effect of lateral walls on peristaltic ﬂow through an asymmetric...

Applied Bionics and Biomechanics 8 (2011) 295–308DOI 10.3233/ABB-2010-0011IOS Press

295

Effect of lateral walls on peristaltic flowthrough an asymmetric rectangular duct

Kh.S. Mekheimera,∗, S.Z.-A. Hussenyb and A.I. Abd el Lateefc

aMathematical Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo, EgyptbMathematical Department, Faculty of Science, King AbdulAziz University, Jeddah, Saudi ArabiacBasic Science Department, Higher Institute of Engineering, Shorouk City, Cairo, Egypt

Abstract. Peristaltic transport of an incompressible viscous fluid due to an asymmetric waves propagating on the horizontalsidewalls of a rectangular duct is studied under long-wavelength and low-Reynolds number assumptions. The peristaltic wavetrain on the walls have different amplitudes and phase. The flow is investigated in a wave frame of reference moving withvelocity of the wave. The effect of aspect ratio, phase difference, varying channel width and wave amplitudes on the pumpingcharacteristics and trapping phenomena are discussed in detail. The results are compared to with those corresponding to Poiseuilleflow.

Keywords: Lateral walls, rectangular duct, peristaltic pumping

1. Introduction

It is well known that peristalsis is an importantmechanism for mixing and transporting fluids, whichis induced by a progressive wave of contraction orexpansion moving on the wall of a channel/tube. Suchmechanism occurs in the urine transport from kidneyto the bladder, food mixing and chyme movement inthe intestine, lymph transport in the lymphatic vesseland in vasomotion of small blood vessels, transportof spermatozoa in cervical canal, movement of eggsin the female fallopian tube and many others. Sev-eral investigators are engaged to discuss the peristalticmechanism theoretically under various approxima-tions [1–20]. Recently, physiologists observed that the

∗Corresponding author: Kh.S. Mekheimer, Mathematical Depar-tment, Faculty of Science, Al-Azhar University, Nasr City 11884,Cairo, Egypt. E-mail: kh [email protected].

intra-uterine fluid flow due to myometrical contrac-tions is peristaltic type motion and the myometricalcontractions may occur in both symmetric and asym-metric directions, De Vries et al. [21]. In view of this,Eytan and Elad [22] have developed a mathematicalmodel of wall-induced peristaltic fluid flow in a two-dimensional channel with wave trains having a phasedifference moving independently on the upper andlower walls to simulate intrauterine fluid motion in asagittal cross-section of the uterus. They have obtaineda time dependent flow solution in a fixed frame byusing lubrication approach. These results have beenused to evaluate fluid flow pattern in a non-pregnantuterus. They have also calculated the possible parti-cle trajectories to understand the transport of embryobefore it gets implanted at the uterine wall. More-over, Mishra and Rao [23] investigated fluid mechanicseffects of peristaltic transport in a two-dimensionalasymmetric channel under the assumptions of long

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296 Kh.S. Mekheimer et al. / Effect of lateral walls on peristaltic flow through an asymmetric rectangular duct

wavelength and low Reynolds number in a wave frameof reference. The channel asymmetry is produced bychoosing the peristaltic wave train on the walls tohave different amplitude and phase due to the vari-ation of channel width, wave amplitudes and phasedifferences. More recently Kothandapani and Srinivas[24] investigated the peristaltic pumping of incom-pressible Newtonian fluid in an inclined asymmetricchannel through a porous medium. Also, Ali and Hayat[25] investigated the peristaltic motion of an incom-pressible micropolar fluid in an asymmetric channel.In fact, the sagittal cross section of the uterus maybe better approximated by a tube of rectangular crosssection than a two-dimensional channel. The effectsof the side walls on the flows in ducts with suctionand injection are examined by Erdogan [26]. Tsan-garis and Vlachakis [27] obtained an exact solutionof the Navier-Stokes equations for a pulsating flowin a rectangular duct in order to explain the bloodflow in fiber membranes used for the artificial kid-ney. Recently, Reddy et al. [28] studied the flow of aviscous fluid due to symmetric peristaltic waves prop-agating on the horizontal side walls of a rectangularduct under the assumptions of long wavelength andlow Reynolds number and only studied the pump-ing characteristic. Literature survey indicates that suchattempt is the only one for a symmetric peristalticwaves. Therefore, the present analysis is fundamen-tal and is a second attempt in this direction. Keepingin view of the studies mentioned above, the aim of thispaper is to investigate the fluid mechanics effects ofperistaltic transport in a rectangular duct with asym-metric waves propagating on the horizontal side walls(lateral walls) under the assumptions of long wave-length and low Reynolds number in a wave frameof reference. The asymmetry waves are produced bychoosing the peristaltic wave train on the walls tohave different amplitude and phase due to the vari-ation of channel width, wave amplitudes and phasedifferences.

