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International Journal of Innovative Technology & Creative Engineering
Vol.4 No.2
February 2014
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From Editor's Desk
Dear Researcher, Greetings! Research article in this issue discusses about Peristaltic MHD Flow of a Jeffrey. Let us review research around the world this month; The virtual currency is about more than money – the real innovation is what people are doing with the technology it is based on. BITCOIN has been called many things, from the future of money to a drug dealer's dream and everything else in between. But beyond creating the web's first native currency, the true innovation of Bitcoin's mysterious designer, Satoshi Nakamoto, is its underlying technology, the "block chain". That fundamental concept is being used to transform Bitcoin – and could even replace it altogether. So what is the block chain? It is a ledger of transactions that keeps Bitcoin secure and allows all users to agree on exactly who owns how many bitcoins. Each new block requires a record of recent transactions along with a string of letters and numbers, known as a hash, which is based on the previous block and produced using a cryptographic algorithm. The floating school built for Nigeria's water world.In Makoko, Nigeria, a water world of homes above the sea is no futuristic fantasy. For generations, people have lived in this sprawling slum on stilts above the lagoon that surrounds the country's largest city, Lagos. This floating school aims to bring much-needed education to the children of Makoko. At the same time, its architects say it is an exploration of how African cities can cope with the breakneck pace of urbanisation, and the flooding and sea level rise brought on by climate change. The school was built by local builders out of reclaimed materials from the area: it uses the empty plastic barrels abundant in Lagos as floats, for example. Its three levels include an open play space on the bottom, an enclosed classroom area in the middle and workshop space on the top. The school opened in spring 2013 and can accommodate up to 100 students. CLEAN energy inspired by the stars is the dream of scientists pursuing nuclear fusion, in which atomic nuclei fuse together and release energy. In a first for laser-driven fusion, scientists at a US lab say they have reached a key milestone called fuel gain: they are producing more energy than the fuel absorbed to start the reaction. But the National Ignition Facility (NIF) in Livermore, California, is still a long way from sparking a self-sustaining fusion reaction with an overall gain in energy – a process called ignition. Currently, the reactor as a whole needs more energy to operate than the amount that is produced. India unveils its first home-grown astronaut capsule. INDIA is about to take one small step towards human space flight. Last week the country's space agency unveiled a prototype of its first crew capsule, a 4-metre-high module designed to carry two people into low Earth orbit.Built by Hindustan Aeronautics in Bangalore, the prototype capsule cannot be hermetically sealed and so cannot take people into space. But if the rocket launch is a success, ISRO should be able to remotely test some in-flight controls and see how the module survives the stresses of re-entry and landing at sea. It has been an absolute pleasure to present you articles that you wish to read. We look forward to many more new technologies related research articles from you and your friends. We are anxiously awaiting the rich and thorough research papers that have been prepared by our authors for the next issue. Thanks, Editorial Team IJITCE
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Contents
Effect of Heat Transfer on the Peristaltic MHD Flow of a Jeffrey Fluid in An Inclined Channel by V.P. Rathod and Mahadev. M................................................................................................................................................................[180]
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Effect of Heat Transfer on the Peristaltic MHD Flow of a Jeffrey Fluid in
An Inclined Channel V.P. Rathod* and Mahadev. M.
Department of studies and Research in Mathematics, Gulbarga University, Gulbarga-585106, Karanataka, India
Corresponding Author E-mail address: [email protected]
Abstract— Peristaltic transport of a conducting Jeffrey fluid in an inclined channel with heat transfer is studied under long wave length and low Reynolds number approximations. An exact solution is presented for the temperature field. The expressions for axial velocity, axial pressure gradient, Stream line and coefficient of heat transfer have been obtained analytically. The effect of various parameters on the flow characteristics are discussed with the help of graph.
Key words — Jeffrey fluid, MHD flow, heat transfer, inclined channel.
