INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.1 JANUARY 2014
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UK: Managing Editor International Journal of Innovative Technology and Creative Engineering 1a park lane, Cranford London TW59WA UK E-Mail: editor@ijitce.co.uk Phone: +44-773-043-0249 USA: Editor International Journal of Innovative Technology and Creative Engineering Dr. Arumugam Department of Chemistry University of Georgia GA-30602, USA. Phone: 001-706-206-0812 Fax:001-706-542-2626 India: Editor International Journal of Innovative Technology & Creative Engineering Dr. Arthanariee. A. M Finance Tracking Center India 17/14 Ganapathy Nagar 2nd Street Ekkattuthangal Chennai -600032 Mobile: 91-7598208700
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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.1 JANUARY 2014
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International Journal of Innovative Technology & Creative Engineering Vol.4 No.1 January 2014
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From Editor's Desk Dear Researcher, Greetings! Research article in this issue discusses about Pulsatile flow of a Jeffrey fluid. Let us review research around the world this month; Green Drive, is showing off the free app this week at the Consumer Electronics Show in Las Vegas. EnLighten feeds off real-time traffic data supplied by cities, then uses a phone's GPS and accelerometer to determine its user's location and velocity. Ginsberg aims to sell his patented technology to carmakers, so that it can be built into cars. That would avoid draining smartphone batteries and allow for integration into other car systems. It's hardly Chopin. But a soft robot can bend and change the shape of its four rubbery fingers fast enough to hit keys and play a simple tune on a piano. The fingers, which are secured to the edges of the keys, change shape in just 50 milliseconds when air is pumped into them. A computer-controlled valve regulates the pressure of each finger, or actuator, with the appropriate timing. The new actuators have already been used in a pneumatic glove developed for the rehabilitation of stroke sufferers. The system could also be used in robotic surgery, to create soft tools for handling fragile internal organs.The team is now looking at increasing the force that can be applied with the fingers. The next generation of soft robots may not only move faster – they may be more powerful as well.
The plant racks in a vertical farm can be fed nutrients by water-conserving, soil-free hydroponic systems and lit by LEDs that mimic sunlight. And they need not be difficult to manage: control software can choreograph rotating racks of plants so each gets the same amount of light, and direct water pumps to ensure nutrients are evenly distributed. Advances in vertical farms could trickle through from other sources, too. The US Defense Advanced Research Projects Agency is using an 18-storey vertical farm in College Station, Texas, to produce genetically modified plants that make proteins useful in vaccines. Adversity also plays its part: the tsunami-sparked nuclear accident in Fukushima, Japan, in 2011 is leading to innovation in vertical farming because much of the region's irradiated farmland can no longer be used. 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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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.1 JANUARY 2014
Contents Pulsatile flow of a Jeffrey fluid in a channel bounded by porous lined plates with suction and injection by S.Sreenadh, P.Govardhan and Y.V.K.Ravi Kumar.............................................................................................[172]
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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.1 JANUARY 2014
Pulsatile flow of a Jeffrey fluid in a channel bounded by porous lined plates with suction and injection 1
2
S.Sreenadh
1
P.Govardhan
1
Y.V.K.Ravi Kumar
2
Department of Mathematics, S. V. University, Tirupati – 517502, INDIA Practice School Division, Birla Institute of Technology and Science (BITS) – Pilani, INDIA (yvkravi@gmail.com , corresponding author)
Abstract— Pulsatile flow of a Jeffrey fluid in a channel bounded by porous lined plates with suction and injection is studied in this paper. The steady and unsteady velocities are obtained. The effect of various parameters on the flow phenomenon is discussed through graphs.. Key words — pulsatile flow, Jeffrey fluid, suction, injection.
