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Proceeding of 2nd International Conference on Mathematics and Information Sciences, 9-13 Nov. 2011, Sohag, Egypt
Heat and mass transfer analysis on the flow of non-
Newtonian micropolar fluid with uniform suction/blowing,
heat generation, chemical reaction and Thermophoresis
effects.
R. A. Mohamed
1, S. Z. Rida
2, A. A. M. Arfa
3 and M. Said
4
Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt
Corresponding author: Email: [email protected]
Abstract: In this paper, the problem of heat and mass transfer on the flow of non-Newtonian micropolar
fluid with uniform suction/blowing, heat generation, radiation, thermophoresis and chemical reaction is
presented and discussed. The Homotopy Analysis Method (HAM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. The
effects of various physical parameters such as material parameter, suction parameter, heat
generation/absorption parameter, Prandtl number, radiation parameter, thermophoretic parameter, chemical
reaction parameter and Schmidt number on the velocity profile temperature profile and
concentration profile are studied and shown in several plots. Keywords: Non-Newtonian micropolar fluid, Heat and mass transfer , Uniform suction/blowing, Heat
generation, Chemical reaction ,Thermophoresis and HAM.
1.Introduction
Micropolar fluids are fluids with microstructure belonging to a class of fluid with non-symmetrical
stress tensor referred to as polar fluids. Physically
they represent fluids consisting of randomly
oriented particles suspended in a viscous medium. The classical theories of continuum mechanics are
inadequate to explicate the microscopic
manifestations of microscopic events, a new stage in the evolution of fluid dynamic theory is in
progress. Eringen presented the earliest formulation
of a general theory of fluid microcontinua taking into account the inertial characteristics of the
substructure particles which are allowed to undergo
rotation. Eringen's actual theory of a fluid
microcontinuum was presented in 1964 in his paper on simple micro fluids [1]. This theory has been
extended by Eringen [2] to take into account
thermal effects. The theory of micropolar fluids and its extension thermo micropolar fluids [3] may form
suitable non-Newtonian fluid models which can be
used to explain the flow of colloidal fluids, liquid crystals, polymeric suspensions, animal blood, etc.
The theory of micropolar fluids developed by Eringen [1-3] describes some physical systems
which do not satisfy the Navier-Stokes equations.
This general theory of micropolar fluids deviates
from that of Newtonian fluids by adding two new variables to the velocity. These variables are
microrotations that are spin and microinertia tensors
describing the distributions of atoms and molecules inside the microscopic fluid particles. This theory
may be applied to the explanation for the
phenomenon of the flow of colloidal fluids, liquid crystals, polymeric suspensions, animal blood, etc.
An excellent review of micropolar fluids and their
applications was given by Ariman et al.[4]. Gorla
[5] discussed the steady state heat transfer in a micropolar fluid flow over a semi-infinite plate, and
the analysis is based on similarity variables. Rees
and Pop [6] studied the free convection boundary layer flow of micropolar fluid from a vertical flat
plate. Singh [7] has studied the free convection flow
of a micropolar fluid past an infinite vertical plate using the finite difference method.
American Academic & Scholarly Research Journal
Special Issue - January 2012 © 2012 NSP
86
The combined heat and mass transfer problems with
chemical reactions are of importance in many processes, and therefore have received a
considerable amount of attention in recent years. In
processes, such as drying, evaporation at the surface
of a water body, energy transfer in a wet cooling tower and the flow in a desert cooler, the heat and
mass transfer occurs simultaneously. Chemical
reactions can be codified as either homogeneous or heterogeneous processes. A homogeneous reaction
is one that occurs uniformly through a given phase.
In contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of a
phase. A reaction is said to be the first order if the
rate of reaction is directly proportional to the
concentration itself. In many chemical engineering processes, a chemical reaction between a foreign
mass and the fluid does occur. These processes take
place in numerous industrial applications, such as the polymer production, the manufacturing of
ceramics or glassware, the food processing [8] and
so on. Das et al.[9] considered the effects of a first order chemical reaction on the flow past an
impulsively started infinite vertical plate with
constant heat flux and mass transfer.