I. INTRODUCTION
Peristaltic pumping is a form of fluid transport which occurs in a biological system. The study of peristaltic motion has gained considerable interest because of its extensive role in transporting many physiological fluid in the body in various situations such as urine transport from the kidney to bladder, transport of the spermatozoa in the ducts afferents of male reproductive tract, the moment of chyme in the gastrointestinal tract, swallowing of food through esophagus and vasomotion of small blood vessels. Many mechanical devices have been designed on the peristaltic pumping to transport fluids without internal moving parts, for example, the transport of slurries, sensitive or corrosive fluids, sanitary fluid and noxious fluids in the nuclear industries. The literature on this topic is quite extensive. Mention may be made to some recent theoretical and experimental contributions Shapiro and Jaffrin [1], Hayat and Sajid [2], El Misery et al.[3],Yin and Fung[4,5], Pzrikidis [6], Shukla and gupta[7], Vjravelu et al.[8,9], Raju and Devanathan[10] in the field for Newtonian fluids. Such approximation is true in the ureter but it fails to give an adequate understanding of food mixing and chyme moment in the intestine, fuel slurries, flow of plasma, flow of mercury amalgams, and lubrication with heavy oils and greases etc. Also, the assumptions that most of the physiological fluid behave like Newtonian fluid is not true in reality. With all these facts in mind, it is clear that non Newtonian fluid plays an indispensable role in peristaltic flow problem.
A number of researchers have been discussed the effect of magnetic field on peristaltic flow Mekheimer [11], Harikrishana and Subbareddyy [12], Suryanarayana Reddy et al. [13], Subba Reddy and Gangadhar [14], Elshahed and Haroun [15], Kothadapani [16, 17], Affi and Gad[118], Krishana kumari and Raman Murthy [19], Nadeem and Akbar [21], due to its applications in bio engineering and medical devices. Specially, magnetic wound or cancer treatment causes hyperthermia, bleeding reduction during surgeries, and targeted transport of drugs using magnetic particles as drug carries are few examples.
The interaction of peristalsis in connection with heat transfer has also received some attention Nadeem and Akbar [21], Srinivas and Gayatri [22], Hayat and Heena [23], Hayat and Ali [24], Hayat et al.[25] as it might be relevant in processes like hemodialysis and oxygenation. Rathod and Mahadev [26] have studied the effect of thickness of porous material on the peristaltic pumping of Jeffrey fluid with non-erodible porous ling wall. Effects of magnetic field and an endoscope on peristaltic motion is studied by Rathod and Asha [27]. Rathod and Pallavi [28] have discussed the influence of wall properties on MHD peristaltic transport of dusty fluid. The present article studies the effect of heat
transfer on peristaltic MHD flow of Jeffrey fluid in an
inclined channel. The governing equations of Jeffrey
fluid in Cartesian co-ordinate have been modeled. The
equations are simplified using long wave length and low
Reynolds number approximations. The closed form of
solution for velocity field and temperature are obtained.
The influence of various parameters on the flow
characteristics, the temperature and heat transfer
coefficient are discussed through graph.
2. FORMULATION OF THE PROBLEM Consider the peristaltic pumping of a Jeffrey
fluid with heat transfer in an inclined channel of half- width ‘a’. A longitudinal train of progressive sinusoidal wave takes place on the upper and lower wall of the channel. We further assume that the fluid is electrically
conducting. A uniform magnetic field 0B is applied in
the transverse direction to the flow. The Reynolds number is taken small so that the induced magnetic field is neglected. For simplicity, we restrict our discussion to the half width of the channel as shown in figure .(1)
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Figure.1. Physical model
The wall deformation is given by
2
( , ) ( )H x t a bCos X ct
(1)
where b is amplitude of the waves and λ is the wave length. The constitutive equations for an incompressible Jeffrey fluid are
T PI S (2)
2
1
( )1
S
(3)
where T and S are Cauchy stress tensor and extra stress tensor, p is the pressure, I is the identity tensor, is
the dynamic viscosity, 1 is the ratio of relaxation to retardation times, 2 is the retardation time, is the shear
rate and dots over the quantities denote differentiation.
Under the assumptions that the channel length is an integral multiple of the wave length and the pressure
difference across the ends of the channel is a constant, the flow is inherently unsteady in the laboratory frame
( , )X Y and become steady in the wave frame ( , )x y which is moving with velocity ‘c’ along the wave. The
transformation between these two frames is given by
, , ,x X ct y Y u U v V (4)
where U and V are velocity components in the laboratory frame and u and v are the velocity components in the wave frame. In the many physiological situations it is proved experimentally that the Reynolds number of the flow is very small. So, we assume that the wavelength is infinite, the flow is of Poiseuille type at each local cross-section. Introducing the non-dimensional quantities
2
2 2
2
1 0 0
1 0
2 2 2, , , , , ,
2, , , , , , ,
, ( ) , ,( )p
x y u v a a px y u v p
a c c c
a a b ga ca ct S S G R Pr
c a c k
cEc T T T T u v
c T T y x
where R is the Reynolds number, G is the gravitational parameter, Ec is the Eckert number and Pr is the Prandtl
number. The equations governing the flow become
0,u u
x y
(5)
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2( 1) ( 1) sin
xyxxSSu u p
R u v u gx y x x y
(6)
3 2( 1) cos
xy yyS Sv v pR u v g
x y x x y
(7)
2 222 22 2 2
2 2( 1) 2
u u v uPrR u v N N
x y x y x y x y
(8)
where 2
1
21 ( ) ,
1xx
c v uS u
a x y x
2
1
21 ( ) ,
1yy
c v uS u
a x y y
2
1
11 ,
1xy
c v u vS u
a x y y x
and
2
2
10
1
1
xyS u
y y
The non dimensional boundary conditions are
0u
y
; 0 0at y
y
(9)
1u ; 1 at y h (10)
Using long wavelength approximation and dropping terms of order and higher.
