slide 1: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
Page | 53
Numerical treatment and Global Error Estimation of a MHD
Flows of an Oldroyd 6-Constant Nano-Fluid through a non-
Darcy Porous medium with Heat and Mass Transfer
Abeer A. Shaaban
Department of Mathematics Faculty of Education Ain Shams University Roxy Cairo Egypt
Department of Management Information Systems Faculty of Business Administration in Rass Qassim University Qassim
KSA
Abstract — Explicit Finite-Difference method was used to obtain the solution of the system of the non-linear ordinary
differential equations which transform from the non-linear partial differential equations. These equations describe the steady
magneto-hydrodynamic flow of an oldroyd 6-constant non-Newtonian nano-fluid through a non-Darcy porous medium with
heat and mass transfer. The numerical formula of the velocity the temperature the concentration and the nanoparticles
concentration distributions of the problem were illustrated graphically. The effect of Darcy number D
a
Forchheimer number
F
s
magnetic field parameter M local temperature Grashof number G
r
local nanoparticle Grashof B
r
Prandtl number P
r
Dufour number N
d
Brownian motion parameter N
b
Thermophoresis parameter N
t
Lewis number L
e
Sort number L
d
Chemical reaction parameter R
c
and Chemical reaction order m on those formula were discussed at the values of material
parameters
specially in the case of pure Coutte flow. Then the effects of modified pressure gradients on
those formulas were discussed in the case of pure Poiseuille flow and the generalized Couette flow. Also an estimation of the
global error for the numerical values of the solutions is calculated by using Zadunaisky technique.
Keywords — Finite-difference method Heat and Mass transfer MHD flows Non-Darcy Porous medium Oldoryd 6-
Constant non-Newtonian nano-Fluid.
I. NOMENCLATURE
Chemical reaction parameter deﬁned by Eq. 42
Chemical Reaction rate constant A
The dimensionless concentration
Local nanoparticle Grashof number
deﬁned by Eq. 42
B
r
The time t Forchheimer number c
The fluid temperature T The concentration of the fluid C
The temperature at lower plate
The concentration at lower plate
The temperature at upper plate
The concentration at upper plate
The velocity vector Nanoparticle susceptibility
Darcy number deﬁned by Eq. 42 D
a
Brownian diffusion coefficient
D
B
Greek symbols
Thermophoretic diffusion coefficient
D
T
The nanoparticles phenomena
Electrical field E
The dissipation function The external force F
The dimensionless nanoparticles Forchheimer number deﬁned by Eq. 42 F
s
The dimensionless temperature Gravitational acceleration G
The non-Newtonian parameters deﬁned by Eq. 36
Local temperature Grashof number
deﬁned by Eq. 42
G
r
The magnetic field
Gradient operator The current density J
Laplacian operator
Thermal conductivity
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the dynamic viscosity of ﬂuid Permeability constant
the magnetic permeability
Thermal diffusion ratio K
T
The kinematic viscosity
Lewis number deﬁned by Eq. 42
The specific heat capacity at constant pressure
Sort number deﬁned by Eq. 42
The density of the fluid
f
Chemical Reaction order M
The density of the particle
p
Magnetic field parameter deﬁned by Eq.
42
M
heat capacity of the ﬂuid
f
c
Brownian motion parameter deﬁned by Eq.
42
N
b
effective heat capacity of the nanoparticle material
p
c
Dufour numbe deﬁned by Eq. 42 N
d
Electrical conductivity of the fluid
The thermophoresis parameter deﬁned by
Eq. 42
N
t
The Cauchy Stress tensor The fluid pressure P
Volumetric thermal and solute expansion coefficients of the
base fluid
Prandtl number deﬁned by Eq. 42 P
r
II. INTRODUCTION
The study of non-Newtonian fluids has gained much attention recently in view of its promising applications in engineering
and industry. Such fluids exhibit a non-linear relationship between the stresses and the rate of strain. Due to non-linear
dependence the analysis of the behavior of the non-Newtonian fluids tends to be much more complicated and subtle in
comparison with that of Newtonian fluids. Flow of fluids with complex microstructure e. g. molten polymer polymer
solutions blood paints greases oils ketchup etc. cannot be described by a single model of non-Newtonian fluids. Many
models that exist are based either on natural modifications of established macroscopic theories or molecular considerations.
In general the equations of motion for non-Newtonian fluids are of higher order than the Navier-Stokes equations and thus
one need conditions in addition to the usual adherence boundary condition 28.
Guillope and Saut 2 has established existence results for some shearing motions of viscoelastic fluids of Oldroyd type.
Some exact solutions of an Oldroyd 3-constant fluid are studied in 4 5 11 13. Baris 21 investigated the steady flow
of an Oldroyd 6-constant fluid between intersecting planes using the series expansion method. Hayat et al. 23 studied the
Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field by using the Homotopy analysis method.