All previous scientific investigations of transportwithin the biological systems they assume. The peri-staltic flow in an asymmetric two-dimensional channelExplain the motion of intrauterine fluid in a sagittalcross section of the uterus. In fact, the sagittal cross sec-tion of the uterus may be better approximated by a tubeof rectangular cross section than a two-dimensionalchannel. And the use of this method of treatment willhelp to improve the overall performance of the artificialheart and blood vessel.

2. Formulation of the problem

Consider the motion of an incompressible viscousfluid in a duct of rectangular cross section with width2 d3 and height d1 + d2 as shown in Fig. 1. Cartesiancoordinate system (X, Y, Z) is with X, Y and Z axescorresponding to axial, lateral, and vertical directions,respectively, of a rectangular duct. The duct walls areflexible, and an infinite train of sinusoidal waves prop-agate with constant velocity c only along the wallsparallel to the XY plane in the axial direction. Theperistaltic waves on the walls are given by

Z=H1=d1 + a1 cos2π

λ(X− ct) upper wall

Z=H1=−d2 −b1 cos

(2π

λ(X−ct)

)+φ) lower wall

(1)

where a1, b1 are the amplitudes of the waves, λ isthe wave length, d1 + d2 is the width of the channel,the phase difference, φ varies in the range 0 � φ � π,φ = 0 corresponds to symmetric channel with wavesout of phase and φ = π the waves are in phase, andfurther a1, b1, d1, d2 and φ satisfies the condition a2

1 +b2

1 + 2a1b1 cos(d1 + d2)2. Introducing a wave frame(x, y, z) moving with velocity c away from the fixedframe (X, Y,Z) by the transformation between the twoframes is given by

x = X− ct, y = Y, z = Z, u = U − c,

w = W, v = V, p(x, z) = P(X,Z, t) (2)

where (u, v,w) and (U,V,W) are the velocity com-ponents, and p and P are the pressures in wave andfixed frames of reference, respectively. Using the non-dimensional quantities

x= x

λ, y= y

d3, z= z

d1, u= u

c,w= w

cδ, v= d3

cλδ3 v

t= ct

λ, h1 = H1

d1, h2 = H2

d1, d= d2

d1, a= a1

d1, b= b1

d1,

Re= cd1

ν, β= d1

d3, p= d2

1p

µcλ. (3)

in the Navier-Stokes equations and equation of con-tinuity, which governs the flow, we get (dropping thebars)

Kh.S. Mekheimer et al. / Effect of lateral walls on peristaltic flow through an asymmetric rectangular duct 297

Fig. 1. Geometry of the problem

Reδ

[u∂u

∂x+ δβ2v

∂u

∂y+ w

∂u

∂z

]= −∂p

∂x+ δ2 ∂

2u

∂x2 + β2 ∂2u

∂y2 + ∂2u

∂z2 (4)

Reδ4[u∂v

∂x+ δβ2v

∂v

∂y+ w

∂v

∂z

]= −∂p

∂y+ δ3

(δ2 ∂

2v

∂x2 + β2 ∂2v

∂y2 + ∂2v

∂z2

)(5)

Reδ3[u∂w

∂x+ δβ2v

∂w

∂y+ w

∂w

∂z

]= −∂p

∂z+ δ4 ∂

2w

∂x2 + δ2(β2 ∂

2w

∂y2 + ∂2w

∂z2

)(6)

∂u

∂x+ δβ2 ∂v

∂y+ ∂w

∂z= 0 (7)

where β is an aspect ratio and Re is a Reynolds num-ber, the aspect ratio β < 1 means that height is lesscompared to width, and β = 0 corresponds to a twodimensional channel. When β = 1, the rectangularduct becomes a square duct and for β > 1, the height ismore compared to width. Under lubrication approach(negligible inertiaRe→ 0 and long wavelength δ � 1)the Eqs. (4)–(7) reduce to