I. INTRODUCTION Viscous fluid flow through and past porous media is attracting the attention of scientists and engineers because of its wider applications in various branches of science and technology. The movement of ground water in soil, the seepage of water through earth fills and concrete dams, the movement of oil fields can described using the knowledge of flow through porous media. The petroleum industry has been showing a lot of interest in these problems in connection with the crude oil production from the under ground reservoirs. These reservoirs consists of porous materials like lime stone and dolomite where oil is preserved. Oil can be obtained by drilling wells down into the reservoir. In order to have a better oil production, it is necessary to use the knowledge of flow through porous media. The oil available in the porous reservoir is a complex fluid. The properties of such fluid have impact on the oil production. The behavior of the oil may be Newtonian or non-Newtonian. In view of this, it is interesting to study non- Newtonian fluid flow through and past porous media. Further there are many important applications in biomechanics also (vide Fung and Tang, [1,2]. Muakat [3] made theoretical and experimental studies on porous flow using Darcy law. Darcy law is observed to be valid for low speed flows and agrees with several experiments modeled in one dimensional motion. Yih [4] suggested the modified Darcy law for describing unsteady flow through porous media. Following this law, Rudraiah et al. [5], analyzed several time dependent
flows through and past permeable beds. Radhakrishnamacharya [6] investigated the pulsatile flow of a dusty fluid containing small solid particles uniformly, through a two-dimensional constricted channel. The effect of non-Newtonion nature of blood and pulsatality on flow through a stenosed tube is analyzed by Chaturani and Samy [7]. The pulsatile flow in a porous channel is important in understanding the process of dialysis of blood in an artificial kidney. Recently, Chandra and Prasad [8] discussed the pulsatile flow problems with periodic acceleration and varying cross section of tubes. Wang [9] studied the interesting problem of pulsatile flow in a porous channel bounded by rigid walls. The pulsatile flow between permeable walls is important in understanding the blood flow in the circulatory system where the nutrients are supplied to tissues of various organs and waste products are removed. Vajravelu et al.,[10]made a detailed study on pulsatile flow between permeable beds. Avinash et.al [11] studied the pulsatile flow of a viscous stratified fluid of variable viscosity between permeable beds is studied. The interaction of peristaltic flow with pulsatile flow through a porous medium is discussed by Afifi and Gad [12]. In this paper an exact solution for the pulsatile flow of Jeffrey fluid in porous lined plates is obtained. The flow between the permeable layers is governed by Jeffrey model where as the flow in the lower and upper beds are governed by Darcy law. The velocity distributions in the porous and non-porous regions are determined. II. MATHEMATICAL FORMULATION OF THE PROBLEM
Consider the pulsatile flow of a Jeffrey fluid in a channel bounded by porous lined plates (Fig.1). The thickness of the porous lining on each of the plates is . The fluid is injected into the channel from the lower porous layer with a velocity V and is sucked out into the upper porous layer with the
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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.1 JANUARY 2014
u (u B1 Q1 ) y k1
same velocity. The permeabilities of lower and upper beds are k1 and k2 . The flow between the permeable beds is governed by the Jeffrey model whereas the flow in the porous medium is described by modified Darcy’s law
u uB2
u (u B 2 Q2 ) y k2
at y =
(5)
at y = h -
(6)
at y = h -
(7)
u, v are velocities of the fluid, viscosity
is the coefficient of
p is the pressure, is the slip parameter,k1,k2a are permeabilities of the lower and upper permeable beds, u B1 , u B 2 are slip velocities in lower and upper permeable beds and 1 is the Jeffrey parameter. Fig.1. Physical Model
Separating equations (1) – (7) into steady part denoted by a bar (-) and unsteady part denoted by a tilde ( we get
The following assumptions are made in the analysis of the problem
Steady part:
u 0 x
The flow is laminar and fully developed The permeable beds are homogeneous The flow is driven by unsteady pressure gradient. We assume that 1 p A Be it
V
u 2u A 2 y 1 1 y
u u B1
x
(9) at y = (10)
where A and B are constants and ‘ ’ is the frequency. In view of the above assumptions, the basic equations and boundary conditions of the flow take the following form.
