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1. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol 3, No 5, 2012
MHD Peristaltic Flow of a Couple Stress Fluids with Heat and Mass
Transfer through a Porous Medium
N.T. Eldabe,
Department of Mathematics, Faculty of Education, Ain Shams University,Cairo, Egypt
S.M. Elshaboury,
Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt
Alfaisal A. Hasan,
Department of Basic and Applied Sciences, Faculty of Engineering and Technology, Arab Academy for Science,
Technology and Maritime Transport, Cairo, Egypt
M.A. Elogail*,
Department of Basic and Applied Sciences, Faculty of Engineering and Technology, Arab Academy for Science,
Technology and Maritime Transport, Cairo, Egypt
*
E-mail of the corresponding author: mustafaelogail@gmail.com
Abstract
In the present article, we have studied the effects of heat and mass transfer on the MHD flow of an incompressible,
electrically conducting couple stress fluid through a porous medium in an asymmetric flexible channel over which a
traveling wave of contraction and expansion is produced, resulting in a peristaltic motion. The flow is examined in a
wave frame of reference moving with the velocity of the wave. Formulas of dimensionless velocity, temperature and
concentration are obtained analytically under assumptions of long wavelength and low Reynolds number. The effects
of various parameters of interest such as the couple stress fluid parameter, Darcy number, Hartmann number and
Schmidt number on these formulas were discussed and illustrated graphically through a set of figures.
Key words: peristalsis, Couple stress fluid, Porous medium, MHD flow, Heat transfer, Mass transfer.
1. Introduction
Peristalsis is a form of transporting fluids in which an induced wave causes the propagation of the
flexible walls of a channel/tube. This mechanism is seen in many biological systems such as the transportation of
urine from kidney to bladder, movement of chyme in the gastroin testinal tract, blood circulation in the small blood
vessels, and in the ducts efferentes of the male reproductive tract. Also in industry the phenomenon of peristaltic
pumping are used in many useful applications like transportation of sanitary fluids, blood pump in heart lung
51

2. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol 3, No 5, 2012
machines and also in transporting of corrosive and toxic liquids to avoid contamination with the outside
environment.
Recently, Elshehawy et al. [2] studied the peristaltic transport in an asymmetric channel through a
porous medium. Srinivas and Kothandapani [12] examined the heat transfer analysis for peristaltic flow in an
asymmetric channel. Mekheimer and Abdelmaboud [7] discussed the influence of heat transfer and magnetic field on
peristaltic transport of a Newtonian fluid in a vertical annulus.
Since most of physiological and industrial fluids are non-Newtonian fluids, Numerous attempts were
done by researchers to study various models of non-Newtonian fluids. We might mention some of the recent studies
that dealt with the peristaltic flow of non-Newtonian fluids. Hayat et al. [4] and they have analyzed the effect of an
induced magnetic field on the peristaltic transport of a Carreau fluid. Haroun [3] has studied the peristaltic transport
of fourth grade fluid in an inclined asymmetric channel. Wang et al. [15] have examined the magnetohydrodynamic
peristaltic motion of a Sisko fluid in symmetric or asymmetric channel. Nadeem and Akram [9] have studied the
peristaltic flow of Williamson fluid in an asymmetric channel. Eldabe et al. [1] have made an interesting analysis on
the mixed convective heat and mass transfer in a non-Newtonian fluid obeying the biviscosity model at a peristaltic
surface with temperature dependent viscosity. Abdelmaboud and Mekheimer [6] analyzed the transport of second
order fluid through a porous medium. Nadeem et al. [8] have discussed the influence of heat and mass transfer on
peristaltic flow of a third order fluid in a diverging tube.
The couple stress fluid is a special case of the non-Newtonian fluids where these fluids are consisting
of rigid randomly oriented particles suspended in a viscous medium and their sizes are taken into account. This
model can be used to describe human and animal blood, infected urine from a diseased kidney and liquid crystals.
There have only few attempts for studying the peristaltic flow of a couple stress fluids, first discussed by Stokes [13].
From the recent attempts dealing with the couple stress model, we refer to Mekheimer [5], as he has investigated the
problem of the peristaltic transport of a couple sterss fluid in a uniform and non-uniform channel. Also Nadeem and
Akram [10] have investigated the peristaltic flow of a couple stress fluid under the effect of induced magnetic field in
an asymmetric channel, and Sobh [11] has studied the effect of slip velocity on peristaltic flow of a couple stress
fluid in uniform and non uniform channels.
In dealing with heat and mass transfer problems, we observe a phenomenon called diffusion - thermo
effect (Duffor effect) in which an energy flux could be generated by the concentration gradients in addition to that
generated by the temperature gradients, also on the other hand mass fluxes could be created by heat gradients which
is known by themal- diffusion effect (Sort effect).
To the best of our knowledge no attempt has been reported yet to discuss the peristaltic transport of a
couple stress fluid in the presence of heat and mass transfer. The aim of the present study is to investigate the effect
52

3. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol 3, No 5, 2012
of the mixed convective heat and mass transfer on the peristaltic transport of a couple stress fluid in an asymmetric
channel through a porous medium in the presence of magnetic field while taking into consideration the viscous
dissipation effect during this work. So that this study is useful in filling that gap.
2. Mathematical Analysis
Consider the flow of an incompressible, electrically conducting, couple stress fluid through a porous
medium in an asymmetric two dimensional channel having width ݀ଵ ൅ ݀ଶ with a sinusoidal waves travelling
down its walls with a constant speed c. A rectangular coordinate system ሺܺ, ܻሻ is choosen such that ܺ-axis lies
along the direction of wave propagation and ܻ-axis transverse to it. The fluid is subjected to a constant transverse
ሬԦ
magnetic field ‫ ܤ‬ൌ ሺ0, ‫ܤ‬଴ , 0ሻ . The effects of induced magnetic field is neglected by considering small magnetic
Reynolds number, also the effect of Julian dissipation is neglected.
The geometry of the channel walls is given by
ܻ ൌ ‫ܪ‬ଵ ሺܺ, ‫ݐ‬ሻ ൌ ݀ଵ ൅ ܽଵ ܿ‫ ݏ݋‬ቂ ሺܺ െ ܿ‫ݐ‬ሻቃ
ଶగ
ఒ
upper wall (1)
ܻ ൌ ‫ܪ‬ଶ ሺܺ, ‫ݐ‬ሻ ൌ െ݀ଶ െ ܾଵ ܿ‫ ݏ݋‬ቂ ሺܺ െ ܿ‫ݐ‬ሻ ൅ ߶ቃ
ଶగ
ఒ
lower wall (2)
Where ܽଵ and ܾଵ are amplitudes of the waves . ߣ is the wavelength, ‫ݐ‬ is the time, ݀ଵ ൅ ݀ଶ is the
width of the channel and the phase difference ߶ varies in the range 0൑߶൑ߨ , where ܽଵ , ܾଵ , ݀ଵ , ݀ଶ
and ߶ satisfies ܽଵ ൅ ܾଵ ൅ 2ܽଵ ܾଵ ܿ‫ ߶ݏ݋‬൑ ሺ݀ଵ ൅ ݀ଶ ሻଶ
ଶ ଶ
in order for the channel walls not to collide with each
other. At the lower wall of the channel the temperature is ܶଵ and the concentration is ‫ܥ‬ଵ while at the upper wall the
temperature is ܶ଴ and the concentration is ‫ܥ‬଴ . See Fig. (20).
In Laboratory frame ሺܺ, ܻሻ, the Governing equations are given by:
൅ డ௒ ൌ 0
డ௎ డ௏
డ௑
(3)
డమ ௎ డమ ௎ డర ௎ డర ௎ డర ௎
ߩቀ ൅ܷ ൅ ܸ ቁൌെ ൅ߤቀ ൅ ቁ െ ߟ ቀ ൅ 2 ൅ ቁ െ ߪ ‫ܤ‬଴ ܷ െ ܷ
డ௎ డ௎ డ௎ డ௉ ଶ ఓ
డ௧ డ௑ డ௒ డ௑ డ௑ మ డ௒ మ డ௑ ర డ௑ మ డ௒ మ డ௒ ర ௞
(4)
డమ ௏ డమ ௏ డర ௏ డర ௏ డర ௏
ߩቀ ൅ ܷ ൅ ܸ ቁൌെ ൅ߤቀ ൅ ቁ െ ߟ ቀ ൅ 2 ൅ ቁ െ ܸ
డ௏ డ௏ డ௏ డ௉ ఓ
డ௧ డ௑ డ௒ డ௒ డ௑ మ డ௒ మ డ௑ ర డ௑ మ డ௒ మ డ௒ ర ௞
(5)
డమ ் డమ ் డ௎ ଶ డ௏ ଶ డ௏ ଶ
ߦቀ ൅ ܷ ൅ ܸ ቁ ൌ ቀ ൅ ቁ ൅ ߥ ൤2 ቀ ቁ ൅ 2 ቀ ቁ ൅ ቀ ൅ ቁ ൨൅
డ் డ் డ் ఑భ డ௎
డ௧ డ௑ డ௒ ఘ డ௑ మ డ௒ మ డ௑ డ௒ డ௒ డ௑
ଶ ଶ
డమ ௏ డమ ௏ డమ ௎ డమ ௎ డమ ஼ డమ ஼
ሾቀ మ ൅ మቁ ൅ ቀ మ ൅ మቁ ሿ ൅ ቀ ൅ ቁ
ఎ ஽೘ ௞೅
ఘ డ௑ డ௒ డ௑ డ௒ ௖ೞ డ௑ మ డ௒ మ
(6)
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4. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol 3, No 5, 2012
డమ ஼ డమ ஼ ஽೘ ௞೅ డమ ் డమ ்
ቀ డ௧ ൅ ܷ డ௑ ൅ ܸ డ௒ቁ ൌ ‫ܦ‬௠ ቀడ௑ మ ൅ డ௒ మ ቁ ൅ ቀడ௑ మ ൅ డ௒ మ ቁ
డ஼ డ஼ డ஼
ത
்
(7)
in which ܷ, ܸ are the velocity components in the laboratory frame ሺܺ, ܻሻ , ‫ ݐ‬is the time, P is the pressure, ߩ is
ത
the density, ܶ is the temperature of the fluid, ‫ ܥ‬is the concentration of the fluid, ܶ is the mean value of ܶ଴ and
ܶଵ , ‫ܦ‬௠ is the coefficient of mass diffusivity, ߤ is the coefficient of viscosity of the fluid, ߥ is the kinematic
viscosity of the fluid, ߦ is the specific heat at constant pressure, ݇ is the permeability parameter, ߪ is the electrical
conductivity of the fluid, ߢଵ is the thermal conductivity of the fluid, ܿ௦ is the concentration susceptibility and ߟ is a
constant associated with the couple stress.