Muthucumarswamy and Ganesan [11] and Muthucumarswamy [10] studied the first order
homogeneous chemical reaction on the flow past an
infinite vertical plate. Recently, Kandasamy et al.[12] discussed the heat and mass transfer effect
along a wedge with a heat source and concentration
in the presence of suction/injection taking into
account the chemical reaction of the first order. The study of heat generation or absorption in
moving fluids is important in problems dealing with
chemical reactions and those concerned with dissociating fluids. Possible heat generation effects
may alter the temperature distribution;
consequently, the particle deposition rate in nuclear reactors, electronic chips and semiconductor wafers.
In fact, the literature is replete with examples
dealing with the heat transfer in laminar flow of
micropolar fluids. The study of radiation effects on the various types
of flows is quite complicated. In the recent years,
many authors have studied radiation effects on the boundary layer of radiating fluids past a plate.
Influence of chemical reaction and thermal radiation
on the heat and mass transfer in MHD micropolar flow over a vertical moving porous plate in a porous
medium with heat generation was studied by R.A.
Mohamed and S.M. Abo-Dahab [13]. Raptis [14]
studied the flow of a micropolar fluid past continuously moving plate by the presence of
radiation. The radiation effect on heat transfer of a
micropolar fluid past unmoving horizontal plate through a porous medium was studied by Abo-
Eldahab and Ghonaim [15]. Kim and Fedorov [16]
investigated the transient mixed radiative
convection flow of a micropolar fluid past a moving semi-infinite vertical porous plate.
Thermophoresis is a mechanism of migration of
small particles in direction of decreasing thermal gradient [17]. It is an effective method for particle
collection [18]. The velocity acquired by the
particle is called thermophoretic velocity and the force experienced by the suspended particle is
called thermophoretic force [19]. Thermophoresis
causes small particles to deposit on the cold
surfaces. It has many applications in aerosol technology, deposition of silicon thin films, and
radioactive particle deposition in nuclear reactor
safety simulations. For more detail on the topic, the readers may consult the studies [20-25]. Also
convective free mixed and forced convection flows
play an important role in petroleum extraction, in soils, storage of agricultural products, porous
material heat exchanger etc [26-32].
Since there are some limitations with the common
perturbation methods, and also because the basis of the common perturbation method is upon the
existence of a small parameter, developing the
method for different applications is very difficult. Therefore, many different methods have recently
introduced to eliminate the small parameter. The
Homotopy Analysis method (HAM) is one of the
well-known methods to solve the nonlinear equations. This method has been first introduced in
1992 by Liao [33-38]. The method has been used by
many authors in a wide variety of scientific and engineering applications to solve different types of
governing differential equations: linear and
nonlinear, homogeneous and non-homogeneous, and coupled and decoupled as well.
The purpose of present paper is to study the
problem of non-Newtonian micropolar fluid flow
with uniform suction/blowing, heat generation, radiation, thermophoresis and chemical reaction and
to investigated the effect of the various
dimensionless parameters of these non-Newtonian micropolar fluid on the velocity, temperature and
concentration. by means of an analytic technique,
namely the Homotopy Analysis Method (HAM).
2. Mathematical description
87
Consider the two-dimensional stagnation point
flow of an incompressible non-Newtonian micropolar fluid impinging perpendicular on a
permeable wall and flowing away along the -axis. And using the boundary layer approximation and
neglecting the dissipation, the equation of energy
for temperature with heat generation or absorption and thermal radiation, the equation of mass for
concentration with thermophoresis and chemical reaction. The simplified two-dimensional equations
governing the flow in the boundary layer of a steady, laminar, and incompressible micropolar
fluid are governed by:
(1)
(2)
(3)
(4)
(5)
where is the microrotation or angular velocity
whose direction of rotation is in the plane,
is the viscosity of the fluid, is the density, is the specific heat capacity at constant pressure of the
fluid, is the thermal conductivity of the fluid,
is the heat generation/absorption coefficient and
, and are the microinertia per unit mass, spin gradient viscosity, and vortex viscosity,
respectively, which are assumed to be constant.
The appropriate physical boundary conditions of Eqs. (1) – (5) are
(6)
where is a constant and .The case
indicates the vanishing of the ant symmetric part of the stress tensor and denotes
weak concentration of microelements, which will be considered here. Using the transform function we have
(7)
After using the transformation (7), for micropolar fluid, there are two equations in which one is for
angular velocity or microrotation and physically it
is important in micropolar fluid. In this study, we
have two equations and which
equals to (see Ziabakhsh. Z, et al.[39])
So Eqs. (2) and (3) reduce to the single equation as
Eq. (8a)
(8a)
(8b)
(8c) subject to the boundary conditions
(8d)
where is the material
parameter, is the suction parameter, and
primes denote differentiation with respect to .
is the Prandtl number and is the
heat Generation/absorption parameter is
the Schmidt number, is the chemical
reaction parameter is the
thermophoretic parameter and is the heat
radiation parameter. For micropolar boundary layer
flow, the wall skin friction .