Eqs. (6) - (8) reduces to
21( 1)
1
u pM u GSin
y y x
(11)
0p
y
(12)
22
2
10
uEc
Pr y y
(13)
where
22 0B a
M
3. SOLUTION OF THE FROBLEM Solving the Eq. (11) with the boundary conditions (9), we get
2
22(( 1 ) )
( 1 )
P Gsin P Gsin Mu cosh M y
MM cosh Mh
(14)
where p
Px
.
Integrating the equation (14) and using the conditions at = 0y , we get the stream function as
2
2 2
( sin )sinh 1{ cosh[ 1 ] ( sin ) *
1
(cosh 1 sinh (1 ) 1 (1 ) }
P G hMy hM M P G
M
hM h M Tanh M
(15)
Substituting Eq. (14) into Eq. (13) subject to the boundary conditions (10), the temperature is
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2 2 2 2 2 4 2
4
2 2
1{2 (2 ( sin ) ( )(1 )) (4 ( sin ) )
8
cosh(2 1 ) ( sin ) cosh(2 1 )sech( 1 ) }
M M Br p G h y M Br p GM
hM Br p G My Mh
(16)
where Br EcPr is the Brinkman number.
The coefficient of heat transfer is given by
x yZ h . (17)
The volume flux ‘q’ through each cross section in the wave frame is given by
0
h
q udy (18)
2
2
1 ( ) ( 1 )
1
P Gsin Tanh M hP Gsin M h
M M
(19)
The expression for pressure gradient from Eq. (19) is given by 2 21 (( ) ( ) ) . ( 1 )
( 1 ) 1
M q M M Gsin h Gsin Tanh M hp
x Tanh M h Mh
(20)
The instantaneous volume flow rate ( , )Q x t in the laboratory frame between the central line and the wall is
0
( , ) ( 1)
h
Q x t u dy q h (21)
Averaging the Eq. (21) over one period yields the time mean flow rate (time averaged flow rate) Q as
0
11
T
Q Qdt qT
(22)
The pressure difference and frictional force across one wave length in an inclined channel is given by
1
0
pp dx
x
(23)
1
0
pF h dx
x
(24)
4. Results and Discussion
To Study the behavior of the distribution of the
axial velocity u, numerical calculations for several values
of Jeffrey fluid parameter , Hartman number M , angle
of inclination , gravitational parameter G are carried
out. Fig. 2(a) shows that an increase in results in
increase of velocity distribution. The effect of on the
velocity distribution can be seen through Fig. 2(b). It reveals that the axial velocity decreases with increasing . Fig. 2(c) displays the effect of G on the velocity
distribution for fixed values of other parameters. It is observed that the velocity increases with increase of G. The effect of the M on the velocity distribution is illustrated in Fig. 2(d). It is evident that, increase the value of M has a tendency to slow down the fluid motion and fluid moves like a block, which shows some sort of rigidity. This is because of the presence of the transverse magnetic field creates a resistive force similar to the drag force that acts in the opposite direction of the fluid motion, thus causing the velocity of the fluid to decrease.
The effect of heat transfer on peristalsis is illustrated in Fig.3. Fig. 3(a) is made to see the variation
temperature for various values of Brinkman number
Br . It is observed that the temperature profiles are
almost parabolic and the increase of Br , the temperature distribution increases. From Fig. 3(b), it can be noticed that the temperature decreases with an increase of M. Figs. 3(c) and 3(d) are plotted to see the influence of Jeffrey fluid parameter and angle of inclination on the temperature distribution. It is observe
that an increase in Jeffrey fluid parameter affects the
temperature profile in an opposite way to that of angle of inclination.
Fig. 4 is plotted to study the effect of Jeffrey fluid
parameter , Hartman number M , angle of inclination
, gravitational parameter G on the pressure gradient
Figs. 4(a) and 4(c) indicate that the pressure gradient increases with increase of M and G. From Figs. 4(b) and 4(d), it is observe that pressure gradient decreases with
increase of and .