Introducing a wave frame ሺ‫ݕ ,ݔ‬ሻ moving with the velocity ܿ away from the fixed frame ሺܺ, ܻሻ by the
transformation
‫ ݔ‬ൌ ܺ െ ܿ‫ ݕ , ݐ‬ൌ ܻ , ‫ ݑ‬ൌ ܷ െ ܿ , ‫ ݒ‬ൌ ܸ , ‫݌‬ሺ‫ݔ‬ሻ ൌ ܲሺܺ, ‫ݐ‬ሻ (8)
in which ሺ‫ݒ ,ݑ‬ሻ are the components of the velocity in the wave frame.
and defining the following non dimensional quantities
‫ ̅ݔ‬ൌ , ‫ ݕ‬ൌ
ത , ‫ ݑ‬ൌ , ‫ ̅ݒ‬ൌ , ‫ ̅ݐ‬ൌ ‫݄ ,ݐ‬ଵ ൌ భ , ݄ଶ ൌ మ , ߜ ൌ భ , ‫ ܯ‬ൌ ට ‫ܤ‬଴ ݀ଵ , ߮ ൌ
ത .
௫ ௬ ௨ ௩ ௖ ு ு ௗ ఙ ஼ି஼బ
ఒ ௗభ ௖ ௖ ఒ ௗభ ௗభ ఒ ఓ ஼భ ି஼బ
మ
݀ൌ , ܽ ൌ , ܾ ൌ , ‫ ̅݌‬ൌ , ܴ݁ ൌ , ߠ ൌ , ܵ௥ ൌ , ‫ܦ‬௔ ൌ , ߛ ଶ ൌ ݀ଵ
ௗమ ௔భ ௕భ ௗభ ௣ ఘ௖ௗభ ்ି்బ ఘ஽೘ ௞೅ ሺ்భ ି்బ ሻ ௞ ఓ ଶ
ௗభ ௗభ ௗభ ఓ௖ఒ ఓ ்భ ି்బ ഥ
் ఓሺ஼భ ି஼బ ሻ మ
ௗభ ఎ
௖మ
ܲ௥ ൌ , ܵ௖ ൌ , ‫ܦ‬௙ ൌ , ‫ܧ‬௖ ൌ
ఘ ఔ క ఓ ఘ஽೘ ௞೅ ሺ஼భ ି஼బ ሻ
఑భ ఘ ஽೘ ఓ క ௖ೞ ሺ்భ ି்బ ሻ క ሺ்భ ି்బ ሻ
(9)
where Re is the Reynolds number, ‫ ܯ‬is the Hartmann number, ‫ܦ‬௔ is the Darcy number, ܲ௥ is the Prandtl number,
ܵ௖ is the Schmidt number, ܵ௥ is the Soret number, ‫ܧ‬௖ is the Eckert number, ‫ܦ‬௙ is the Dufour number and ߛ is
the couple stress fluid parameter.
The non-dimensional time averaged flows ‫ ݍ‬and ܳ in the wave and in the laboratory frames respectively are related
by
ܳ = 1൅݀൅‫ݍ‬ (10.a)
‫ ݍ‬ൌ ‫ ׬‬మ ‫ݕ݀ ݑ‬
௛
௛ భ
in which (10.b)
3. Analytic Solutions
Using the above transformations (8) and the non-dimensional quantities (9) with the assumptions of long wave
length and low Reynolds number. Eqs. (4) - (7) can be written after dropping the bars in the following form:
െ డ௫ ൅ డ௬మ െ ఊమ డ௬ర െ ቀ‫ܯ‬ଶ ൅ ஽ ቁ ሺ‫ ݑ‬൅ 1ሻ ൌ 0
డ௣ డమ ௨ ଵ డర ௨ ଵ
(11)
ೌ
54

5. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol 3, No 5, 2012
ൌ0
డ௣
డ௬
(12)
ଵ డమ ఏ డమ ௨ ଶ డమ ఝ
൅ ‫ܧ‬௖ ሺ ሻଶ ൅ ሺ ሻ ൅ ‫ܦ‬௙ ൌ 0
డ௨ ா೎
௉ೝ డ௬ మ డ௬ ఊమ డ௬మ డ௬మ
(13)
ଵ డమ ఝ డమ ఏ
൅ ܵ௥ ൌ0
ௌ೎ డ௬ మ డ௬ మ
(14)
and assuming that the components of the couple stress tensor at the walls to be zero [14], the corresponding
boundary conditions in dimensionless form are given by
డమ ௨
‫ ݑ‬ൌ െ1, డ௬మ ൌ 0, ߠ ൌ 0 , ߮ ൌ 0 at ‫ ݕ‬ൌ ݄ଵ ൌ 1 ൅ ܽ cos 2 ߨ ‫ݔ‬ሻ
ሺ (15)
డమ ௨
‫ ݑ‬ൌ െ1 , ൌ 0 , ߠ ൌ 1 , ߮ ൌ 1 at ‫ ݕ‬ൌ ݄ଶ ൌ െ݀ െ ܾ cos 2 ߨ ‫ ݔ‬൅ ߶ሻ
ሺ
డ௬ మ
(16)
From Eq. (12) ‫݌ ് ݌‬ሺ‫ݕ‬ሻ and therefore ൌ
డ௣ ௗ௣
డ௫ ௗ௫
.