Hayat et al. 25 studied the steady flow of a magneto-hydrodynamic Oldroyd 6-constant fluid by the motion of an infinite
plate using the Homotopy analysis method. Wang et al. 30 investigated the non-linear magnetohydrodynamic problems of
an Oldroyd 6-constant fluid by using analytical method and the finite-difference discretization method. Hayat et al. 24
studied the effect of the slip condition on flows of an Oldroyd 6-constant fluid. Rana et al. 14 studied the Hall effects on
hydromagnetic flow of an Oldroyd 6-constant fluid between concentric cylinders by the finite difference method. Hayat et al.
26 investigated the exact solution of a thin film flow of an Oldroyd 6-constant fluid over a moving belt by the Homotopy
perturbation.
Investigation of nanofluid flow has received special focus in the past due to its relevance in numerous industrial applications.
The researchers not only discovered unexpected thermal properties of nanofluids but also proposed mechanisms behind the
enhanced thermal properties of nanofluids and thus identified unusual opportunities to develop them as next generation
coolants for computers and safe coolants for nuclear reactors. A combination of nanofluid with biotechnological components
can provide potential applications in agriculture pharmaceuticals and biological sensors. Various types of nanomaterials
including nanoparticles nanowires nanofibers nanostructures and nanomachines are used in biotechnological applications.
The commercialization of nano-biotechnological products seems to have a potential future and within next a few years many
new products of this nature are likely to be used. Nano and micro-fluidics is a new area which has potential for engineering
applications especially for the development of new biomedical devices and procedures 1.
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The magnetic nanofluids possess both liquid and magnetic properties. These fluids have key importance in modulators
optical switches optical gratings and tunable optical fiber filters. The magnetic nanoparticles are significant in medicine
construction of loud speakers sink float separation cancer therapy and tumor analysis. Nowadays the sustainable energy
generation is one of the most important issues across the globe. Perhaps the solar energy is also one of the best sources of
renewable energy with minimal environment impact. Solar power is a direct way obtaining heat water and electricity from
the nature. Researchers concluded that heat transfer and solar collection processes can be improved through the addition of
nanoparticles in the fluids. Fluid heating and cooling are required in many industrial fields such as power manufacturing and
transportation. Effective cooling techniques are needed for cooling any sort of high energy device. Common heat transfer
fluids such as ethylene glycol water and engine oil have limited heat transfer capabilities due to their low heat transfer
properties and thus cannot meet the modern cooling requirements 1.
The terminology of nanofluid was first used by Choi 22 when he experimentally discovered a significant improvement in
the heat transfer performance through the addition of small amount nanometer sized particles in the base fluids. This addition
also causes scattering of the incident radiation which allows a deeper absorption within the fluid. Recently Trieb and Nitsch
3 proposed the idea of solar thermal collectors by using nanofluids to directly absorb the solar radiation.
The phenomenon of thermal conductivity enhancement by dispersing nanoparticles in the liquid was also observed by
Masuda et al. 7. Buongiorno 9 recognized that the two main effects namely the Brownian motion and thermophoretic
diffusion of nanoparticles contribute to the massive increase in the thermal conductivity of the liquids. He also proposed the
modifications in the transport equations due to these effects.
Vajravelu et al. 12 investigated heat and mass transfer properties of three-layer fluid flow in which nano-fluid layer is
squeezed between two clear viscous fluid. Farooq et al. 27 studied heat and mass transfer of two-layer flows of third-grade
nano-fluids in a vertical channel. Abou-zeid et al. 16 obtained numerical solutions and Global error estimation of natural
convection effects on gliding motion of bacteria on a power-law nano-slime through a non-Darcy porous medium. El-Dabe et
al. 17 investigated magneto-hydrodynamic non-Newtonian nano-fluid flow over a stretching sheet through a non-Darcy
porous medium with radiation and chemical reaction.
Magneto-hydrodynamic MHD is concerning the mathematical and physical scaffold that introduces magnetic-dynamics in
electrically conducting fluids e.g. in plasmas and liquid metals . The applications of Magneto-hydrodynamic
incompressible viscous flow in science and engineering involving heat and mass transfer under the influence of chemical
reaction is great importance to many areas of science and engineering. This frequently occurs in agriculture engineering
plasma studies and petroleum industries 15.
Flow through porous media plays an important role in countless practical applications such as ground water flows enhanced
oil recovery processes contamination of soils by hazardous wastes pollution movement 20. An understanding of the
dynamics of fluids in a porous medium is of principal interest because these flows are quite prevalent in nature. Such flows
have attracted the attention of number of scholars due to their applications in many branches of science and technology viz.
in the fields of agricultural engineering to study the underground water resources seepage of water in river beds in
petroleum technology to study the movement of natural gas oil and water through the oil reservoirs. In the widely used
continuum approach to transport processes in a porous media the differential equation governing the macroscopic fluid
motion is based on the experimentally established Darcy law 8 which accounts for the drag exerted on the fluid by porous
medium 29.