Basic equations u v 0 x y
(1)
u u 1 p u V t y x 1 1 y 2
(2)
p 0 y
(3)
2
du (u B1 Q1 ) dy k1
at y =
at y = (11)
at y = h u u B2 du at y = h (u B 2 Q 2 ) dy k2 k A k A where Q 1 1 (1 1 ) , Q 2 2 (1 1 )
(12) (13)
Unsteady part:
u~ 0 x u~ u~ 2 u~ V Be it t y 1 1 y 2
u~ u~B1
(14) (15)
at y = ε
(16)
du~ ~ ~ (uB1 Q1 ) at y = ε dy k1
Boundary conditions
u uB1
(8)
(4)
u~ u~B 2
173
(17) at y = h - ε
(18)
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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.1 JANUARY 2014
du~ ~ ~ (u B 2 Q2 ) dy k2
at y = h – ε
u u B2
(19)
k 2 B(1 1 ) it ~ k1 B(1 1 ) it , ~ where Q Q2 e e 1
at y = 1
-
(24)
1 1 du 2 (u B 2 R) at y = 1dy 22
Non – Dimensionalization of flow quantities
Vh
(25)
h , k1
where
R
2
h ( 1 , 2 are dimensionless parameters). k2
The following non – dimensional quantities are introduced to make the basic equations and the boundary conditions dimensionless
is the Reynolds number,
1
Steady part:
u B1 * u , , u B1 u A1h A1h ( ) ( ) v v
u B2
*
y*
u B2 x x* , A1h h ( ) v
Unsteady Part:
du~ (26) 0 dx d 2 u~ du~ du~ R ( 1 ) ( 1 ) (1 1 )e it (27) 1 1 dy dt dy 2
y, * Q1 , * Q2 , h Q1 A1 h Q2 A h 1 (
*
*
v
)
(
Unsteady part: ~ u~ u~ * u B1 * ~ u 2 B1 h 2 B1 h B1 ( ) ( )
~
(
h2
~* Q2 (
~ u~ f ( y)eit ~ ~ ~ f ( y) R(1 1 ) f ( y) i(1 1 ) f ( y) (1 1 )
)
Letting
where A1 A .
h
~*
v
~* , Q 1 )
~ Q2 h 2 B1
( ,
~ Q1 h 2 B1
y*
)
u~ t * u~B 2 2B 2 , ~t * 2 , h B1 h ( ) ( )
~ u~B1 f1eit
~
(28)
~ f 2eit
Using , uB 2 The boundary conditions become
~ ~ f f1
at y =
(29)
~ ~ 1 1 df 1 ( f1 ) at y = dy 12 ~ ~ at y = 1 f f2 ~ ~ 1 1 df 2 ( f 2 ) at y = 1 dy 22
)
y * , where B1 B . h h
(30) (31) (32)
III. SOLUTION OF THE PROBLEM
In view of the above dimensionless quantities Eqs.(8) to (19) take the following form, neglecting the asterisks (*), we get,
Steady part:
Solving Eq.(21) subject to boundary conditions (22)-(25) we get the velocity field as Steady part
We get the velocity field as
u C1 C2 e (11 ) y y du 0 dx
(20)
The slip velocities are
d 2u du (1 1 ) R (1 1 ) R 2 dy dy
(21)
u B1
u u B1
(22)
at
y=
1 1 du 1 (u B1 R) dy 12
u B1
(33) and u B 2 are given by
( D3 E1 D1E2 ) ( D2 E2 D4 E1 ) , u B2 (34) ( D1D4 D3 D2 ) ( D1D4 D3 D2 )
Unsteady part: at y =
(23)
Solving equation (28) subject to boundary conditions (10) –(13), we get
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INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.1 JANUARY 2014
~ i f ( y) Ae m1 y Be m2 y
(35)
The unsteady part of the velocity is given by
~ u~ f ( y)eiwt
Separating real and imaginary parts, we get
u~ ( A12Cost B12Sint ) i( B12Cost A12Sint )
The slip velocities are given by
Am e m1 Bm2 e m2 1 1 it ~ u~B1 f1e it ( 1 )e 2
1
2
1 1 Am1e ~ u~B 2 f 2 e it ( 2
m1 (1 )
2
2
1
Bm 2 e m2 (1 )
2
)e it
5. Deductions (i) Taking k1 = k2 = k (i.e. in equation (33 ) and (35) we obtain the velocity field for pulsatile flow of Jeffrey fluid between porous lined plates with equal permeability as follows: Steady state velocity ̅ = c1 + c2 (
+y
(36) Unsteady part:
̃
(
) (37)
(ii) When the permeabilities k1 and k2 tend to zero in equations (36) and (37) we obtain the velocity field for the pulsatile flow of the Jeffrey fluid in a channel bounded by rigid walls as follows Steady part
̅
(38)
The slip velocities ̃ Unsteady part
̃
(
and ̃
are zero.