The closed form solution of Eq. (11) is given by
ಲ ೤ ಲఱ ೤ ಲ ೤ ಲల ೤ ೏೛
ି ఱ ି ల ଵାቀ ାெమ ቁ஽ೌ
‫ ݑ‬ൌ ‫ܣ‬ଵ ݁ √మ ൅ ‫ܣ‬ଶ ݁ √మ ൅ ‫ܣ‬ଷ ݁ √మ ൅ ‫ܣ‬ସ ݁ √మ െ ೏ೣ
ଵାெమ ஽ೌ
(17)
‫ܣ‬ହ ൌ ටߛ ଶ െ ‫ ଺ܣ‬ൌ ටߛ ଶ ൅ ‫ ଻ܣ‬ൌ ඥߛ ଶ ‫ܦ‬௔ ሺെ4 ൅ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ‫ܦ‬௔
஺ళ ஺ళ
஽ೌ ஽ೌ
where , and
(18)
and using appropriate boundary conditions (15) and (16) to evaluate ‫ܣ‬ଵ , ‫ܣ‬ଶ , ‫ܣ‬ଷ and ‫ܣ‬ସ , we get
ಲఱ ሺ೓భ శ೓మ ሻ
೏೛ ೏೛
௘ ஽ೌ ሺఊమ ஽ೌ ା ஺ళ ሻ ஽ ሺఊమ ஽ೌ ା ஺ళ ሻ
‫ܣ‬ଵ ൌ ‫ܣ‬ଶ ൌ
√మ
೏ೣ ೏ೣ ೌ
ಲఱ ೓భ ಲఱ ೓మ ಲఱ ೓భ ಲఱ ೓మ
,
ଶ቎௘ √మ ା ௘ √మ ቏ ሺଵାெమ ஽ೌ ሻ ஺ళ ଶ቎௘ √మ ା ௘ √మ ቏ ሺଵାெమ ஽ೌ ሻ ஺ళ
ಲల ሺ೓భ శ೓మ ሻ
೏೛ ೏೛
௘ ஽ೌ ሺି ఊమ ஽ೌ ା ஺ళ ሻ ஽ ሺି ఊమ ஽ೌ ା ஺ళ ሻ
‫ܣ‬ଷ ൌ ‫ܣ‬ସ ൌ
√మ
೏ೣ ೏ೣ ೌ
ಲల ೓భ ಲల ೓మ ಲల ೓భ ಲల ೓మ
,
ଶ቎௘ √మ ା ௘ √మ ቏ ሺଵାெమ ஽ೌ ሻ ஺ళ ଶ቎௘ √మ ା ௘ √మ ቏ ሺଵାெమ ஽ೌ ሻ ஺ళ
(19)
ௗ௣
ௗ௫
Using Eqs. (10.b) , (17) and (19) to get an explicit form for , we get
ሺ൫ଵାெమ ஽ೌ ൯ሺ௤ା௛భ ି௛మ ሻሻ
ൌെ
ௗ௣
ௗ௫ √మ ൫ം మ ವೌ శ ಲ ൯ ೅ೌ೙೓൤ ಲఱ ሺ೓ ష೓ ሻ൨ ಲ
√మ ൫ംమ ವೌ ష ಲళ ൯ ೅ೌ೙೓൤ ల ሺ೓భ ష೓మ ሻ൨
ళ మ√మ భ మ మ√మ
஽ೌ ሺ௛భ ି௛మ ሻቌଵ ି ା ቍ
ಲళ ಲఱ ሺ೓భష೓మ ሻ ಲళ ಲల ሺ೓భ ష೓మ ሻ
(20)
55

6. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol 3, No 5, 2012
Using Eqs. (14) and (17) into Eq. (13), the solution of Eq. (13) in terms of ߠ in closed form is given by
ா ௉ ௬ మ ஺భర ௘ √మ ೤ ಲఱ ா೎ ௉ೝ ஺మ ൫ ஺ళ ିଷ ఊమ ஽ೌ ൯ି ௘ √మ ೤ ಲల ா೎ ௉ೝ ஺మ ൫ ஺ళ ାଷ ఊమ ஽ೌ ൯
ߠ ൌ ‫ ଼ܣ‬൅ ‫ܣ‬ଽ ‫ ݕ‬െ ሺିଵା஺
೎ ೝ
െሺ మ ర
ሻ
భయ ሻ ஽ೌ ଼ ሺିଵା஺భయ ሻ ఊమ ஽ೌ
ா೎ ௉ೝ ሺቀି ௘ ష√మ ೤ ಲఱ ஺మ ା ௘ ష√మ ೤ ಲల ஺మ ቁ ஺ళ ା ଷ ఊ మ ቀ ௘ ష√మ ೤ ಲఱ ஺మ ା ௘ ష√మ ೤ ಲల ஺మ ቁ ஽ೌ
൅
భ య భ య
଼ ሺିଵା஺భయ ሻ ఊమ ஽ೌ
೤ ሺಲఱ షಲలሻ ೤ ሺషಲఱ శಲల ሻ ೤ ሺಲఱ శಲల ሻ ೤ ሺಲఱ శಲల ሻ
‫ۍ‬ଶ ቌ ௘ √మ ஺మ ஺య ା ௘ √మ ஺భ ஺ర ቍ ா೎ ௉ೝ ஺భబ ‫ۍ ې‬ଶ ቌ ௘ ష √మ ஺భ ஺య ା ௘ √మ ஺మ ஺ర ቍ ா೎ ௉ೝ ஺భభ ‫ې‬
‫ێ‬ ‫ێ ۑ‬ ‫ۑ‬
െ‫ێ‬ ሺିଵା஺భయ ሻ ሺ஺ఱ ି஺ల ሻమ ஽ೌ ‫ۑ‬൅‫ێ‬ ሺିଵା஺భయ ሻ ሺ஺ఱ ା஺ల ሻమ ஽ೌ
‫. ۑ‬
‫ێ‬ ‫ێ ۑ‬ ‫ۑ‬
‫ۏ‬ ‫ۏ ے‬ ‫ے‬
(21)
and upon substituting Eq. (21) into Eq. (14) and solving ,we obtain a solution for ߮ in closed form as
೤ ሺయ ಲఱశಲల ሻ
ష
஺భమ ஺భర ௬ మ ௘ √మ ೤ ಲఱ ஺భమ ஺మ ൫ ஺ళ ିଷ ఊమ ஽ೌ ൯
߮ ൌ ‫ܣ‬ଵହ ൅ ‫ܣ‬ଵ଺ ‫ ݕ‬൅ ሺିଵା஺ ൅ െ ሾെ4 ሺ 4 ݁ ଶ √ଶ ௬ ஺ఱ ‫ܣ‬ଶ ‫ܣ‬ଷ
మ ஺భమ ஽ೌ ௘ √మ
భయ ሻ ஽ೌ ଼ ሺିଵା஺భయ ሻ ఊమ ஽ೌ ଼ ሺିଵା஺భయ ሻ ஺య
ళ
ሺߛ ଶ െ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ ‫ ଻ܣ‬൅ 4 ݁ √ଶ ௬ ሺ஺ఱ ା஺లሻ ‫ܣ‬ଵ ‫ܣ‬ସ ሺߛ ଶ െ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ ‫ ଻ܣ‬൅ 4 ݁ √ଶ ௬ ஺ఱ ‫ܣ‬ଵ ‫ܣ‬ଷ ሺߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ ‫ ଻ܣ‬൅
௬ ሺଷ ஺ఱ ି஺ల ሻ ௬ ሺଷ ஺ఱ ା஺ల ሻ
4 ݁ √ଶ ௬ ሺଶ ஺ఱ ା஺ల ሻ ‫ܣ‬ଶ ‫ܣ‬ସ ሺߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ ‫ ଻ܣ‬൅ ݁ √ଶ ߛ ଶ ‫ܣ‬ଶ ሺെ4
ଷ ൅ 3 ‫ ଻ܣ‬ሻ ൅ ݁ √ଶ ߛ ଶ ‫ܣ‬ଶ ሺെ4
ସ ൅ 3 ‫ ଻ܣ‬ሻ ൅
೤ ሺ ಲఱ శಲల ሻ
݁ ߛ ଶ ‫ܣ‬ଵ ሺ4 ൅ 3 ‫ ଻ܣ‬ሻ ሻ ൅ ሺ െ8 ݁ ଶ√ଶ ௬ ஺ఱ ‫ܣ‬ଶ ‫ܣ‬ଷ ‫ܣ ଻ܣ‬ଵ଻ െ 8 ݁ √ଶ ௬ ሺ ஺ఱା஺ల ሻ ‫ܣ‬ଵ ‫ܣ‬ସ ‫ܣ ଻ܣ‬ଵ଻ െ
ଶ
√మ
௬ ሺଷ ஺ఱ ି஺ల ሻ
8 ݁ √ଶ ௬ ஺ఱ ‫ܣ‬ଵ ‫ܣ‬ଷ ‫ܣ ଻ܣ‬ଵଽ െ 8 ݁ √ଶ ௬ ሺ ଶ ஺ఱ ା஺ల ሻ ‫ܣ‬ଶ ‫ܣ‬ସ ‫ܣ ଻ܣ‬ଵଽ ൅ ݁ √ଶ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ‫ܣ‬ଶ ሺെ8 ൅ 3 ‫ ଻ܣ‬ሻ ൅
ଷ
య ೤ ሺ ಲఱ శಲల ሻ ೤ ሺ ಲఱ శಲలሻ
݁ √మ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ‫ܣ‬ଶ ሺെ8 ൅ 3 ‫ ଻ܣ‬ሻ ൅ ݁
ସ √మ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ‫ܣ‬ଵ ሺ8 ൅ 3 ‫ ଻ܣ‬ሻ ሻ ‫ܦ‬௔ ൅
ଶ
೤ ሺ ಲఱ శಲలሻ ೤ ሺయ ಲఱ షಲల ሻ య ೤ ሺ ಲఱ శಲల ሻ
ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻଶ ൬ െ ݁ √మ ‫ܣ‬ଵ ൅ ݁
ଶ √మ ‫ܣ‬ଶ ൅ ݁
ଷ
√మ ‫ܣ‬ଶ ൰ ‫ܦ‬௔ ሿ .
ସ
ଶ
(22)
where, ‫ܣ‬ଵ଴ ൌ ሺെ2 ൅ ሺെ2 ‫ܯ‬ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ‫ܦ‬௔ ሻ , ‫ܣ‬ଵଵ ൌ ሺ2 ൅ ሺ2 ‫ܯ‬ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ‫ܦ‬௔ ሻ , ‫ܣ‬ଵଶ ൌ
‫ܧ‬௖ ܲ௥ ܵ௖ ܵ௥
‫ܣ‬ଵଷ ൌ ‫ܦ‬௙ ܲ௥ ܵ௖ ܵ௥ , ‫ܣ‬ଵସ ൌ ሺ‫ܣ‬ଵ ‫ܣ‬ଶ ൅ ‫ܣ‬ଷ ‫ܣ‬ସ ሻ ሺ1 ൅ ‫ܯ‬ଶ ‫ܦ‬௔ ሻ , ‫ܣ‬ଵ଻ ൌ ሺߛ ଶ ‫ܣ‬ହ ‫ ଺ܣ‬െ 2‫ܯ‬ଶ ሺെߛ ଶ ൅
‫ܣ‬ହ ‫ ଺ܣ‬ሻሻ, ‫ܣ‬ଵ଼ ൌ ሺߛ ଶ ‫ܣ‬ହ ‫ ଺ܣ‬െ 2‫ܯ‬ଶ ሺߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻሻ , ‫ܣ‬ଵଽ ሺെߛ ଶ ‫ܣ‬ହ ‫ ଺ܣ‬൅ 2‫ܯ‬ଶ ሺߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻሻ .
(23)
Using Appropriate boundary conditions from (15) and (16) into (21) and (22) and solving, we get
ି ௘ √మ ಲఱ ೓భ ா೎ ௉ೝ ஺మ ൫஺ళ ି ଷ ఊమ ஽ೌ ൯ ା ௘ √మ ಲల ೓భ ா೎ ௉ೝ ஺మ ൫஺ళ ା ଷ ఊమ ஽ೌ ൯
‫ ଼ܣ‬ൌ െ ሾ ሾ െ
ଵ ିଵ మ ర
௛భ ି ௛మ ሺଶሺ ିଵା ஺భయ ሻ ஽ೌ ସ ఊమ
56

7. Innovative Systems Design and Engineering www.iiste.org
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ሺಲఱ షಲల ሻ ೓భ ሺషಲఱ శಲల ሻ ೓భ ሺಲఱ శಲల ሻ ೓భ ሺಲఱ శಲల ሻ ೓భ
ష
൅ െ
ሺସ ௘ √మ ஺మ ஺య ା ସ ௘ √మ ஺భ ஺ర ሻ ா೎ ௉ೝ ஺భబ ሺସ ௘ √మ ஺భ ஺య ା ସ ௘ √మ ஺మ ஺ర ሻ ா೎ ௉ೝ ஺భభ
ሺ஺ఱ ି஺ల ሻమ ሺ஺ఱ ା஺ల ሻమ
௘ ష √మ ሺಲఱ శಲల ሻ ா೎ ௉ೝ ሺ ௘ √మ ಲల ೓భ ஺మ ൫஺ళ ି ଷ ఊమ ஽ೌ ൯ ି ௘ √మ ಲఱ ೓భ ஺మ ൫஺ళ ା ଷ ఊమ ஽ೌ ൯ሻ
భ య
െ 2 ‫ܧ‬௖ ܲ௥ ‫ܣ‬ଵସ ݄ଵ ሿ ݄ଶ ൅
ଶ
ସ ఊ మ
௘ √మ ಲఱ ೓మ ா೎ ௉ೝ ஺మ ൫஺ళ ି ଷ ఊ మ ஽ೌ ൯ା ௘ √మ ಲల ೓మ ா೎ ௉ೝ ஺మ ൫஺ళ ା ଷ ఊమ ஽ೌ ൯ା ቀ ௘ ష √మ ಲఱ ೓మ ஺మ ା ௘ ష √మ ಲల ೓మ ஺మ ቁ൫ ଷ ఊమ ஽ೌ ି ா೎ ௉ೝ ஺ళ ൯
݄ଵ ሾെ1 െ െ
మ ర భ య
ሺ଼ ሺିଵା஺భయ ሻ ఊమ ஽ೌ ሻ
ሺಲఱ షಲల ሻ ೓మ ሺషಲఱ శಲల ሻ ೓మ ሺಲఱ శಲల ሻ ೓మ ሺಲ శಲలሻ ೓మ
ష ఱ
൅ െ
ሺଶ ௘ √మ ா೎ ௉ೝ ஺మ ஺య ஺భబ ା ଶ ௘ √మ ா೎ ௉ೝ ஺భ ஺ర ஺భబ ሻ ଶ ௘ √మ ா೎ ௉ೝ ஺భ ஺య ஺భభ ାଶ ௘ √మ ா೎ ௉ೝ ஺మ ஺ర ஺భభ
ሺିଵା஺భయ ሻ ሺ஺ఱ ି஺ల ሻమ ஽ೌ ሺିଵା஺భయ ሻ ሺ஺ఱ ା஺ల ሻమ ஽ೌ
మ
ா೎ ௉ೝ ஺భర ௛మ
ሺିଵା஺భయ ሻ ஽ೌ
]] .
(24)
௘ √మ ಲఱ ೓భ ா೎ ௉ೝ ஺మ ൫ଵା஺ళ ା൫ெమ ିఊమ ൯ ஽ೌ ൯ ௘ √మ ಲల ೓భ ா೎ ௉ೝ ஺మ ൫ିଵା஺ళ ା൫ିெమ ାఊమ ൯ ஽ೌ ൯ మ
‫ܣ‬ଽ ൌ ሾ ൅ െ ሺିଵା ஺ ൅
ିଵ మ ర ଶ ா ௉ ஺భర ௛భ
೎ ೝ
ଶ ௛భ ሺିଵା஺భయ ሻሺ஺ళ ି ఊమ ஽ೌ ሻ ሺିଵା஺భయ ሻሺ஺ళ ା ఊమ ஽ೌ ሻ భయ ሻ ஽ೌ
ሺಲఱ షಲల ሻ ೓భ ሺಲ షಲల ሻ ೓భ
ష ఱ
ሾா೎ ௉ೝ ஺ళ ሺቀି ௘ ష√మ ಲఱ ೓భ ஺మ ା௘ ష√మ ಲల ೓భ ஺మ ቁାଷఊమ ஽ೌ ቀ௘ ష√మ ಲఱ ೓భ ஺మ ା ௘ ష√మ ಲల ೓భ ஺మ ቁሻሿ
െ ൅
భ య భ య ሺ஺మ ஺య ௘ √మ ା ஺భ ஺ర ௘ √మ ሻସா೎ ௉ೝ ஺భబ
ସ ሺିଵା ஺భయ ሻ ఊమ ஽ೌ ሺିଵା஺భయ ሻ ሺ஺ఱ ି஺ల ሻమ ஽ೌ
ሺಲఱ శಲల ሻ ೓భ ሺಲఱ శಲలሻ ೓భ
ష
௘ √మ ಲఱ ೓భ ா೎ ௉ೝ ஺మ ൫ିଵି஺ళ ା൫ିெమ ାఊమ ൯ ஽ೌ ൯
െ ௛ ሾଶሺିଵା ஺ ሺ ൅
ሺ஺భ ஺య ௘ √మ ା ஺మ ஺ర ௘ √మ ሻସா೎ ௉ೝ ஺భభ ଶ ିଵ మ
ሺିଵା஺భయ ሻ ሺ஺ఱ ା஺ల ሻమ ஽ೌ భ ି ௛మ భయ ሻ ஽ೌ ି஺ళ ା ఊమ ஽ೌ
ሺಲఱ షಲల ሻ ೓భ ሺಲ ష ಲల ሻ ೓భ
ష ఱ
௘ √మ ಲల ೓భ ா೎ ௉ೝ ஺మ ൫ିଵା஺ళ ା൫ିெమ ାఊమ ൯ ஽ೌ ൯
െ െ 2 ‫ܧ‬௖ ܲ௥ ‫ܣ‬ଵସ ݄ଵ ൅
ర ሺ஺మ ஺య ௘ √మ ା ஺భ ஺ర ௘ √మ ሻସ ா೎ ௉ೝ ஺భబ ଶ
஺ళ ା ఊమ ஽ೌ ሺ஺ఱ ି஺ల ሻమ
ሺಲఱ శಲల ሻ ೓భ ሺಲఱ శಲలሻ ೓భ
ష
െ ݁ ି √ଶ ሺ஺ఱ ା஺ల ሻ ௛భ ‫ܧ‬௖ ܲ௥ ሺ ݁ √ଶ ஺ల ௛భ ‫ܣ‬ଵ ሺ‫ ଻ܣ‬െ 3 ߛ ଶ ‫ܦ‬௔ ሻ െ
ሺ஺భ ஺య ௘ √మ ା ஺మ ஺ర ௘ √మ ሻସா೎ ௉ೝ ஺భభ ଵ ଶ
ሺ஺ఱ ା஺ల ሻమ ସ ఊమ
௘ √మ ಲఱ ೓మ ா೎ ௉ೝ ஺మ ൫ଵା஺ళ ା൫ெమ ିఊమ ൯ ஽ೌ ൯ ௘ √మ ಲల ೓మ ா೎ ௉ೝ ஺మ ൫ିଵା஺ళ ା൫ିெమ ାఊమ ൯஽ೌ ൯
݁ √ଶ ஺ఱ ௛భ ‫ܣ‬ଶ ሺ‫ ଻ܣ‬൅ 3ߛ ଶ ‫ܦ‬௔ ሻሻ݄ଶ ൅ ݄ଵ ሺെ1 ൅
ଷ
మ
൅ ర
൅
ଶ ሺିଵା஺భయ ሻሺ஺ళ ି ఊమ ஽ೌ ሻ ଶ ሺିଵା஺భయ ሻሺ஺ళ ା ఊమ ஽ೌ ሻ
ா೎ ௉ೝ ሺሺି ௘ ష √మ ಲఱ ೓మ ஺మ ା ௘ ష √మ ಲల ೓మ ஺మ ሻ ஺ళ ା ଷ ఊమ ா೎ ௉ೝ ሺ ௘ ష √మ ಲఱ ೓మ ஺మ ା ௘ ష √మ ಲల ೓మ ஺మ ሻ ஽ೌ ሻ ௛మ
െ ሺିଵ ା ஺ భరሻ ஽ െ
భ య భ య ா ௉ ஺
೎ ೝ మ
଼ ሺିଵ ା஺భయ ሻ ఊమ ஽ೌ భయ ೌ
ሺಲఱ షಲల ሻ ೓మ ሺಲ షಲల ሻ ೓మ ሺಲఱ శಲల ሻ ೓మ ሺಲ శಲల ሻ ೓మ
ష ఱ ష ష ఱ
൅
ሺ஺మ ஺య ௘ √మ ା ஺భ ஺ర ௘ √మ ሻ ଶ ா೎ ௉ೝ ஺భబ ሺ஺భ ஺య ௘ √మ ା ஺మ ஺ర ௘ √మ ሻ ଶ ா೎ ௉ೝ ஺భభ
ሺିଵା஺భయ ሻ ሺ஺ఱ ି஺ల ሻమ ஽ೌ ሺିଵା஺భయ ሻ ሺ஺ఱ ା஺ల ሻమ ஽ೌ
)]] .