(9)
Using as a characteristic velocity, the
skin friction Coefficient, can be defined as
(10)
By using this definition we have
(11)
where is the local Reynolds number, The
heat transfer from the surface to the fluid is
computed by application of Fourier’s law
(12)
Introducing the transformed variables, the
expression for becomes and the heat transfer
coefficient, in terms of the Nuselt number, , can
be expressed as
(13)
then we have The definition of the local mass flux and the local
Sherwood number are respectively given by
(14)
(15)
3. Application of ( HAM) to a problem
88
According to the boundary condition (8d), it is
nature that , and can be expressed by the function
(16) In the following form:
where are coefficients. The rule of
solution expression provides us with a starting
point. It is under the rule of solution expression that initial approximations, auxiliary linear operators,
and the auxiliary functions are determined. So,
according to the rule of solution expression, we choose the initial guess and auxiliary linear operator
in the following form:
(20)
As the initial approximations of ,
and we choose
As the auxiliary linear operator, we have the
following property: (22a)
(22b) (22c)
where are constants .based on, we are led to define the non linear operators:
(23)
(24)
(25) 3.1 Zeroth-order deformation equations
Let denotes the embedding parameter
and indicates non-zero auxiliary parameters. We then construct the following equations:
(26)
subject to the boundary conditions:
(27)
Obviously, when , the above (HAM ) deformation equations (26) have the
solutions:
(28)
if increases from then and
vary from and
By Taylor’s theorem and
using equations.(28),
can be expanded in a power series of as
follows:
In which is chosen in such a way that these three
series (29), (30) and (31) are convergent at ,
we have, using equations (28), the solutions series:
3.2 High order deformation equation
89
For the sake of simplicity, we define the vectors:
Differentiation the zeroth-order deformation
equations times with respect to then
setting and finally dividing them by we
obtain the th- order deformation equations
(34)
subject to the boundary conditions:
(35)
Where
(36c)
And
(37)
Let denote the particular
solutions of equations (34). Using
we have the general solution:
(38)
where to are constants that can be obtained by applying the boundary conditions in equations. (35) as discussed by Liao the rule of coefficient
ergodicity and the rule of solution existence play
important roles in determining the auxiliary
function and ensuring that the high-order deformation equations are closed and have
solutions. In many cases, by means of the rule of
solution expression and the rule of coefficient ergodicity, auxiliary functions can be uniquely
determined. So we define the auxiliary function
which for both velocity field and temperature is true and same. It is in the following form:
4. Convergence of the (HAM) solution
Liao [35] proved in general that, as long as a
solution series given by homotopy analysis method is not divergent, it must converge to the exact
solution of non linear problems under investigation.
The convergence of the solution series depends
upon the choice of initial approximations, the auxiliary linear operators and the nonzero auxiliary
parameters. Once if the initial guess approximations
and the auxiliary linear operators have been selected then the convergence of the solution series will
strictly depend upon the auxiliary parameter only.
Therefore, the convergence of the solution series is determined by the values of such kind of
parameters. The admissible values of parameter is
determined by the so-called curves. In order to
find the allowed value of to make the series (32)
convergent we have plotted the curves
corresponding to Our analysis
shows that the admissible value of for and
are and , respectively.
5. Results and discussion
To study the behavior of the velocity ,
temperature and concentration
curves are drawn for various values of the parameters that describe the flow. The results
of analytical computation are displayed in figures from Fig.1 to Fig 15. Results are obtained for
. Fig. 1 and Fig. 2. display results
for the velocity It is seen that increases
with increasing the suction parameter and
decreases with increasing the material parameter respectively. Fig. 3 and Fig. 4 display results for the temperature distribution, it is seen that
decreases with increasing the suction
parameter and increases with increasing the heat
generation/absorption parameter respectively. Fig.
5 and Fig. 6 describes the behavior of the temperature distribution with changes in the values
of the material parameter and radiation
parameter it is seen that the temperature
distribution decreases with increasing the
material parameter , but it increase with
increasing the radiation parameter . The effect of
prandtl number on the dimensionless
temperature distributions is displayed in Fig.