The objective of this work is to investigate the numerical solution by using Explicit Finite Difference method 6 for the
system of non-linear differential equations which arises from magneto-hydrodynamic flows of an Oldroyd 6-constant Nano-
fluid with a magnetic field through a non-Darcy porous medium with heat and mass transfer. We obtained the distributions of
the velocity the temperature the concentration and the Nanoparticles. Numerical results are found for different values of
various non-dimensional parameters in the case of pure Coutte flow
. The effects of modified pressure
gradients on those formulas were discussed in the case of pure Poiseuille flow and the generalized Couette flow. The results
are shown graphically and discussed in detail. Also the global error estimation for the error propagation is obtained by
Zadunaisky technique 19.
III. FLUID MODEL
For an Oldroyd 6-constant fluid the Cauchy stress tensor is given by 30
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1
. 2
Where
. 3
is the indeterminate part of the stress due to the constraint of incompressibility S
is the extra stress tensor
is the
first Rivlin-Ericksen tensor L is the velocity gradient is the velocity vector and
are the six material
constants.
The contra variant convected derivative
satisfying the principle of material frame in difference in terms of the material
is defined by the following equation
. 4
in which the superscript T is used for the transpose operation.
It should be noted that when
the model reduces to the classical linearly viscous model of Newtonian
fluid. For
a 3-constant model of an Oldroyd-B fluid is described while for
a
Maxwell model is formulated. For
the model describes a second-grade fluid.
IV. MATHEMATICAL FORMULATION
We consider magnetohydrodynamic flows of an Oldroyd 6-constant Nano-fluid through a non-Darcy porous medium
between two parallel plates see FIG. 1. We take in our consideration the presence of heat and mass transfer with chemical
reaction.
FIGURE 1: MAGNETOHYDRODYNAMIC FLOWS OF AN OLDROYD 6-CONSTANT NANO-FLUID THROUGH A NON-
DARCY POROUS MEDIUM BETWEEN TWO PARALLEL PLATES
V. BASIC EQUATIONS
The basic equations governing the flow of an incompressible fluid are the following equations 10 18
The continuity equation
5
The momentum equation
6
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The energy equation
7
The concentration equation
8
The nanoparticles concentration equation
. 9
Where
denotes the material time derivative
and .
For a simple one-dimensional shearing flow considered in this work the extra stress tensor and the velocity are of the form
. 10
Also the constant magnetic field the temperature the concentration and the nanoparticles are of the form
11
For steady motion it is clear that with the simplified dependence 10-11 the continuity equation 5 is satisfied identically
and equations 1-3 and 6 become
12
13
14
15
16
17
18
19
. 20
by addition of two equations 15 and 18 we have
. 21
from equations 17 and 19 we get
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. 22
Substituting 18 – 22 in 14 and 16 we have
23
. 24
where
25
. 26
Defining the modified pressure
27
and using 12 – 14 and 24 we arrive at
28
. 29
The dissipation function can be written as follows
. 30
By using the definitions 10 we have
. 31
Then equations 7 - 9 can be written as follows:
32
33
. 34
The boundary conditions of the problem are given by:
35
We shall now write the field equations 28 32 – 34 and the boundary conditions 35 in terms of a set of dimensionless
variables and for this purpose we choose H and U as the characteristic length and velocity and introduce the following
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dimensionless variables of both:
. 36
Thus the system of our non-linear differential equations 28 32 – 34 and the boundary conditions 35 can be rewritten in
a non-dimensional forms:
37
38
39
. 40
with the dimensionless boundary conditions:
41
Where the dimensionless parameters are defined by
. 42
For convenience we shall drop the bars that identify the dimensionless quantities. The system of non-linear ordinary
differential equations 37 – 40 with the boundary conditions 41 will be solved numerically by using the explicit finite-
difference method. And we computed the global error for the solutions of the problem.
VI. NUMERICAL SOLUTION
The equations 37 – 40 can be written after applied explicit finite difference schemes 6 as:
43
44
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45
. 46
Where the index refers to and the . According to the boundary conditions 41 we can solved equations
43 – 46 numerically then a Newtonian iteration method continues until either of goals specified by accuracy goal or
precision goal is achieved.
VII. ESTIMATION OF THE GLOBAL ERROR
We used Zadunaisky technique 19 for calculating the global error which can be explained in the following steps:
1. We fined the interpolating polynomial of
from the values of them which came from the
explicit finite-difference method. Then we named it
and we found the interpolating
functions of
and we named them as:
. 47
2. We calculate the defect functions
which can be written as follows:
48
3. We add the defect functions
to the original problem.