)
(39)
The slip velocities ̃ and ̃ are zero. Further with , the results (38) and (39) reduce the corresponding ones of Wang [9].
RESULTS AND DISCUSSION From equation (33) we have calculated the steady part of the velocity as a function of y for different values of permeability parameter with fixed 0.5, R 2, 0.5, 0.02 and is shown in Fig.2. We observe that the
velocity decreases with the increase in h . This k may be due to the increase in the permeability ‘k’ of the upper and lower beds. Also, as the increases, the gap between the velocity profiles becomes smaller i.e. there is not much change in the velocity due to variation for large values of . The variation of steady part of the velocity with y is calculated from equation (33) for different values of permeability parameter with fixed 0.1, R 2, 0.5 , 0.02 and is shown in Fig.3. It is noticed that the velocity decreases with the increase in . From Fig.2 and Fig.3 it is noticed that the magnitude of the velocity increases with decrease in the slip parameter . Also, as the increases, the gap between the velocity profiles becomes smaller. The variation of u with y is calculated from equation (33) for different values of the Reynolds number R and for fixed 2, 25, 0.5, 0.02 and is shown in Fig.4. We observe that the velocity increases with the increase in R. From equation (33) we have calculated the steady part of the velocity as a function of y for different values of Jeffrey parameter with fixed 2, 25, R 2 , 0.02 and is shown in Fig.5. It is noticed that the velocity increases with the increase in the Jeffrey parameter . Similar behavior due to is noticed by Srinivas et al. (2008) and Hayat et al. (2008) for the peristaltic transport of viscous fluid in a flexible channel. From equation (33) we have calculated the steady part of the velocity as a function of y for different values of the thickness of porous lined plate with fixed 2, 25, R 2 , 0.5 and is shown in Fig.6. We observe that the velocity decreases with the increase in . We have calculated the unsteady part of the velocity as the function of y from equation (35) for different values of wt with fixed 5, 5, R 0 , 0.02 , M=1, 1 and is shown in Fig.7.We observe the velocity decreases with the increase in wt . We have calculated the unsteady part of the velocity ( u ) as the function of y from equation (35) for different values of with fixed wt
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0.5, 25, R 10 , 0.02 , M = 1, 1 and is shown in Fig.8.We observe the velocity decreases with the increase in wt . From Fig.7. and Fig.8. for fixed wt and , as R increases the maximum velocity moves closure to the upper permeable bed. We have calculated the unsteady part of the velocity ( u ) as the function of y from equation (35) and are shown in Figs.9 and 10. For fixed wt , the velocity decreases with the increment in . As R increases the maximum velocity moves closure to the upper permeable bed. From equation (35) we have calculated the unsteady part of the velocity as a function of y for different values of Reynolds number R and Jeffrey parameter with fixed 5, M=1, 0.02, wt
Fig. 2. Steady state velocity profiles for 0.5, R 2, 0.5 and 0.02
4
, 10 and are shown in Fig.11. and Fig.12. It is noticed that the velocity increases with the increase in . As R increases the maximum velocity moves closure to the upper permeable bed. The variation of u with y is calculated from equation (35) for different values of the Reynolds number R and for fixed 2, 25 , M=1, 0.02, wt , 0.5 and is shown in Fig.13. We 4 observe that the velocity decreases with the increase in R. From equation (35) we have calculated the unsteady part of the velocity as a function of y for different values of the thickness of porous lined plate with fixed 2, 10 , M = 1, R 2, wt
Fig. 3. Steady state velocity profiles for with fixed 0.1, R 2, 0.5 and 0.02
4
, 0.5 and is shown in Fig.14. We observe that the velocity decreases with the increase in .