β2 ∂2u

∂y2 + ∂2u

∂z2 = dp

dx(8)

∂u

∂x+ ∂w

∂z= 0 (9)

The corresponding no-slip boundary conditions are

u = −1 at y = ±1 (10)


u = −1 at z = h1, z = h2, (11)

where h1=1+a cos(2πx), h2 =−d−b cos(2πx+ φ),0 ≤ a ≤ 1, a = 0 for straight duct and a= 1 corre-sponds to total occlusion. Introducing x= x, y=βy,z= z in Eqn .(8), we get

∂2u

∂y2 + ∂2u

∂z2 = dp

dx(12)

Substituting u = u− 1 + 1 12dpdx

(h1 − z)(h2 − z), in(12), we get

∂2u

∂y2 + ∂2u

∂z2 = 0 (13)

The corresponding boundary conditions are

u = −1

2

dp

dx(h1 − z)(h2 − z) at y = ± 1

β(14)

u = 0 at z=h1, z=h2 (15)

The solution of (8) using Eqs. (12)–(15), valid in−1 ≤ y ≤ 1, h2 ≤ z ≤ h1, satisfying the correspond-ing boundary conditions (10) and (11) is given by

u=−1 + dp

dx

((h1 − z)(h2 − z)

2− 2

(h1 − h2)

×∞∑n=1

((−1)n−1) cosh(αnβy)

sin αn(z−h2)

α3n cosh

(αnβ

)⎞⎠ (16)

where αn = (nπ)/(h1 − h2) The volumetric flow ratein the rectangular duct in the wave Frame (in verticalhalf) is given by

q=∫ h1

h2

∫ 1

0(u) dy dz=−(h1−h2)+dp

dx

(−(h1−h2)3

12

+ 4β

(h1 − h2)

∞∑n=1

(1 − (−1)n) tanh αnβ

α5n

)(17)

The instantaneous flux in the laboratory frame is

Q =∫ h1

h2

∫ 1

0(u+ 1)dy dz = q+ (h1 − h2) (18)

The average fluxQ(x, t) over one period (T = α/c)of the peristaltic wave is

Q = 1

T

∫ h1

h2

∫ 1

0(Q)dy dz = q+ 1 + d (19)

From Eqs (17) and (19), the pressure gradient isobtained as

dp

dx= (Q− 1 − d) + (h1 − h2)

− (h1−h2)3

12 + ∑∞n=1

4β(1−(−1)n) tanh αnβ

(h1−h2)α5n

(20)

Integrating Eq. (20) with respect to x over one wave-length, we get the pressure rise (drop) over one cycleof the wave as

p = (Q− 1 − d)I1 + I2

where

I1=∫ 1

0

1

−(h1−h2)3

12 + 4β(h1−h2)

∑∞n=1

(1−(−1)n) tanh αnβ

α5n

dx

I2 =∫ 1

0

(h1 − h2)

−(h1−h2)3

12 + 4β(h1−h2)

∑∞n=1

(1−(−1)n) tanh αnβ

α5n

dx

(21)

The stream function associated with the flowdepends on all the three coordinates. We can find thestreamlines in xz plane (fixing a lateral station y)andwith the condition. ψ = 0 at z = 1/2(h1 + h2) willtake the form

ψ(x, y) = 1

2(h1 + h2) − z+ dp

dx

{1

24(h1 + h2 − 2z)(h2

1 − 4h1h2 + h22 + 2(h1 + h2)z− 2z2)

+∞∑n=1

(1 − (−1)n) cosh(αnβy)

α4n cosh

(αnβ

) (√1

2((−1)n + 1) cos (αn(z− h2))

)}(22)

In the limitβ → 0 (keeping a fixed and d → ∞), therectangular duct reduces to a two-dimensional chan-nel and our results reduce to those given by Mishraand Rao [23], also when (d1 = d2, a1 = b1) and φ = 0(symmetric walls) our results reduce to those givenby Reddy, et al. [28]. Finally, for the symmetric wallsand β → 0 our results are the same as those given byShapiro et al. [1].