4. We solved the pseudo-problem new problem by the same method which is used for solving the main problem and we
will have the new solutions.
5. We calculate an estimation of the global error from the formulas
49
In this relation
is the approximate solutions of the new problem the pseudo-problem at the point
and
is the
exact solutions of pseudo-problem at
.
The values of global error for the solutions of the problem which solved by the explicit finite difference method are shown in
table 1. The error in table 1 based on using 26 points to find interpolating polynomials
of degree 25. In order to
achieve the above task we used the Mathematica package.
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TABLE 1
DIFFERENT VALUES OF THE DIMENSIONLESS PHYSICAL QUANTITIES
VIII. NUMERICAL RESULTS AND DISCUSSION
In this paper we generalized the problem of MHD flows to include the non-Newtonian Nano-fluid obeying Oldroyd 6-
constant model through a porous medium of non-Darcy type with heat and mass transfer. The system of non-linear ordinary
differential equations 37 – 40 with the boundary conditions 41 was solved numerically by using an Explicit finite
Difference method. The functions are obtained and illustrated graphically as shown in figures a-1 – a-5
b-1 – b-13 c-1 – c-13 and d-1 – d-13 for different values of the parameters of the problem in the case of pure Couette
flow
at the values of non-Newtonian parameters
. And figures e-1 – e-4 show the
effects of modified pressure gradients on those formula in the case of pure Poiseuille flow
at the values of non-
Newtonian parameters
. And figures f-1 – f-4 show the effects of modified pressure gradients on those
formula in the case of generalized Couette flow
at the values of non-Newtonian parameters
.
FIG. A-1: Profiles of the velocity uy for a pure Couette flow with various values of Br for a system have
the particulars M0.1 Da1 Fs0.5 Gr0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The dimensionless
velocity u
The dimensionless distance y
Br0.1
Br10
Br15
Br30
a18 a22
The global error by using finite difference technique
y u Y1 error e1n Y3 error e3n SY5 error e
5n
Y7 error e7n
0 0.00E+00 0.00E+00 1.00E+00 0.00E+00 1.00E+00 0.00E+00 1.00E+00 0.00E+00
0.08 0.084677 2.45938E-06 1.0654 1.06548E-05 0.877711 4.73414E-06 0.832759 9.45491E-06
0.16 0.167804 4.11468E-06 1.10363 1.91716E-05 0.765806 8.26993E-06 0.681824 1.69842E-05
0.24 0.249488 5.13257E-06 1.11623 2.58221E-05 0.662794 1.08326E-05 0.546264 2.28215E-05
0.32 0.329882 5.63641E-06 1.10429 3.07583E-05 0.567513 1.25695E-05 0.425427 2.71033E-05
0.4 0.40918 5.70977E-06 1.06848 3.40436E-05 0.479077 1.35718E-05 0.318913 2.98941E-05
0.48 0.487619 5.39798E-06 1.00906 3.56739E-05 0.39684 1.38901E-05 0.226562 3.12032E-05
0.56 0.565479 4.70768E-06 0.925913 3.55924E-05 0.320364 1.3545E-05 0.148452 3.09962E-05
0.64 0.64308 3.60483E-06 0.818507 3.37017E-05 0.249402 1.25357E-05 0.0848959 2.92048E-05
0.72 0.720789 2.019E-06 0.685922 2.98767E-05 0.183879 1.0847E-05 0.0364469 2.57363E-05
0.8 0.799017 9.48686E-08 0.526812 2.39868E-05 0.123881 8.45801E-06 0.00391268 2.04927E-05
0.88 0.878223 2.40736E-06 0.339383 1.59365E-05 0.0696485 5.37175E-06 -0.0116299 1.34196E-05
0.96 0.958919 2.76915E-06 0.121352 5.76248E-06 0.0215698 1.761E-06 -0.00881092 4.69398E-06
1 1.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
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FIG. A-2: Profiles of the velocity uy for a pure Couette flow with various values of Gr for a system have
the particulars M0.1 Da1 Fs0.5 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. A-3: Profiles of the velocity uy for a pure Couette flow with various values of Da for a system have
the particulars M0.1 Fs0.5 Nt0.3 Gr0.1 Br 0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1
Rc0.5 m2.
FIG. A-4: Profiles of the velocity uy for a pure Couette flow with various values of Fs for a system have
the particulars M0.1 Da1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5
m2.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The dimensionless
velocity u
The dimensionless distance y
Gr1
Gr3
Gr5
Gr10
a18 a22
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
The dimensionless
velocity u
The dimensionless distance y
Da0.01
Da0.04
Da0.1
Da0.4
a18 a22
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
The dimensionless
velocity u
The dimensionless distance y
Fs1
Fs20
Fs50
Fs100
a18 a22
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FIG. A-5: Profiles of the velocity uy for a pure Couette flow with various values of M for a system have
the particulars Da1 Fs0.5 Nt0.3 Gr0.1 Br 0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1
Rc0.5 m2.
