surface heat flux. The similar
problem in a viscous fluid was studied by
Ramachandran et al. (1988) by considering
both cases of an arbitrary surface
temperature and arbitrary surface heat flux.
They found that dual solutions exist for
certain range of the buoyancy parameter,
for opposing flow case only. Devi et al.
(1991) extended this problem to the
unsteady case, while Lok et al (2005)
investigated the case when the plate
immersed in a micropolar fluid. They also
fund that dual solution exist for the
opposing flow only. Azizah et al. (2010)
have showed that dual solutions also exist
for the prescribed surface heat flux case,
for both assisting and opposing flows.
Motivated by the above
investigations, the present paper considers
a steady mixed convection flow of a
micropolar fluid near the stagnation point
on a vertical plate with prescribed heat flux
in presence of steady magnetic field. The
objective of the present study is to show
that dual solutions also exist for the
prescribed surface heat flux case, for both
assisting and opposing flows in presence of
steady magnetic field.
2. Mathematical formulation
A steady, two dimensional flow of
an incompressible electrically conducting
micropolar fluid near the stagnation point
on a vertical flat plat place in the plane
0y of a Cartesian system of coordinates
Oxy in presence of a magnetic field 0B
applied in the normal direction to the walls
is considered. The fluid occupies the half
plane 0y . It is assumed that the free
stream velocity )(xU and the surface heat
flux )(xqw are proportional to the distance
x from the stagnation point, i.e.,
axxU )( and bxxqw )( , where a and
b are the constants. Under these
assumptions, the simplified governing
equation equations are given by
Equation of continuity is
0
y
v
x
u (1)
Equation of momentum is
uBTTgy
N
y
u
dx
dUU
y
uv
x
uu
2
02
2
)(
(2)
Equation of angular momentum is
y
uN
y
N
y
Nv
x
Nuj 2
2
2
(3)
Equation of energy is
2
2
y
T
y
Tv
x
Tu
(4)
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The appropriate physical the boundary conditions of equations
are
wq
y
T
y
unNvu
,,0,0 at 0y
TTNxUu ,0),( as y (5)
where u and v are the components of
velocity in x and y direction respectively,
is the dynamic viscosity, is the
vortex viscosity, is the fluid density,
is the spin gradient viscosity, is the thermal diffusivity, is
the thermal
expansion coefficient, g acceleration due
to gravity, T is the fluid temperature, j is
the micro inertia density, N is the micro
rotation vector, 0B is the external magnetic
field and wq is the surface heat flux. It
should be noted that n is a constant such
that 10 n (Azizah et al., 2010). The
case n=0 is called strong concentration by
Guram and Smith (1980), which indicates
N=0 near the wall, represents concentrated
particle flows in which the microelements
close to the wall surface are unable to
rotate (Jena and Mathur, 1981). The case
n=1/2 indicates the vanishing of anti
symmetric part of the stress tensor and
denotes weak concentrations (Ahmadi,
1976). The case n=1, as suggested by
Peddieson (1972), is used for the modeling
of turbulent boundary layer flows. The
case n=1/2 for weak concentration
(Azuzah et al., 2010). Here we follow by
assuming jKj )2/1()2/(
(Azuzah et al., 2010). This assumption is
invoked to allow the field of equations
predicts the correct behaviour in the
limiting case when the microstructure
effects become negligible and the total spin
N reduces to the angular velocity (Ahmadi,
1976, Yucel, 1989).
The governing equations subject to
the boundary conditions can be expressed
in a simpler form by introducing the
following transformations:
x
vy
u
, (6)
2/12/13
2
12
1
)()(),(),()(,
v
a
q
TTkxg
v
aNxfavy
v
a
w
(7)
Where is the thermal conductivity, is the stream function.