180

160

140

120

100

dpdx

80

60

40

20

0

0.0 0.1 0.2 0.3 0.4 0.5X

Q

0.6 0.7

0.0

12

9

6

3

0

-3

-6

0.2 0.4 0.6

0.8 0.9 1.0

∆p

β = 0.1

β = 1

β = 0.5

β = 1

β = 2

Fig. 2. (i) Variation of dp/dx versus x for different values of β with a = b = 0.6 and (i) φ = 0 (dot lines), (ii) φ = π/4 (solid lines). (ii) Variationof p versus Q for different values of β with a = b = 0.6 and (i) φ = 0 (solid lines with closed mark), (ii) φ = π/4 (solid lines with openmark), (iii) φ = π (dot lines).


10

20000

10000

-10000

-20000

-30000

0

5

0

-5

-10

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

∆p∆p

β = 0.01

β = 0.5

β = 1

β = 2

β = 50

β = 100

β = 150

Q

Q

(i)

(ii)

Fig. 3. Variation ofp verusQwith a = b = 0.6 and φ = π/4 (i) for small values of (ii) For large β, and the corresponding graphs for Poiseuille(a = b = 0) flow are given in dot lines.


350

Q = 0.291

300

250

200

150∆p∆p

100

50

4

2

0

-2

-4

0

-50

0 20

0 1 2 3 4

40 60 80 100 120β

β

140 160 180

Q = 0.293

Q = 0.295

Q = 0.297

Q = 0

Q = 0.2

Q = 0.3

Q = 0.4

Q = 0.29

(i)

(ii)

Fig. 4. Variation of p versus β with a = b = 0.7 and φ = π/4 (i) for different values of flux Q and (ii) enlargement of (i) near β = 0.


16000

14000

φ

12000

10000

8000

4000

6000

2000

0

–2000

–40000 25 50 75 100

β

∆p

125 150 175 200

= 0

φ = 0.3 π

φ = 0.6 π

φ = 0.9 π

Fig. 5. variation of p versus β with a = b = 0.7, d = 2 and Q = 0.1 (solid lines) and (Q = −0.1 dot lines) for different phase.

3. Discussion

This section describes the influence of variousemerging parameters of the problem on dp/dx, p,Qand trapping phenomena. Moreover, to understand theeffect of lateral asymmetric walls on the pumping char-acteristic one have to analyze the pressure-flux relationgiven in Eq. (21) for different values of the aspectratio β and the dimensional phase shift φ. Due to thecomplexity of expression (21), the integrations and theseries involved are not integrable in closed form, sonumerical integrations are performed and the graph-ical results are presented. Physically, as β increasesfrom small value to a value greater than one, a transi-tion takes place from rectangular duct with dominationof width to the one with height passing through a cross-section of square duct. In Fig. 2(i) the variation ofdp/dxwithin a wavelength for different values of the phasedifferent φ and the aspect ratio β is presented. It isshown that as φ increases, i.e moves from symmet-ric to an asymmetric wall waves, the pressure gradientdecreases, while as β increases the pressure gradientincreases. Figure 2(ii) and (3) depicts the pumpingcurves with a fixed value of a = b = 0.6 and for differ-ent values of β and φ. For small values of β, any twopumping curves intersect at a point in the first quad-rant as seen in Fig. 3(i). As long as this point remain

in the first quadrant, to the left of this point pumpingincreases and to the right both pumping and free pump-ing p = 0 decreases with increasing β, i.e. verticaldimension a is large compared to horizontal dimensiond. This point of intersection will stay at the first quad-rant asφ increases but with less pressure rise, while thispoint move to the fourth quadrant for large values of βas shown in Fig. 3(ii) and so, both pumping and freepumping increases with increasing βwhile it decreasesasφ increases. Our results for the pumping characteris-tics (p versusQ)are comparing to those of Poiseuille(without peristaltic a = b = 0) flow in dot lines inFig. 3(i), (ii). The variation ofp with β for differentvalues ofQ is illustrated in Fig. 4(i), (ii). It is observedthatp increases for small values ofQ. Fig. 4(ii) is anenlargement of Fig. 4(i), showing that the curves forvarious Q do not meet at a single point near β = 0.0.Figure 5 depicts the variation ofp with β and for dif-ferent values of φ. This figure shows thatp decreasesasφ increases while it increases asβ increases for smallvalues ofφ. The variation ofQwith β for different val-ues ofp and a, b (amplitudes of peristaltic waves onthe upper and lower walls) is depicted in Fig. 6, wherethe flow rateQ decreases forp = -1 and increases forp= 4 only for small values of β. Also, Fig.6(i), (ii)shows that the free pumping flux rate Q attain a min-imum around β � 0.25 and then have approximately


2.0

∆p = -1

∆p = 0

∆p = 2

∆p = 4

a = b = 0.9

a = b = 0.6

a = b = 0.3

1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

–2.0

–2.5

–3.0

–3.5

–4.0

–4.5

–5.0

–5.5

–6.0

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 25 50

Q

75β

β

100 125

0 1 2 3 4

Q

(i)

(ii)

Fig. 6. (i) Variation of Q versus β with a= b= 0.7, d = 2 and φ=π/2 for different values of p, and (ii) variation of Q versus β with freepumping (p = 0) for different a, b at d = 1 and φ=π/5.