(25)
ሺయ ಲఱ శಲల ሻ ೓భ
ష
௘ √మ ಲఱ ೓భ ஺మ ஺భమ ൫ ஺ళ ିଷ ఊమ ஽ೌ ൯
‫ܣ‬ଵହ ൌ ሾሾെ ൅ ሺെ4 ሺ 4 ݁ ଶ √ଶ ஺ఱ ௛భ ‫ܣ‬ଶ ‫ܣ‬ଷ ሺߛ ଶ െ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ ‫ ଻ܣ‬൅
ଵ మ ஽ೌ ஺భమ ௘ √మ
௛భ ି ௛మ ଼ ሺିଵା஺భయ ሻ ఊమ ஽ೌ ଼ ሺିଵା஺భయ ሻ ஺య
ళ
4݁ √ଶ ሺ஺ఱା஺ల ሻ ௛భ ‫ܣ‬ଵ ‫ܣ‬ସ ሺߛ ଶ െ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ‫ ଻ܣ‬൅ 4 ݁ √ଶ ஺ఱ ௛భ ‫ܣ‬ଵ ‫ܣ‬ଷ ሺߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ‫ ଻ܣ‬൅ 4݁ √ଶሺଶ ஺ఱ ା஺ల ሻ௛భ ‫ܣ‬ଶ ‫ܣ‬ସ ሺߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ‫ ଻ܣ‬൅
ሺଷ ஺ఱ ି஺ల ሻ ௛భ ଷ ሺ ஺ఱ ା஺ల ሻ ௛భ ሺ ஺ఱ ା஺ల ሻ ௛భ
݁ √ଶ ߛ ଶ ‫ܣ‬ଶ ሺെ4
ଷ ൅ 3 ‫ ଻ܣ‬ሻ ൅ ݁ √ଶ ߛ ଶ ‫ܣ‬ଶ ሺെ4
ସ ൅ 3 ‫ ଻ܣ‬ሻ ൅ ݁ √ଶ ߛ ଶ ‫ܣ‬ଵ ሺ4
ଶ
൅ 3 ‫ ଻ܣ‬ሻሻ ൅
57

8. Innovative Systems Design and Engineering www.iiste.org
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ሺെ8 ݁ ଶ √ଶ ஺ఱ ௛భ ‫ܣ‬ଶ ‫ܣ‬ଷ ‫ܣ ଻ܣ‬ଵ଻ െ 8 ݁ √ଶ ሺ ஺ఱା஺ల ሻ ௛భ ‫ܣ‬ଵ ‫ܣ‬ସ ‫ܣ ଻ܣ‬ଵ଻ െ 8 ݁ √ଶ ሺ ଶ ஺ఱ ା஺లሻ ௛భ ‫ܣ‬ଶ ‫ܣ‬ସ ‫ܣ ଻ܣ‬ଵଽ െ
ሺଷ ஺ఱ ି஺ల ሻ ௛భ
8 ݁ √ଶ ஺ఱ ௛భ ‫ܣ‬ଵ ‫ܣ‬ଷ ‫ܣ ଻ܣ‬ଵଽ ൅ ݁ √ଶ ߛ ଶ ‫ܣ‬ଶ ሺെ8
ଷ ൅ 3 ‫ ଻ܣ‬ሻ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ൅
ଷሺ ஺ఱ ା஺ల ሻ ௛భ ሺ ஺ ା஺ ሻ ௛
ఱ ల భ ଶ ଶ
݁ √ଶ ߛ ଶ ‫ܣ‬ଶ ሺെ8
ସ ൅ 3 ‫ ଻ܣ‬ሻ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ൅ ݁ √ଶ ߛ ‫ܣ‬ଵ ሺ8 ൅ 3 ‫ ଻ܣ‬ሻ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻሻ ‫ܦ‬௔ ൅
ሺ ஺ఱ ା஺ల ሻ ௛భ ሺଷ ஺ఱ ି஺ల ሻ ௛భ ଷ ሺ ஺ఱ ା஺ల ሻ ௛భ
‫ܣ‬ଵସ ݄ଵ
ଶ
ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ቆ െ ݁ √ଶ ‫ܣ‬ଵ
ଶ
൅݁ √ଶ ‫ܣ‬ଶ ൅ ݁ √ଶ ‫ܣ‬ଶ ቇ ‫ܦ‬௔ ሻ
ଶ
െ ሿ ݄ ൅
ଷ ସ
ሺെ1 ൅ ‫ܣ‬ଵଷ ሻ ‫ܦ‬௔ ଶ
ሺଷ ஺ఱ ା஺ల ሻ ௛మ
ି
݁ √ଶ ஺ఱ ௛మ ‫ܣ‬ଶ ‫ܣ‬ଵଶ ሺ ‫ ଻ܣ‬െ 3 ߛ ଶ ‫ܦ‬௔ ሻ ‫ܦ‬௔ ‫ܣ‬ଵଶ ݁ √ଶ
݄ଵ ሺെ1 ൅ െ ሺെ4 ሺ 4 ݁ ଶ √ଶ ஺ఱ ௛మ ‫ܣ‬ଶ ‫ܣ‬ଷ ሺߛ ଶ െ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ ‫ ଻ܣ‬൅
ଶ
8 ሺെ1 ൅ ‫ܣ‬ଵଷ ሻ ߛ ଶ ‫ܦ‬௔ 8 ሺെ1 ൅ ‫ܣ‬ଵଷ ሻ ‫ܣ‬ଷ
଻
4݁ √ଶ ሺ஺ఱ ା஺లሻ ௛మ ‫ܣ‬ଵ ‫ܣ‬ସ ሺߛ ଶ െ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ‫ ଻ܣ‬൅ 4݁ √ଶ ஺ఱ ௛మ ‫ܣ‬ଵ ‫ܣ‬ଷ ሺߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ‫ ଻ܣ‬൅ 4݁ √ଶ ሺଶ஺ఱ ା஺ల ሻ ௛మ ‫ܣ‬ଶ ‫ܣ‬ସ ሺߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ‫ ଻ܣ‬൅
ሺଷ ஺ఱ ି஺ల ሻ ௛మ ଷ ሺ ஺ఱ ା஺ల ሻ ௛మ ሺ ஺ఱ ା஺ల ሻ ௛మ
݁ √ଶ ߛ ଶ ‫ܣ‬ଶ ሺെ4
ଷ ൅ 3 ‫ ଻ܣ‬ሻ ൅ ݁ √ଶ ߛ ଶ ‫ܣ‬ଶ ሺെ4
ସ ൅ 3 ‫ ଻ܣ‬ሻ ൅ ݁ √ଶ ߛ ଶ ‫ܣ‬ଵ ሺ4
ଶ
൅ 3 ‫ ଻ܣ‬ሻሻ ൅
ሺെ8 ݁ ଶ √ଶ ஺ఱ ௛మ ‫ܣ‬ଶ ‫ܣ‬ଷ ‫ܣ ଻ܣ‬ଵ଻ െ 8 ݁ √ଶ ሺ ஺ఱ ା஺ల ሻ ௛మ ‫ܣ‬ଵ ‫ܣ‬ସ ‫ܣ ଻ܣ‬ଵ଻ െ 8 ݁ √ଶ ሺ ଶ ஺ఱ ା஺ల ሻ ௛మ ‫ܣ‬ଶ ‫ܣ‬ସ ‫ܣ ଻ܣ‬ଵଽ െ
ሺଷ ஺ఱ ି஺ల ሻ ௛మ
8 ݁ √ଶ ஺ఱ ௛మ ‫ܣ‬ଵ ‫ܣ‬ଷ ‫ܣ ଻ܣ‬ଵଽ ൅ ݁ √ଶ ߛ ଶ ‫ܣ‬ଶ ሺെ8
ଷ ൅ 3 ‫ ଻ܣ‬ሻ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ൅
ଷሺ ஺ఱ ା஺ల ሻ ௛మ ሺ ஺ ା஺ ሻ ௛
ఱ ల మ ଶ ଶ
݁ √ଶ ߛ ଶ ‫ܣ‬ଶ ሺെ8
ସ ൅ 3 ‫ ଻ܣ‬ሻ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ൅ ݁ √ଶ ߛ ‫ܣ‬ଵ ሺ8 ൅ 3 ‫ ଻ܣ‬ሻ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻሻ ‫ܦ‬௔ ൅
ሺ ಲఱ శಲల ሻ ೓మ ሺయ ಲఱ షಲల ሻ ೓మ య ሺ ಲఱ శಲల ሻ ೓మ
ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻଶ ൬ െ ݁
మ
‫ܣ‬ଵ ൅ ݁ ‫ܣ‬ଶ ൅ ݁ ‫ܣ‬ଶ ൰ ‫ܦ‬௔ ሻ െ ሺିଵା஺ ሿ
ଶ ଶ భర ஺ ௛మ
√మ √మ √మ
ଷ ସ
భయ ሻ ஽ೌ
.