The transformed ordinary differential equations are:
01)1( 2 hKfRffffK (8)
0)2(2
1
fgKgfgfg
K (9)
01
ffPr
(10)
subject to the boundary conditions (5) which become
1)0(),0()0(,0)0(,0)0( fngff
0)(,0)(,1)( gf as (11)
where we have taken avj / as a reference length (Nazar et al
2004). Here prime denote
differentiation with respect to , /K is the material parameter,
/vPr is Prandtl
number and 2/5
Re/ xxGr , a
BR
2
00 is the buoyancy parameter. Further
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)/( 24 kvxqgGr wx is the local Grashof number and vUxx /Re is
the local Reynolds
number. Here is a constant with 0 and 0 correspond to the
opposing and assisting flows respectively, while 0 is for pure
forced convection flow. When 0K (viscous
fluid), the present problem reduces to those of Ramachandran et
al (1988) and when 00 B ,
the present problem reduces to those of Azizah et al.
(2010).
The physical quantities of interest are the skin friction
coefficient fC and the local
Nusselt number xNu , which are defined as
)(,
22
TTk
xqNu
UC
w
w
x
w
f
(12)
Where the wall shear stress w and the wall heat flux wg are
given by
00
,
y
w
y
wy
TkqkN
y
uk (13)
Using (6), we obtain
)0(/1Re/),0(])1(1[Re2
1 2/12/1 xxx NufKnC f (14)
3. Results and Discussion
In this paper, we have attempted to
develop a mathematical analysis for
investigating a steady, two dimensional
flow of an incompressible micropolar fluid
near the stagnation point on a vertical plate
with prescribed surface heat flux in
presence of magnetic field. The systems of
non linear equations (8-10) subject to the
boundary conditions (11) have been solved
numerically by applying Runge-Kutta
Shooting techniques for some values of the
governing parameters, namely the material
parameter K, buoyancy parameter , Prandtl number Pr and
Magnetic
parameter R, while we consider n=1/2
throughout the paper, which represents
weak concentration of fluids particles at
the surface. The dual solutions were
obtained by setting two different values of
boundary layer thickness which
satisfies the boundary condition (11) and
produce two different velocity,
temperature and microrotation profiles as
shown in Figs 1-3. The variations of the
skin friction coefficient )0(f and the
surface temperature )0( with the mixed
convection or buoyancy parameter for different values of Pr and
K=1 are
presented in Figs. 4 and 5, respectively.
These figures show that dual solutions
exist for both assisting )0( and
opposing )0( flows, For the assisting
flows, the solution exists for all values of
, while for the opposing flow it exists up
to a critical value, say c . Between the two
solutions as presented in Figs. 4 and 5, we
expect that the upper branch solution is
stable and most physically relevance, while
the lower branch solution is not, since it is
the only solution for the forced convection
case )0( .
4. Conclusion
The problem of mixed convection flow towards a vertical plate
with a
prescribed surface heat flux immersed in
an incompressible micropolar fluid in
presence of a magnetic field has been
studied numerically. The governing partial
differential equations were transformed
into ordinary differential equations using
similarity transformation, before being
solved numerically by shooting method.
We discussed the effects of the material
parameter, buoyancy parameter, the
Prandtl number, Magnetic parameter on
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the fluid flow and heat transfer
characteristics. We found that dual
solutions exist for both assisting and
opposing flows. The solutions for the
assisting flow )0( could be obtained
for all values of the buoyancy parameter ,
while for the opposing flow )0( , the
solutions were obtained only in the range
of )0( c where c is the minimum
value of for which the solution exists. It is also found that
micropolar fluids as well
as fluids with larger Prandtl number
increase the range of for which the solution exists.
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-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
f'
Fig. 1. Variation velocity profiels )(f against for various
values of K taking Pr=0.7,
1 , R=0.1
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-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Fig. 2. Variation temperature profiles )( against for various
values of K taking
Pr=0.7, 1 , R=0.1
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0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
->
Fig. 3. Variation of microrotation profiles )(g against for
various values of K taking
Pr=0.7, 1 , R=0.1
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5
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Fig. 4. Variation velocity profiles )(f against for various
values of Pr taking K=1,
1 , R=0.1
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0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Fig. 5. Variation temperature profiles )( against for various
values of Pr taking
K=1, 1 , R=0.1
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