8.1 Couette flow
Considering the Couette flow of an Oldroyd 6-constant fluid between two infinite parallel plates for which the lower plate is
fixed and the top plate moves with the velocity
it is assumed that the flow is driven only by the motion of the top
plate. The modified pressure gradient in the flow direction vanishes instant
. Figures a-1 – a-5 b-1 – b-13 c-1 – c-
13 and d-1 – d-13 will show the effect of the problem parameters on the problem solutions in this case.
Figures a-1 – a-5 show the distributions of the velocity profile u at different values of some parameters of the problem. It is
clear that the velocity increases by increasing each of local temperature Grashof number G
r
local nanoparticle Grashof B
r
and Darcy number D
a
. But the velocity decreases by increasing each of Forchheimer number F
s
magnetic field parameter M.
Figures b-1 – b-13 describe the distributions of the temperature profile θ at different values of some parameters of the
problem. It is noted that as the local nanoparticle Grashof B
r
local temperature Grashof number G
r
and Darcy number D
a
increase the temperature increases in the region and it returns decrease to . The temperature
increases by increasing each of Sort number L
d
Lewis number L
e
Brownian motion parameter N
b
Dufour number N
d
Thermophoresis parameter N
t
Prandtl number P
r
and Chemical reaction parameter R
c
. And as Forchheimer number F
s
increases the temperature decreases in the region and it returns increase to . Also as magnetic
field parameter M increases the temperature decreases in the region and it returns increase to .
Finally the temperature decreases by increasing Chemical reaction order m.
FIG. B-1: Profiles of the temperature θy for a pure Couette flow with various values of Br for a system
have the particulars M0.1 Da1 Fs0.5 Gr0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5
m2.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
The dimensionless
velocity u
The dimensionless distance y
M10
M30
M50
M100
a18 a22
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Br0.1
Br10
Br15
Br30
a18 a22
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FIG. B-2: Profiles of the temperature θy for a pure Couette flow with various values of Gr for a system
have the particulars M0.1 Da1 Fs0.5 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5
m2.
FIG. B-3: Profiles of the temperature θy for a pure Couette flow with various values Ld for a system
have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Rc0.5
m2.
FIG. B-4: Profiles of the temperature θy for a pure Couette flow with various values of Le for a system
have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Ld0.1 Rc0.5
m2.
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Gr1
Gr3
Gr5
Gr10
a18 a22
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Ld1
Ld2
Ld3
Ld6
a18 a22
-0.2
0.3
0.8
1.3
1.8
2.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Le0.01
Le1
Le3
Le5
a18 a22
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FIG. b-5: Profiles of the temperature θy for a pure Couette flow with various values of Nb for a system
have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Le2 Nt0.3 Ld0.1 Rc0.5
m2.
FIG. b-6: Profiles of the temperature θy for a pure Couette flow with various values of Nd for a system
have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nt0.3 Ld0.1 Rc0.5
m2.
FIG. b-7: Profiles of the temperature θy for a pure Couette flow with various values of Nt for a system
have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nd2 Ld0.1 Rc0.5
m2.
-0.2
0.3
0.8
1.3
1.8
2.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Nb0.1
Nb0.5
Nb2
Nb5
a18 a22
-0.2
0.3
0.8
1.3
1.8
2.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Nd0.1
Nd1
Nd3
Nd5
a18 a22
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Nt0.1
Nt0.5
Nt1
Nt1.5
a18 a22
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FIG. b-8: Profiles of the temperature θy for a pure Couette flow with various values of Da for a system
have the particulars M0.1 Fs0.51 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2
Ld0.1 Rc0.5 m2.
FIG. b-9: Profiles of the temperature θy for a pure Couette flow with various values of Fs for a system
have the particulars M0.1 Da1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1
Rc0.5 m2.
FIG. b-10: Profiles of the temperature θy for a pure Couette flow with various values of M for a system
have the particulars Da1 Fs0.51 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1
Rc0.5 m2.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Da0.01
Da0.04
Da0.1
Da0.4
a18 a22
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Fs1
Fs20
Fs50
Fs100
a18 a22
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
M10
M30
M50
M100
a18 a22
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FIG. b-11: Profiles of the temperature θy for a pure Couette flow with various values of Pr for a system
have the particulars Da1 Fs0.51 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1
Rc0.5 m2.
FIG. b-12: Profiles of the temperature θy for a pure Couette flow with various values of Rc for a system
have the particulars Pr0.7 Da1 Fs0.51 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2
Ld0.1 m2.