(26)
ሺయ ಲఱ శಲల ሻ ೓భ
ష ሺ ಲఱ శಲల ሻ ೓భ ሺ ఱ ಲఱ శಲల ሻ ೓భ ሺ య ಲఱ షಲల ሻ ೓భ య ሺ ಲఱ శಲల ሻ ೓భ
‫ܣ‬ଵ଺ ൌ ሺെ4 ൬െ݁ ‫ܣ‬ଵ െ ݁ ‫ܣ‬ଶ ൅ ݁ ‫ܣ‬ଶ ൅ ݁ ‫ܣ‬ଶ ൰ ‫ ଻ܣ‬൅
஺భమ ௘ √మ ଶ
√మ √మ √మ √మ
଼ ሺିଵା஺భయ ሻ ஺మ ሺ௛భ ି௛మ ሻ ଶ ଷ ସ
ళ
(16݁ ଶ √ଶ ஺ఱ ௛భ ‫ܣ‬ଶ ‫ܣ‬ଷ ሺെߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ+(16݁ √ଶ ሺ஺ఱ ା஺ల ሻ ௛భ ‫ܣ‬ଵ ‫ܣ‬ସ ሺെߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ െ ሺ16݁ √ଶሺ஺ఱ ା஺ల ሻ௛భ ‫ܣ‬ଵ ‫ܣ‬ସ ሺߛ ଶ ൅
‫ܣ‬ହ ‫ ଺ܣ‬ሻ െ
ሺయಲఱ షಲల ሻ೓భ యሺಲఱ శಲల ሻ೓భ ሺಲఱ శಲల ሻ೓భ
ሺ16 ݁ √ଶ ሺଶ ஺ఱା஺ల ሻ ௛భ ‫ܣ‬ଶ ‫ܣ‬ସ ሺ ߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ ൅ ݁ √మ ‫ܣ‬ଶ ‫ܣ‬ଵହ ൅ ݁
ଷ √మ ‫ܣ‬ଶ ‫ܣ‬ଵହ െ ݁
ସ √మ ‫ܣ‬ଵ ‫ܣ‬ଵ଺ െ
ଶ
ሺ ఱ ಲఱ శಲల ሻ ೓భ ሺ ಲఱ శಲల ሻ ೓భ ሺ ఱ ಲఱ శಲల ሻ ೓భ
݁ ‫ܣ‬ଶ ‫ܣ‬ଵ଺ ሻ ‫ܦ‬௔ ൅ ሺ3 ݁ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ‫ܣ‬ଵ ൅ 3 ݁ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ‫ܣ‬ଶ ൅
ଶ
√మ √మ √మ
ଶ ଶ
ሺ య ಲఱ షಲల ሻ ೓భ య ሺ ಲఱ శಲల ሻ ೓భ
3 ݁ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ‫ܣ‬ଶ ൅ 3 ݁ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶሻ ‫ܣ‬ଶ െ 8 ݁ ଶ √ଶ ஺ఱ ௛భ ‫ܣ‬ଶ ‫ܣ‬ଷ ‫ܣ‬ଵ଻ െ
√మ √మ
ଷ ସ
8 ݁ √ଶ ሺ஺ఱ ା஺ల ሻ ௛భ ‫ܣ‬ଵ ‫ܣ‬ସ ‫ܣ‬ଵ଻ ൅ 8 ݁ √ଶ ஺ఱ ௛భ ‫ܣ‬ଵ ‫ܣ‬ଷ ‫ܣ‬ଵ଼ ൅ 8 ݁ √ଶ ሺଶ ஺ఱ ା஺ల ሻ ௛భ ‫ܣ‬ଶ ‫ܣ‬ସ ‫ܣ‬ଵ଼ ሻ‫ܦ‬௔ ሻ ൅
ଶ
ሺ య ಲఱ శಲల ሻ ೓మ
మ ష ሺ ಲఱ శಲల ሻ ೓మ ሺ ఱ ಲఱ శಲల ሻ ೓మ ሺ య ಲఱ షಲల ሻ ೓మ
൅ ሺିଵା஺ ൅ ሺ4 ‫ܣ‬ଵଶ ሺሺ݁ ‫ܣ‬ଵ ൅ ݁ ‫ܣ‬ଶ െ ݁ ‫ܣ‬ଶ െ
஺భమ ஺భర ௛భ ௘ √మ ଶ
√మ √మ √మ
଼ ሺିଵା஺భయ ሻ ஺మ ሺି௛భ ା௛మ ଶ ଷ
భయ ሻ ஽ೌ ሺି௛భ ା௛మ ሻ ళ ሻ
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య ሺ ಲఱ శಲల ሻ ೓మ ሺ య ಲఱ శಲల ሻ ೓మ
݁ ‫ܣ‬ଶ ሻ ‫ ଻ܣ‬൅ 8݁ ߛ ଶ ሺ‫ܣ‬ଵ ‫ܣ‬ଶ ൅ ‫ܣ‬ଷ ‫ܣ‬ସ ሻ ݄ଶ ሻ െ ‫ܦ‬௔ ሺെ16݁ ଶ √ଶ ஺ఱ ௛భ ‫ܣ‬ଶ ‫ܣ‬ଷ ሺെߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻAଵଶ െ
ଶ
√మ √మ
ସ
16݁ √ଶ ሺ஺ఱ ା஺ల ሻ ௛మ ‫ܣ‬ଵ ‫ܣ‬ସ ሺെߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ Aଵଶ ൅ 16݁ √ଶሺ஺ఱ ା஺ల ሻ௛మ ‫ܣ‬ଵ ‫ܣ‬ଷ ሺߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ Aଵଶ ൅
ሺଷ ஺ఱ ା஺ల ሻ ௛మ ሺଷ ஺ఱ ି஺ల ሻ ௛మ
16 ݁ √ଶ ሺଶ ஺ఱ ା஺ల ሻ ௛మ ‫ܣ‬ଶ ‫ܣ‬ସ ሺ ߛ ଶ ൅ ‫ܣ‬ହ ‫ ଺ܣ‬ሻ Aଵଶ ൅ 32 ݁ √ଶ ߛ ଶ ሺെ1 ൅ ‫ܣ‬ଵଷ ሻ െ ݁ √ଶ ‫ܣ‬ଶ Aଵଶ Aଵହ
ଷ െ
యሺ ಲఱ శಲల ሻ ೓మ ሺ ಲఱ శಲల ሻ ೓మ ሺ ఱಲఱ శಲల ሻ ೓మ ሺ యಲఱ శಲల ሻ ೓మ
݁ ‫ܣ‬ଶ Aଵଶ Aଵହ ൅ ݁ ‫ܣ‬ଵ Aଵଶ Aଵ଺ ൅ ݁ ‫ܣ‬ଶ Aଵଶ Aଵ଺ ൅ 8 ݁ ߛ ଶሺെ8 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ
ଶ
√మ √మ √మ √మ
ସ ଶ
ሺ ಲఱ శಲల ሻ ೓మ ሺ ఱ ಲఱ శಲల ሻ ೓మ
ሺ‫ܣ‬ଵ ‫ܣ‬ଶ ൅ ‫ܣ‬ଷ ‫ܣ‬ସ ሻ ‫ܣ‬ଵଶ ݄ଶ ሻ െ ‫ܦ‬௔ ሺെ3 ݁ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶሻ ‫ܣ‬ଵ ‫ܣ‬ଵଶ െ 3 ݁ ߛ ଶ
ଶ ଶ ଶ
√మ √మ
ሺ ଷ ஺ఱ ି஺ల ሻ ௛మ ଷ ሺ ஺ఱ ା஺ల ሻ ௛మ
ሺെ4‫ܯ‬ଶ ൅ ߛ ଶ ሻ ‫ܣ‬ଶ ‫ܣ‬ଵଶ െ 3 ݁
ଶ
√ଶ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ‫ܣ‬ଶ ‫ܣ‬ଵଶ െ 3 ݁
ଷ
√ଶ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ ‫ܣ‬ଶ ‫ܣ‬ଵଶ െ
ସ
ሺ యಲఱ శಲల ሻ ೓మ
8 ݁ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻሺെ1 ൅ ‫ܣ‬ଵଷ ሻ ൅ 8 ݁ ଶ √ଶ ஺ఱ ௛మ ‫ܣ‬ଶ ‫ܣ‬ଷ ‫ܣ‬ଵ଻ ‫ܣ‬ଵଶ ൅ 8 ݁ √ଶ ሺ஺ఱା஺ల ሻ ௛మ ‫ܣ‬ଵ ‫ܣ‬ସ ‫ܣ‬ଵ଻ ‫ܣ‬ଵଶ െ
√మ
ሺ యಲఱ శಲల ሻ ೓మ
8 ݁ √ଶ ஺ఱ ௛మ ‫ܣ‬ଵ ‫ܣ‬ଷ ‫ܣ‬ଵଶ ‫ܣ‬ଵ଼ െ 8 ݁ √ଶ ሺଶ ஺ఱ ା஺ల ሻ ௛మ ‫ܣ‬ଶ ‫ܣ‬ସ ‫ܣ‬ଵଶ ‫ܣ‬ଵ଼ ൅ሻ‫ܦ‬௔ ሻ ൅ 8 ݁ ‫ܯ‬ଶ ߛ ଶ ሺെ4 ‫ܯ‬ଶ ൅ ߛ ଶ ሻ
ଶ
√మ
ሺ‫ܣ‬ଵ ‫ܣ‬ଶ ൅ ‫ܣ‬ଷ ‫ܣ‬ସ ሻ ‫ܣ‬ଵଶ ݄ଶ .