7. The effect of suction is to decrease temperature
distribution . Fig. 8 and Fig. 9 show that the
concentration distribution decreases with
increasing the material parameter and with
increasing prandtl number . Fig. 10 and Fig. 11
represents the effect of radiation parameter on the
90
concentration profiles while no effect appears
in the velocity . As the radiation parameter
increases the concentration distribution
decreases and it decreases with increasing the heat
generation/absorption parameter too . Fig 12 and Fig. 13 describes the behavior of the concentration
distribution with changes in the values of the
thermophortic parameter and chemical reaction
parameter , respectively. It is seen that
concentration distribution decreases with
increasing both thermophortic parameter and
chemical reaction parameter . Fig. 14 and Fig. 15
show that the concentration distribution
decreases with increasing the suction parameter and decreases with increasing the Schmidt
number far from the wall but increases near from the wall. The governing fundamental are
approximated by a system on non-linear ordinary
differential equations by similarity transformation and it solved analytically by means of an analytic
technique, namely the homotopy analysis method,
results are presented graphically to illustrate the
variation of velocity, temperature and concentration with various values of parameters for the problem,
e.g. suction parameter , prandtl number ,
radiation , Schmidt number , chemical reaction
parameter , heat generation/absorption parameter
, thermophortic parameter and material
parameter . The analytical results indicate that the velocity
increases with increasing and decreases
with increasing but and not affected on
it The temperature distribution increases with
increasing but decrease with
increasing and The concentration
distribution increases with increasing ,
but decrease with increasing ,
and
Fig.1. The for various
Fig.2. The for various
Fig.3. The for various
Fig.4. The for various
0 1 2 3 4 5 6 7 80.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
B = -0.5
B = -0.1
B = 0.0
B = 0.1
B = 0.5
0 1 2 3 4 5 6 7 80.0
0.2
0.4
0.6
0.8
1.0
f'
K=0.0
K=0.5
K=1.0
K=1.5
K=2.0
0 1 2 3 4 5 6 7 80.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
A = -2.0
A = -1.0
A = 0.0
A = 1.0
A = 2.0
0 1 2 3 4 5 6 7 80.0
0.2
0.4
0.6
0.8
1.0
f'
A = -2.0
A = -1.0
A = 0.0
A = 1.0
A = 2.0
91
Fig. 5. The for various
Fig. 6. The for various
Fig. 7. The for various
Fig. 8. The for various
Fig.9. The for various
Fig.10. The for various
0 1 2 3 4 5 6 7 80.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
K = 0.0
K = 1.0
K = 2.0
K = 3.0
K = 5.0
0 1 2 3 4 5 6 7 80.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
R = 0.0
R = 0.5
R = 1.0
R = 1.5
R = 2.0
0 1 2 3 4 5 6 7 8
0.0
0.2
0.4
0.6
0.8
1.0
Pr=0.1
Pr=0.5
Pr=0.71
Pr=1.0
Pr=7.0
0 1 2 3 4 5 6 7 80.0
0.2
0.4
0.6
0.8
1.0
pr = 0.1
pr = 0.5
pr = 0.71
pr = 1.0
pr = 7.0
pr = 10
0 1 2 3 4 5 6 7 80.0
0.2
0.4
0.6
0.8
1.0
B = -2.0
B = -1.0
B = 0.0
B = 1.0
B = 2.0
0 1 2 3 4 5 6 7 80.0
0.2
0.4
0.6
0.8
1.0
K = 0.0
K = 3.0
K = 7.0
K = 10.0
92
Fig.11. The for various
Fig.12. The for various
Fig.13. The for various
Fig.14. The for various
Fig.15. The for various
6. Conclusions
In this paper, the effect of chemical reaction and thermophoresis of a micropolar fluid in the
presence of heat generation or absorption and
thermal radiation are studied by means of an
analytical technique, namely the homotopy analysis method. The governing equations for the problem
are changed to dimension less ordinary differential
equations by similarity transformation. The effect of the various dimensionless parameters are invest-
tigated.
The proposed analytic approach has general meaning and thus may be applied in a similar way
to other unsteady nonlinear problems to get accurate
analytic solutions valid for all dimensionless time.
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0 1 2 3 4 5 6 7 80.0
0.2
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93
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