FIG. b-13: Profiles of the temperature θy for a pure Couette flow with various values of m for a system
have the particulars Pr0.7 Da1 Fs0.51 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2
Ld0.1 Rc0.5.
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Pr0.1
Pr0.5
Pr0.7
a18 a22
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
Rc0.5
Rc2
Rc10
Rc20
a18 a22
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
m1
m2
m3
m4
a18 a22
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Page | 68
Figures c-1 – c-13 illustrate the distributions of the concentration profile S at different values of some parameters of the
problem. It is seen that the concentration decreases by increasing each of local nanoparticle Grashof B
r
local temperature
Grashof number G
r
Sort number L
d
Lewis number L
e
Brownian motion parameter N
b
Dufour number N
d
Thermophoresis
parameter N
t
Prandtl number P
r
and Chemical reaction parameter R
c
. And as Darcy number D
a
increases the concentration
decreases in the region and it returns increase to . But as Forchheimer number F
s
increases the
concentration increases in the region and it returns decrease to . The effect of magnetic field
parameter M on the concentration disappears in the region the concentration returns decrease to at
large values of M. Finally the concentration increases by increasing Chemical reaction order m.
FIG. c-1: Profiles of the concentration Sy for a pure Couette flow with various values of Br for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. c-2: Profiles of the concentration Sy for a pure Couette flow with various values of Gr for a system
have the particulars M0.1 Da1 Fs0.5 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5
m2.
FIG. c-3: Profiles of the concentration Sy for a pure Couette flow with various values of Ld for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Rc0.5 m2.
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
0 0.2 0.4 0.6 0.8 1
The dimensionless
concentration S
The dimensionless distance y
Br0.1
Br10
Br15
Br30
a18 a22
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
0 0.2 0.4 0.6 0.8 1
The dimensionless
concentration S
The dimensionless distance y
Gr1
Gr3
Gr5
a18 a22
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
concentration S
The dimensionless distance y
Ld1
Ld2
Ld3
Ld6
a18 a22
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Page | 69
FIG. c-4: Profiles of the Concentration Sy for a pure Couette flow with various values of Le for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Ld0.1 Rc0.5 m2.
FIG. c-5: Profiles of the concentration Sy for a pure Couette flow with various values of Nb for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Le2 Nt0.3 Ld0.1 Rc0.5 m2.
FIG. c-6: Profiles of the Concentration Sy for a pure Couette flow with various values of Nd for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nt0.3 Ld0.1 Rc0.5 m2.
FIG. c-7: Profiles of the Concentration Sy for a pure Couette flow with various values of Nt for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nd2 Ld0.1 Rc0.5 m2.
-0.4
0.1
0.6
1.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The dimensionless
Concentration S
The dimensionless distance y
Le0.01
Le1
Le3
a18 a22
-0.2
0.3
0.8
1.3
1.8
2.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
concentration S
The dimensionless distance y
Nb0.1
Nb0.5
Nb2
a18 a22
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The dimensionless
Cocentration S
The dimensionless distance y
Nd0.1
Nd1
Nd3
a18 a22
0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The dimensionless
concentration S
The dimensionless distance y
Nt0.1
Nt0.5
Nt1
a18 a22
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FIG. c-8: Profiles of the concentration Sy for a pure Couette flow with various values of Da for a system have the
particulars M0.1 Fs1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. c-9: Profiles of the concentration Sy for a pure Couette flow with various values of Fs for a system have the
particulars M0.1 Da1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. c-10: Profiles of the concentration Sy for a pure Couette flow with various values of M for a system have the
particulars Da1 Fs1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
concentration S
The dimensionless distance y
Da0.01
Da0.04
Da0.1
Da0.4
a18 a22
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
concentration S
The dimensionless distance y
Fs1
Fs20
Fs50
Fs100
a18 a22
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
concentration S
The dimensionless distance y
M10
M30
M50
M100
a18 a22
slide 19: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
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FIG. c-11: Profiles of the concentration Sy for a pure Couette flow with various values of Pr for a system have the
particulars Da1 Fs1 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. c-12: Profiles of the concentration Sy for a pure Couette flow with various values of Rc for a system have the
particulars Pr0.7 Da1 Fs1 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 m2.
FIG. c-13: Profiles of the concentration Sy for a pure Couette flow with various values of m for a system have the
particulars Pr0.7 Da1 Fs1 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5.
Figures d-1 – d-13 display the distributions of the nanoparticle profile at different values of some parameters of the
problem. It is clear that the nanoparticle decreases by increasing each of local nanoparticle Grashof B
r
local temperature
Grashof number G
r
Sort number L
d
Lewis number L
e
Dufour number N
d
Thermophoresis parameter N
t
Prandtl number P
r
and Chemical reaction parameter R
c
. But the nanoparticle increases by increasing Brownian motion parameter N
b.