1.5

(i)

(ii)

1.0

0.5

0.0

–0.5∆p = –0.5

∆p = –0.5

∆p = –0.05

∆p = 0

∆p = 0.5

∆p = 1

∆p = 0

∆p = 0.05

∆p = 0.1

∆p = 1

–1.0

–1.50.0

0 1 2 3 4 5 6d

7 8 9 10

4

3

2

1

0

–1

–2

–3

–4

–5

–6

0.2 0.4 0.6 0.8 1.0

Q

Q

Fig. 7. (i) Variation of Q with φ when d = 2, β = 0.5 for different values of p at a= b = 0.7 (solid lines) and a= 0.7; b = 1.2 (dot lines).(ii) Variation of Q with d for different p with a = 0.7, b = 1.2 β = 0.5, and φ = π/2.


0.5

0.0

–0.5

–1.0

–1.5β = 0.01

β = 1

β = 2

β = 3

–2.0

–2.50.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Q

Fig. 8. Variation of Q versus φ with a = b = 0.7, d = 2 and p = 1 for different β.

constant value and this flow rate decrease as the ampli-tudes of peristaltic waves increases. The variation ofthe time – average fluxQ as a function φ and d (widthof the channel) for different values ofp is presented inFig. 7(i), (ii) respectively. We observe that when φ = 0,the time - average flux Q is maximum and decreasesas φ increases then vanishes for certain value of φ andremains negative until φ becomes 1. Whenp = 0 ,forfree pumping case, we observe Q is zero for φ = 1(i.e. when peristaltic waves are in phase, the cross sec-tion of the channel remains same through out) and ispositive for all 0 ≤ φ < 1.1. Also, Q remains alwayspositive for p < 0 (copumping region) as pressureassists the flow due to peristalsis on the walls. Fig.7(ii)shows that for p ≥ 0 the flux rate decreases as thedistance d between the walls increases due to the reduc-tion in the peristalsis effects. From the curvep= -0.5(p < 0),Q decreases for some d (small) in the begin-ning, but it start increasing for d large as the Poiseuilleflow due to the pressure loss dominates the peristalticflow. Figure 8 show the variation of Q with φ for dif-ferent values of β. It is observed that as φ increases theflow rate flux decreases (more rapidly for small valuesof β), and the change in Q is small for large valuesof β.

4. Trapping

In the wave frame the streamlines in general havea shape similar to the walls as the walls are station-ary. But under certain conditions some streamlines split(due to the existence of a stagnation point) to enclosea bolus of fluid particles in closed streamlines. In thefixed frame the bolus moves as a whole at the wavespeed as if trapped by the wave to see the effectsof lateral station y, aspect ratio β and phase differ-ence φ on the trappings, we prepared Figures 9–11.Figure 9 reveals that in the symmetric (φ = 0) chan-nel the trapping is about the center line and trappedbolus decrease in size as the lateral state y increases.The effect of the aspect ratio β on the trapping isobserved in Fig. 10, where as β increases the bolussize increases and for β = 1, the bolus will move asthe wave wall. The effect of phase shift φ on trap-ping with same amplitudes a = b = 0.5 for ¯Q = 1.4(within the centerline trapping limits) is illustratedin Fig. 11. It is observed that the bolus appearing inthe center region for φ = 0 then moves towards leftand decreases in size as φ increases. For φ =π thebolus disappears and the streamlines are parallel tothe boundary walls [21, 23] and here Q > 0 is pos-


1.0

(i)

(ii)

(iii)

0.5

0.0

–0.5

–1.0

–1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

–0.4 –0.2 0.0 0.2 0.4

–0.4 –0.2 0.0 0.2 0.4

–0.4 –0.2 0.0 0.2 0.4

Fig. 9. Streamlines for a= 0.5, b = 0.5, d = 1.Q = 1.4 and β = 0.1for different y (i) y = 0.3, (ii) y = 0.5, (iii) y = 0.7.