ଶ
(27)
4. Results and discussions
In this section, we shall discuss the influence of various physical parameters of interest on the pressure
gradient ݀‫ ,ݔ݀⁄݌‬the pressure rise ∆ܲ, the temperature profile ߠ and the concentration profile ߮. For this purpose
Figures (1) - (19) were prepared. In all these figures, as ߛ → ∞, this corresponds to the case of considering a
Newtonian fluid.
Figures (1) - (4) illustrate the variations of ݀‫ ݔ݀⁄݌‬for a given wavelength versus ‫ ,ݔ‬where ‫ ∈ ݔ‬ሾ0,1ሿ.
Figure (1) shows that by increasing ‫ ݔ݀⁄݌݀ ,ܯ‬increases in the narrow part of the channel ‫ ∈ ݔ‬ሾ0.27,0.64ሿ and
decreases in the wider part of the channel ‫ ∈ ݔ‬ሾ0,0.27ሿ ∪ ሾ0.64,1ሿ. Figure (2) indicates that the effect of ‫ܦ‬௔ on
݀‫ ݔ݀⁄݌‬is quite opposite to that of ‫ .ܯ‬From Figure (3) it can be seen that an increase in ߛ decreases ݀‫ ݔ݀⁄݌‬in the
narrow part of the channel ‫ ∈ ݔ‬ሾ0.27,0.64ሿ while in the wider part of the channel ‫ ∈ ݔ‬ሾ0,0.27ሿ ∪ ሾ0.64,1ሿ there is
no noticeable difference. Figure (4) indicates that as ߶ increases, a lesser amount of pressure gradient is required in
order to pass the flow in the narrow part of the channel.
Figures (5) - (7) present the variation of the pressure rise ∆ܲ per wavelength against the time
averaged flux ܳ.When pressure difference ∆ܲ ൌ 0 which is the case of free pumping, the corresponding time
averaged flux ܳ is denoted by ܳ ∗ ൐ 0. Here we subdivide the graph into four regions, (I) ∆ܲ ൌ 0 and
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ܳ ൌ ܳ ∗ ൐ 0 (free pumping region ), (II) ∆ܲ ൐ 0 and ܳ൏0 (backward pumping region), (III) ∆ܲ ൐ 0 and
ܳ ൐ ܳ ൐ 0 (peristaltic pumping region), (IV) ∆ܲ ൏ 0 and ܳ ൏ ܳ
∗ ∗
(co-pumping region). Figure (5) depicts that
with increasing ߛ, ∆ܲ decreases in the backward, peristaltic and free pumping regions till it reaches a critical value
ܳ ൌ 1.6 in the co-pumping region where ∆ܲ starts to increase by increasing ߛ. From Figure (6) it is noticed that
by increasing ‫ܦ‬௔ , ∆ܲ decreases in the backward pumping region till it reaches a critical value ܳ ൌ 0.62 in the
peristaltic pumping region after which ∆ܲ increases with decreasing ‫ܦ‬௔ . From Figure (7) we observe that the
effect of ‫ ܯ‬is quite opposite to that of ‫ܦ‬௔ in all pumping regions, however that critical value ܳ ൌ 0.62 remains
unchanged.
Figures (8) - (13) describe the variation of the temperature profile with y for several values of
‫ܦ‬௔ , ‫ܧ‬௖ , ܲ௥ , ߛ and ‫ .ܯ‬From Figures (8), (9), (12) and (13) it is clear that by increasing ‫ܧ‬௖ , ܲ௥ , ‫ܦ‬௔ and ܵ௖ the
temperature profile increases, while from Figures (10) and (11) we observe that the temperature profile decreases
with the increase in ߛ and ‫.ܯ‬
Figures (14) - (19) are plotted to study the effects of ܵ௥ , ‫ܦ‬௙ , ‫ܦ‬௔ , ߛ, ܵ௖ and ‫ ܯ‬on the concentration
profile. Here we have chosen the values of ܵ௥ and ‫ܦ‬௙ such that their product is a constant value, as we assume that
the mean temperature is kept constant. Figure (15) shows that by decreasing ‫ܦ‬௙ and increasing ܵ௥ the
concentration profile decreases, while in Figure (16) it is clear that by increasing ‫ܦ‬௙ and decreasing ܵ௥ the
concentration profile increases. Figures (17) and (19) show that concentration profile decreases with the increase in
‫ܦ‬௔ and ܵ௖ . Figures (14) and (18) illustrates that by increasing ‫ ܯ‬and ߛ the concentration profile increases.
5. Conclusion
In this article, we have presented a mathematical model that describes a MHD peristaltic flow of a
couple stress fluid through a porous medium in an asymmetric channel in presence heat and mass transfer. The
governing equations of the problem were solved analytically under assumptions of long wavelength and small
Reynolds number. A set of graphs were plotted in order to analyze the effects of various physical parameters on these
solutions. The main findings can be summarized as follows:
• The peristaltic pumping region increases as the couple stress parameter ߛ decreases.
• By decreasing the couple stress parameter ߛ, the longitudinal pressure gradient ݀‫ ݔ݀⁄݌‬increases in the
narrow part of the channel while in the wider part there is no appreciable difference.
• By increasing the couple stress parameter ߛ the temperature profile decreases and the concentration profile
increases.
• Increasing the value of ܵ௖ leads to an increase in the temperature profile whereas it causes a decrease in the
concentration profile.
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11. Innovative Systems Design and Engineering www.iiste.org
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• The concentration profile of the fluid decreases with decrease of ‫ܦ‬௙ (or increase in ܵ௥ ) and vise versa.
• By letting ߛ → ∞, ‫ܦ‬௔ → ∞, ‫ܦ‬௙ → 0, ܵ௥ → 0, ܵ௖ → 0, we can get the results obtained for the temperature
profiles by Srinivas and Kothandapani [12].
References
[1] Eldabe, N.T., El-Sayed, M., Ghaly, A., and Sayed, H. (2007). Mixed convective heat and mass transfer
in a non-Newtonian fluid at a peristaltic surface with temperature-dependent viscosity. Arch. Appl. Mech., vol.
78. pp. 599-624.
[2] Elshehawey, E. F., Eldabe, N. T., Elghazy E. M., and Ebaid, A. (2006). Peristaltic transport in an
asymmetric channel through a porous medium. Appl. Math. and Comput., vol. 182, pp. 140-150.
[3] Haroun, M. H. (2007). Non-linear peristaltic flow of a fourth grade fluid in an inclined asymmetric
channel. Comput. Mater. Sci., vol. 39, pp. 324-333.
[4] Hayat, T., Saleem, N., and Ali, N. (2010). Effect of induced magnetic field on peristaltic transport of a
Carreau fluid. Communications in non linear science and Numerical Simulations, vol. 15, pp.
2407-2423.
[5] Mekheimer, Kh. S. (2002). Peristaltic transport of a couple stress fluid in a uniform and non-uniform
channels. Biorehology, vol. 39 , pp. 755-765.
[6] Mekheimer, Kh. S., and Abd elmaboud, Y. (2011). Non-linear peristaltic transport of a second-order
fluid
through a porous medium. Applied Mathematical Modeling, vol. 35, pp. 2695-2710.
[7] Mekheimer, Kh. S., and Abd elmaboud, Y. (2008). The influence of heat transfer and magnetic field on
peristaltic transport of a Newtonian fluid in a vertical annulus:Application of an endoscope. Phys.
Letters
A, vol. 372, pp. 1657-1665.
[8] Nadeem, S., Noreen Sher Akbar, Naheeda Bibi and Sadaf Ashiq (2010) a. Influence of heat and mass
transfer on peristaltic flow of a third order fluid in a diverging tube. Communications in non linear
science and Numerical Simulations, vol. 15, pp. 2916-2931.
[9] Nadeem, S., and Akram, S. (2010) b. Influence of inclined magnetic field on peristaltic flow of a
Williamson fluid model in an inclined symmetric or asymmetric channel. Mathematical and
Computer
Modelling, vol. 52, pp. 107-119.
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Vol 3, No 5, 2012
[10] Nadeem, S., and Akram, S. (2011). Peristaltic flow of a couple stress fluid under the effect of induced
magnetic field in an asymmetric channel. Arch. Appl. Mech., vol. 81, pp. 97-109.
[11] Sobh, A.M. (2008). Interaction of couple stresses and slip flow on peristaltic transport in uniform and
non-uniform channels. Turkish J. Eng. Env. Sci., vol. 32, pp.117-123.
[12] Srinivas, S., and Kothandapani, M. (2008). Peristaltic transport in an asymmetric channel with heat
transfer - A note. International Communication in Heat and Mass Transfer, vol. 35, pp. 514-522.