And as
Darcy number D
a
increases the nanoparticle decreases in the region and it returns increase to .
But as Forchheimer number F
s
increases the nanoparticle increases in the region and it returns decrease to
. The effect of magnetic field parameter M on the nanoparticle disappears in the region and the
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
concentration S
The dimensionless distance y
Pr0.1
Pr0.5
Pr0.7
Pr1
a18 a22
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
concentration S
The dimensionless distance y
Rc0.5
Rc2
Rc10
Rc20
a18 a22
0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The dimensionless
concentration S
The dimensionless distance y
m1
m2
m3
a18 a22
slide 20: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
Page | 72
nanoparticle returns decrease to at large values of M. Finally the nanoparticle increases by increasing Chemical
reaction order m.
FIG. d-1: Profiles of the nano-particle y for a pure Couette flow with various values of Br for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. d-2: Profiles of the nano-particle y for a pure Couette flow with various valuesof Gr for a system have the
particulars M0.1 Da1 Fs0.5 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. d-3: Profiles of the nano-particle y for a pure Couette flow with various values of Ld for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Rc0.5 m2.
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
Br0.1
Br10
Br15
Br30
a18 a22
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
Gr1
Gr3
Gr5
Gr10
a18 a22
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
Ld1
Ld2
Ld3
Ld6
a18 a22
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Page | 73
FIG. d-4: Profiles of the nano-particle y for a pure Couette flow with various values the Le for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Ld0.1 Rc0.5 m2.
FIG. d-5: Profiles of nano-particle y for a pure Couette flow with various values of Nb for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Le2 Nt0.3 Ld0.1 Rc0.5 m2.
FIG. d-6: Profiles of the nano-particle y for a pure Couette flow with various values of Nd for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nt0.3 Ld0.1 Rc0.5 m2.
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
Le0.01
Le1
Le3
Le5
a18 a22
-1.2
-0.7
-0.2
0.3
0.8
1.3
1.8
2.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
Nb0.1
Nb0.5
Nb2
Nb5
a18 a22
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
Nd0.1
Nd1
Nd3
Nd5
a18 a22
slide 22: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
Page | 74
FIG. d-7: Profiles of the nano-particle y for a pure Couette flow with various values of Nt for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nb0.5 Le2 Nd2 Ld0.1 Rc0.5 m2.
FIG. d-8: Profiles of the nano-particle y for a pure Couette flow with various values of Da for a system have the
particulars M0.1 Fs0.5 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. d-9: Profiles of the nano-particle y for a pure Couette flow with various values of Fs for a system have the
particulars M0.1 Da1 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
Nt0.1
Nt0.5
Nt1
Nt1.5
a18 a22
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
Da0.01
Da0.04
Da0.1
Da0.4
a18 a22
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
Fs1
Fs20
Fs50
Fs100
a18 a22
slide 23: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
Page | 75
FIG. d-10: Profiles of the nano-particle y for a pure Couette flow with various values of M for a system have the
particulars Da1 Fs0.5 Nt0.3 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. d-11: Profiles of the nano-particle y for a pure Couette flow with various values of Pr for a system have the
particulars Da1 Fs0.5 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. d-12: Profiles of the nano-particle y for a pure Couette flow with various values of Rc for a system have the
particulars Pr0.7 Da1 Fs0.5 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 m2.
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
M10
M30
M50
M100
a18 a22
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
Pr0.1
Pr0.5
Pr0.7
Pr1
a18 a22
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
Rc0.5
Rc2
Rc10
Rc20
a18 a22
slide 24: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
Page | 76
FIG. d-13: Profiles of the nano-particle y for a pure Couette flow with various values of m for a system have the
particulars Pr0.7 Da1 Fs0.5 Nt0.3 Gr0.1 Br0.1 M0.1 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5.
8.2 Poiseuille flow
Figures e-1 – e-4 describe the effect of pressure gradient
on the velocity the temperature the concentration and the
nanoparticle distributions. It is clear that the velocity the concentration and the nanoparticle decrease with increasing
while the temperature increases with increasing
.
FIG. e-1: Profiles of the velocity uy for a pure Poiseuille flow with various values of dP/dx for a system have the
particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. e-2: Profiles of the temperature θy for a pure Poiseuille flow with various values of dP/dx for a system have
the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
m1
m2
m3
m4
a18 a22
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
The dimensionless
velocity u
dP/dx2
dP/dx5
dP/dx10
dP/dx20
a18 a22
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
dP/dx2
dP/dx5
dP/dx10
dP/dx20
a18 a22
slide 25: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
Page | 77
FIG. e-3: Profiles of the concentration Sy for a pure Poiseuille flow with various values of dP/dx for a system have
the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. e-4: Profiles of the nano-particle y for a pure Poiseuille flow with various values of dP/dx for a system have
the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
8.3 Generalized Couette flow
If the flow of an Oldroyd 6-constant fluid between two parallel plates is driven by both the motion of the top plate and a
constant pressure gradient in the direction parallel to the plates a generalized couette flow is formed for
≠0 U₀1.