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Fig.5 shows the behavior of heat transfer co-
efficient Z. From this figure it is observed that due to peristalsis the heat transfer coefficient is in oscillatory behavior. The absolute value of heat transfer coefficient
decreases with increasing and M while it increases
with increase of Br and .
5. TRAPPING Trapping is an interesting phenomenon in
peristaltic motion. It is basically the formation of an internally circulating bolus of the fluid by closed stream lines. The effect of M on trapping can be seen in Fig. 6. We observed that the size of the bolus reduces with an
increase in M. The effects of and G on the stream
lines are plotted in Figs.7-8. It is observed that the size
of trapping bolus increases with increasing and G.
Figure. 2. The velocity distribution for ( 0.4 , 0.25x ); other parameters are (a) , 0.5, 24
G M
, (b)
2, 0.5, 1M G , (c) , 2, 14
M
, (d) , 0.5, 14
G
.
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Figure. 3.Temperature distribution for ( 0.4 , 0.25x ); other parameters are
(a) , 0.5, 2,4
G M
1 . (b) , 0.5, 1, 34
G Br
.
(c) , 2, 0.5, 34
M G Br
. (d) 2, 0.5, 1, 3.M G Br
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Figure. 4. Pressure gradient versus x for (a) 0.4 , 2, , 33
G
.
(b) 2, 2, 1, 0.4G M . (c) , 2, 1, 0.43
M
.(d) , 2, 1, 0.43
G M
.
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Figure. 5. Coefficient of heat transfer for (a) 0.6, 5, 0.2, 2, 3.M G Br
(b) , 0.2, 2, 3, 0.63
G Br
. (c) , 0.2, 2, 5, 0.63
G M
(d) , 0.2, 3, 5, 0.63
G Br M
.
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Figure. 6. Streamlines for (a) 0.1.M (b) 0.2.M (c) 0.3.M (d) 0.4M ; other
parameters are 00.4, 30 , 4, 0.8G .
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Figure. 7. Streamlines for (a) 2 (b) 3 (c) 4 (d) 5 ; other parameters are 00.4, 30 , 0.4, 0.8M G
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Figure.8. Streamlines for different values of (a) 4G (b) 5G (c) 6G
(d) 7G ; Other parameters are00.4, 30 , 0.4,M 2 .
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[19] Krishnakumari, S.V.H.N. Raman Murthy, M.V. Peristaltic pumping of Jeffrey fluid under the effect of magnetic field in an inclined channel. App. Math Sci. 5(2011), 447- 458. [20] Nadeem, S. Akbar, N.S. Effect of heat transfer on peristaltic transport of MHD Newtonian fluid with variable viscosity: Application to Adomain decomposition method. Comm Nonlinear Sci Numer Simul. 14(2009),3844 - 3855. [21] Nadeem, S. Akbar, N.S. Influence of heat transfer on a peristaltic transport of Hershel Bulkley fluid in a non uniform inclined tube. Comm Nonlinear Sci Numer Simul.14 (2009),4100 - 4113. [22] Srinivas, S. Gayatri, R. Peristaltic transport of a Newtonian fluid in a vertical asymmetric channel with heat transfer and porous medium, 215(2009), 185–196. [23 ]Hayat, T. Hina, N. The influence of wall prosperities on MHD flow of a Maxwell fluid with heat and Mass transfer. Nonlinear Anal. Real world Appl. 11(2010), 3155-3169. [24] Hayat, T. Ali, N. Simultaneous effects of slip and heat transfer on the peristaltic flow. Commun Nonlinear Sci Numer Simulat. 15 (2010), 1526-1537. [25] Hayat, T. Quereshi Mu, Hussain, Q. Effect of heat transfer on the peristaltic flow of an electrically conducting fluid in a porous space. Appl Math Model 33(2009),1862-1873. [26] Rathod V.P. and Mahadev, M. Effect of thickness of the porous material on the peristaltic pumping of a Jefferey fluid with non-erodible porous lining wall. Int J Math Archive. 2(2011), 1-10. [27] Rathod, V. P. and Asha S.K. Effect of magnetic field and an endoscope on peristaltic motion. Adv Appl Sci. Res. 2(2011),102-109. [28] Rathod, V.P and Pallavi Kulkarni. The influence of wall properties on MHD Peristaltic transport of dusty fluid. Adv Appl Sci. Res. 2(2011),265-279.
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