[13] Stokes, V. K. (1966). Couple stresses in fluids. Phys. Fluids, vol. 9, pp. 1709-1715.
[14] Valanis, K. S., and Sun, C. T. (1969). Poiseuille flow of a fluid with couple with applications to blood
flow. Biorheology vol. 6, pp. 85-97.
[15] Wang, Y., Hayat T., Ali, N., and Oberlack, M. (2008). Magnetohydrodynamic peristaltic motion of a
Sisko fluid in a symmetric or asymmetric channel. Physica A, vol. 387, pp. 347–362.
Figure (1). Pressure gradient versus ‫ ݔ‬for ܽ ൌ 0.6, Figure (2). Pressure gradient versus
‫ ݔ‬for ܽ ൌ 0.6,
ܾ ൌ 0.7, ݀ ൌ 1.5, ߶ ൌ , ‫ ݍ‬ൌ െ2, ߛ ൌ 4, ‫ܦ‬௔ ൌ 0.5. ܾ ൌ 0.7, ݀ ൌ 1.5, ߶ ൌ , ‫ ݍ‬ൌ
గ గ
଺ ଺
െ2, ߛ ൌ 4, ‫ ܯ‬ൌ 1.
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Figure (3). Pressure gradient versus ‫ ݔ‬for ܽ ൌ 0.6, Figure (4). Pressure gradient versus ‫ݔ‬
for ܽ ൌ 0.6,
ܾ ൌ 0.7, ݀ ൌ 1.5, ߶ ൌ ଺ , ‫ ݍ‬ൌ െ2, ‫ ܯ‬ൌ 1, ‫ܦ‬௔ ൌ 1. ܾ ൌ 0.7, ݀ ൌ 1.5, ߛ ൌ 2, ‫ ݍ‬ൌ െ2, ‫ ܯ‬ൌ
గ
1, ‫ܦ‬௔ ൌ 1.
Figure (5). Pressure rise versus ܳ for ܽ ൌ 0.7, Figure (6). Pressure rise versus ܳ
for ܽ ൌ 0.7,
ܾ ൌ 1.2, ݀ ൌ 2, ‫ ܯ‬ൌ 0.5, ߶ ൌ , ‫ܦ‬௔ ൌ 2. ܾ ൌ 1.2 , ݀ ൌ 2, ‫ ܯ‬ൌ
గ
ସ
0.5, ߶ ൌ , ߛ ൌ 5.
గ
ସ
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Figure (7). Pressure rise versus ܳ for ܽ ൌ 0. Figure (8). Temperature profile for
ܽ ൌ 0.7, ܾ ൌ 0.8,
ܾ ൌ 1.2, ݀ ൌ 2, ߛ ൌ 2, ߶ ൌ , ‫ܦ‬௔ ൌ 1. ݀ ൌ 1.5, ‫ ݍ‬ൌ െ1.5, ߶ ൌ
గ
ସ
, ‫ ܯ‬ൌ 1, ‫ܦ‬௔ ൌ 2 , ߛ ൌ 4,
గ
ସ
‫ܧ‬௖ ൌ 0.5, ܵ௥ ൌ 0.6, ‫ܦ‬௙ ൌ 0.1, ܵ௖ ൌ 0.5, ‫ ݔ‬ൌ 0.
Figure (9). Temperature profile for ܽ ൌ 0.7, ܾ ൌ 0.8, Figure (10). Temperature profile for ܽ ൌ
0.7, ܾ ൌ 0.8,
݀ ൌ 1.5, ‫ ݍ‬ൌ െ1.5, ߶ ൌ , ‫ ܯ‬ൌ 1, ߛ ൌ 4, ‫ܦ‬௔ ൌ 1 ݀ ൌ 1.5, ‫ ݍ‬ൌ െ1, ߶ ൌ , ‫ ܯ‬ൌ 1, ‫ܦ‬௔ ൌ
గ గ
ସ ସ
1 , ‫ܧ‬௖ ൌ 1,
ܲ௥ ൌ 2, ܵ௥ ൌ 0.6, ‫ܦ‬௙ ൌ 0.1, ܵ௖ ൌ 0.5, ‫ ݔ‬ൌ 0. ܲ௥ ൌ 2, ܵ௥ ൌ 0.6, ‫ܦ‬௙ ൌ
0.1, ܵ௖ ൌ 0.5, ‫ ݔ‬ൌ 0.
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ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol 3, No 5, 2012
Figure (11). Temperature profile for ܽ ൌ 0.7, ܾ ൌ 0.8, Figure (12). Temperature profile for ܽ ൌ
0.7, ܾ ൌ 0.8,
݀ ൌ 1.5, ‫ ݍ‬ൌ െ1, ߶ ൌ , ‫ܦ‬௔ ൌ 2, ߛ ൌ 4, ‫ܧ‬௖ ൌ 1, ݀ ൌ 1.5, ‫ ݍ‬ൌ െ1, ߶ ൌ , ߛ ൌ 4, ‫ ܯ‬ൌ
గ గ
ସ ସ
1, ܲ௥ ൌ 2,
ܲ௥ ൌ 2, ܵ௥ ൌ 0.6, ‫ܦ‬௙ ൌ 0.1, ܵ௖ ൌ 0.5, ‫ ݔ‬ൌ 0. ‫ܧ‬௖ ൌ 1, ܵ௥ ൌ 0.6, ‫ܦ‬௙ ൌ 0.1,
ܵ௖ ൌ 0.5, ‫ ݔ‬ൌ 0.
Figure (13). Temperature profile for ܽ ൌ 0.7, ܾ ൌ 0.8, Figure (14). concentration profile for ܽ ൌ
0.7, ܾ ൌ 1.2,
݀ ൌ 1.5, ‫ ݍ‬ൌ െ1.5, ߶ ൌ , ‫ܦ‬௔ ൌ 1, ߛ ൌ 4, ‫ ܯ‬ൌ 1, ݀ ൌ 1.5, ‫ ݍ‬ൌ െ1.5, ߶ ൌ , ‫ܦ‬௔ ൌ 1, ߛ ൌ
గ గ
ସ ଶ
4, ܲ௥ ൌ 4,
‫ܧ‬௖ ൌ 0.5, ܲ௥ ൌ 4, ܵ௥ ൌ 0.6, ‫ܦ‬௙ ൌ 0.1, ‫ ݔ‬ൌ 0. ‫ܧ‬௖ ൌ 0.8, ܵ௥ ൌ 1, ‫ܦ‬௙ ൌ 0.06,
ܵ௖ ൌ 1, ‫ ݔ‬ൌ 0.
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16. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol 3, No 5, 2012
Figure (15). Concentration profile for ܽ ൌ 0.7, ܾ ൌ 1.2,
݀ ൌ 1.5, ‫ ݍ‬ൌ െ1.5, ߶ ൌ ଶ , ‫ܦ‬௔ ൌ 1, ߛ ൌ 4, ‫ܧ‬௖ ൌ 0.8
గ
ܲ௥ ൌ 4, ‫ ܯ‬ൌ 1, ܵ௖ ൌ 1, ‫ ݔ‬ൌ 0.
Figure (16). concentration profile for ܽ ൌ 0.7, ܾ ൌ 1.2, Figure (17). Concentration profile for ܽ ൌ
0.7, ܾ ൌ 1.2,
݀ ൌ 1.5, ‫ ݍ‬ൌ െ1.5, ߶ ൌ , ‫ܦ‬௔ ൌ 1, ߛ ൌ 4, ‫ܧ‬௖ ൌ 0.8, ݀ ൌ 2, ‫ ݍ‬ൌ െ1.5, ߶ ൌ , ߛ ൌ 4, ‫ܧ‬௖ ൌ
గ గ
ଶ ଶ
1, ‫ ܯ‬ൌ 1
ܲ௥ ൌ 4, ‫ ܯ‬ൌ 1, ܵ௖ ൌ 1, ‫ ݔ‬ൌ 0. ܲ௥ ൌ
4, ܵ௥ ൌ 0.6, ‫ܦ‬௙ ൌ 0.1, ܵ௖ ൌ 1, ‫ ݔ‬ൌ 0.
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17. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol 3, No 5, 2012
Figure (18). concentration profile for ܽ ൌ 0.7, ܾ ൌ 1.2, Figure (19). Concentration profile for ܽ ൌ 0.7, ܾ ൌ
1.2,
݀ ൌ 2, ‫ ݍ‬ൌ െ1.5, ߶ ൌ , ‫ ܯ‬ൌ 1, ‫ܦ‬௔ ൌ 1, ܲ௥ ൌ 4, ݀ ൌ 2, ‫ ݍ‬ൌ െ1.5, ߶ ൌ , ߛ ൌ 4, ‫ܧ‬௖ ൌ
గ గ
ଶ ଶ
1, ‫ ܯ‬ൌ 1,
‫ܧ‬௖ ൌ 1, ܵ௥ ൌ 0.6, ‫ܦ‬௙ ൌ 0.1, ܵ௖ ൌ 1, ‫ ݔ‬ൌ 0 . ܲ௥ ൌ 4, ‫ܦ‬௔ ൌ 1, ܵ௥ ൌ
0.6, ‫ܦ‬௙ ൌ 0.1, ‫ ݔ‬ൌ 0.
Figure (20).
Geometry of the problem
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