Numerical results for such a generalized couette flow are illustrated in figures f-1 – f-4.
FIG. f-1: Profiles of the velocity uy for a generalized Couette flow with various values of dP/dx for a system have
the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
0
0.5
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The dimensionless
Concentration S
The dimensionless distance y
dP/dx2
dP/dx5
dP/dx10
a18 a22
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
dP/dx2
dP/dx5
dP/dx10
dP/dx20
a18 a22
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The dimensionless
velocity u
The dimensionless distance y
dP/dx2
dP/dx5
dP/dx10
dP/dx20
a18 a22
slide 26: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
Page | 78
FIG. f-2: Profiles of the temperature θy for a generalized Couette flow with various values of dP/dx for a system
have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. f-3: Profiles of the concentration Sy for a generalized Couette flow with various values of dP/dx for a system
have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
FIG. f-4: Profiles of the nano-particle y for a generalized Couette flow with various values of dP/dx for a system
have the particulars M0.1 Da1 Fs0.5 Gr0.1 Br0.1 Pr0.7 Nd2 Nb0.5 Nt0.3 Le2 Ld0.1 Rc0.5 m2.
Figures f-1 – f-4 display the variation of the velocity the temperature the concentration and the nanoparticle distributions
for several values of the pressure gradient for
. It is observed that the velocity the concentration and the nanoparticle
decrease with increasing
while the temperature increases with increasing
.
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
temperature θ
The dimensionless distance y
dP/dx2
dP/dx5
dP/dx10
a18 a22
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
Concentration S
The dimensionless distance y
dP/dx2
dP/dx5
dP/dx10
dP/dx20
a18 a22
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
0 0.2 0.4 0.6 0.8 1
The dimensionless
nano-particle
The dimensionless distance y
dP/dx2
dP/dx5
dP/dx10
dP/dx20
a18 a22
slide 27: International Journal of Engineering Research Science IJOER ISSN: 2395-6992 Vol-3 Issue-7 July- 2017
Page | 79
IX. CONCLUSION
In this work we have studied magnetohydrodynamic flows of an Oldroyd 6-constant nano-fluid through a non-Darcy porous
medium with heat and mass transfer. The governing boundary value problem was solved numerically by an Explicit Finite-
Difference method. We concentrated our work on obtaining the velocity the temperature the concentration and the
nanoparticle distributions which are illustrated graphically at different values of the parameters of the problem in three cases
pure Couette flow pure Poiseuille flow and generalized Couette flow. Global error estimation is also obtained using
Zadunaisky technique. We used 26 points to find the interpolating polynomial of degree 25 in interval 01 and the results
are shown in table 1. We notice that the error in table 1 is good enough to justify the use of resulting numerical values.
The main findings from the current study can be summarized as follows:
1- By increasing B
r
G
r
and D
a
the velocity u increases whereas it decreases by increasing each of F
s
and M in the case
of Couette flow at
. And the other parameters doesn’t have any effect on the velocity.
2- By increasing B
r
G
r
and D
a
the temperature θ increases at the first part while it decreases at the
second part . And by increasing L
d
L
e
N
b
N
d
N
t
P
r
and R
c
the temperature increases while by
increasing m the temperature decreases. By increasing F
s
the temperature decreases at the first part
while it increases at the second part . Also By increasing M the temperature decreases at the first part
while it increases at the second part . All these results are given in the case of Couette
flow at
.
3- By increasing Br Gr Ld Le Nb Nd Nt Pr and Rc the concentration S decreases while by increasing m the
concentration increases. By increasing Da the concentration decreases at the first part while it
increases for . But by increasing Fs the concentration increases in the first part while it
decreases at the second part . And by increasing M the concentration doesn’t have any change at the
first part while it decreases at the second part at large values of M. All these results are
given in the case of Couette flow at
.
4- By increasing B
r
G
r
L
d
L
e
N
d
N
t
P
r
and R
c
the nanoparticle decreases while by increasing N
b
and m the
nanoparticle increases. By increasing Da the nanoparticle decreases at the first part while it increases
for . But by increasing Fs the nanoparticle increases in the first part while it decreases
at the second part . And by increasing M the nanoparticle doesn’t have any change at the first part
while it decreases at the second part at large values of M. All these results are given in
the case of Couette flow at
.
5- By increasing
the velocity the concentration and the nanoparticle decrease while the temperature increases in
the case of Poiseuille flow at
.
6- By increasing
the velocity the concentration and the nanoparticle decrease while the temperature increases in
the case of generalized Couette flow at
.
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