Fig..4. Steady state velocity profiles for R with fixed 2, 25, 0.5 and 0.02
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Fig..5. Steady state velocity profiles for with fixed 2, 25, R 2 and 0.02
Fig.8. Unsteady state velocity profiles for t with fixed 0.5, 25, R 10 , 0.02 , M=1 and
0.5
Fig. 6. Steady state velocity profiles for with fixed 2, 25, R 2 and 0.5
Fig.9. Unsteady state velocity profiles for with
0.5
Fig.7. Unsteady state velocity profiles for t with fixed 5, 5, R 0 , 0.02 , M=1 and 1
Fig. 10. Unsteady state velocity profiles for with
1
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Fig. 11. Unsteady state velocity profiles for with fixed 5, R 2 ,M=1, 0.02, wt
10 .
0.5 .
Fig. 12. Unsteady state velocity profiles for
10
fixed 2, 25 ,M=1, 0.02, wt
and
4
fixed 5, R 10 , M=1, 0.02, wt
Fig. 13. Unsteady state velocity profiles for R with
4
with
Fig. 14. Unsteady state velocity profiles for
and
fixed 2, 10 , M=1, R 2, wt
4
0.5 .
4
and
with and
Acknowledgements: Authors thank DST, Govt. of India for providing financial support under major research project to carry this work.
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References [1].Fung.Y. C. and Tang H.T, “Longitudinal discersion of Trater particles in the blood flowing in a pulmonary alveonar Sheep”, ASME, J.Appl, Mech.42 (1975b) 546. [2].Fung,Y. C. and Tang, H.T, “Solute distribution in the flow in a channel bounded by a porous layer”, ASME, J.Appl, Mech.42 (1975a) 531. [3].Muakat, M.,” Flow of homogeneous fluids through porous Media”, Mc Graw Hill book Company. (1937) [4].Yih,S.W.,”Dynamics of non-homogeneous fluids”, Macmillan Co. (1967). [5].Rudraiah, N. et al., “Some flow problems in Porous media”PGSAM series, Bangalore University, Bangalore. (1979). [6].Radhakrishnamacharya, G.,” Pulsatile flow of dusty fluid through a constricted channel” , J. of Applied. Math. Phys. (ZAMP), 29 (1981) [7].Chaturani.P and Samy, R.P.,”Pulsatile flow of casson’s fluid through stenosed articles with applications to blood flow”, Biorheology, 23(5) (1986) 499-511. [8]Chandra, P., Prasad, J.S.V.R.K,”Pulsatile flow in circular tubes of varying cross-section with suction/injection”, J.Aust. Math. Soc. Ser. B.35 (1994) 366-381. [9].Wang, Y.C.,”Pulsatile flow in a porous channel”, Transaction of ASME, J. Appl. Mech. 38 (1971) 553-555. [10].Vajravelu, K..,Ramesh, K.,Sreenadh, S.and Arunachalam, P.V.,” Pulsatile flow between permeable beds”, Int.Jr. of Non-Lin. Mech., 38 (2003) 999-1005. [11].Avinash .K., Ananda Rao .J., Ravi Kumar. Y.V.K., Sreenadh.S.,” Pulsatile flow of a viscous stratified fluid of variable viscosity between permeable beds”, Jr.of Porous Media,14(12),2011,1115 – 1124, [12].Afifi, N.A.S and Gad, N.S.,” Interaction of peristaltic Flow with pulsatile fluid through a porous medium”, App. Math. and Comp., 142 (2003) 167-176.
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