1.0

1.5(i)

(ii)

(iii)

0.5

0.0

–0.5

–1.0

–1.5–0.4 –0.2 0.0 0.2 0.4

–0.4 –0.2 0.0 0.2 0.4

–0.4 –0.2 0.0 0.2 0.4

1.0

1.5

0.5

0.0

–0.5

–1.0

–1.5

1.0

1.5

0.5

0.0

–0.5

–1.0

–1.5

Fig. 10. Streamlines for a= 0.5, b = 0.5, d = 1; Q = 1.4 and for dif-ferent β (i) β = 0.01, (ii) β = 0.1, (iii) β = 1.


(i)

(ii)

(iii)

0.4 0.2 0.0 0.2 0.4

0.4 0.2 0.0 0.2 0.4

0.4 0.2 0.0 0.2 0.4

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Fig. 11. Streamlines for a= b = 0.5; d = 1. Q = 1.4 and β = 0.3 fordifferent γ (i) γ = 0, (ii) γ = π/2, (iii) γ = π.

sible only when p< 0 corresponding to Poiseuilleflow.

5. Concluding remarks

Peristaltic transport of an incompressible viscousfluid due to an asymmetric waves propagating onthe horizontal sidewalls of a rectangular duct is stud-ied under long-wavelength and low-Reynolds numberapproximations. The expressions of the axial veloc-ity, axial pressure gradient and stream function aredeveloped. Numerical integration is used to analyzethe pumping characteristics. Stream lines are plottedto discuss the phenomena of trapping. Such a prob-lem is important in understanding the motion of theintrauterine fluid in a sagittal cross section of the uterus.However, the sagittal cross section of the uterus maybe better approximated by a tube of rectangular crosssection than a two-dimensional channel. The presentstudy discloses some important finding. These are

• The pressure gradient is higher for a square duct(β = 1) than that for a two dimensional channel.

• The pressure gradient (or the pressure rise) for asymmetric walls is higher than that for an asym-metric walls.

• The finite extent of the width of the duct alters thepumping characteristics.

• The pressure rise is higher for β > 1 (a duct witha height large compared to the width) than that forβ = 1 (a square duct) than that forβ < 1 (rectan-gular duct) than that for β = 0 (two-dimensionalchannel).

• The size of the trapped bolus increases with anincrease in the aspect ratio β, while it decreasesas the lateral station y increases.

• Symmetric channel gives bigger trapping zone.• Future works can be done by using numerical

methods to solve the governing equations of theproblem without any approximation.

References

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[4] Kh.S. Mekheimer, Non-linear peristaltic transport of mag-neto-hydrodynamic flow in an inclined planar channel, ArabJ Sci Eng 2A (2003), 183.

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[8] Kh.S. Mekheimer, Peristaltic flow of blood under effect of amagnetic field in a non uniform channels, Appl Math Compu-tation 153 (2004), 763.

[9] T. Hayat, Y. Wang, A.M. Siddiqui and K. Hutter, Peristaltictransport of an Oldroyd-B fluid in a planar channel, MathProblems Eng 4 (2004), 347.

[10] T. Hayat, F.M. Mahomed and S. Asghar, Peristaltic flow ofmagnetohydrodynamic Johnson-Segalman fluid, NonlinearDynamics 40 (2005), 375.

[11] M. Elshahed and M.H. Haroun, Peristaltic transport of John-sonSegalman fluid under effect of a magnetic field, MathProbl Eng 6 (2005), 663.

[12] T. Hayat, M. Khan, S. Asghar and A.M. Siddiqui, A math-ematical model of peristalsis in tubes through a porousmedium, J Porous Media 9 (2006), 55.

[13] T. Hayat and N. Ali, Peristaltically induced motion of a MHDthird-grad fluid in a deformable tube, Physica A 370 (2006),225.

[14] M.H. Haroun, Non-linear peristaltic flow of a fourth gradefluid in an inclined asymmetric channel, Comput Mater Sci39 (2007), 324.

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[16] M.H. Haroun, Effect of Deborah number and phase differenceon peristaltic transport of a third-order fluid in an asymmetricchannel, Nonlinear Sci Numer Simul 12 (2007), 1464.

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[26] M.E. Erdogan, The effects of side walls on axial flow in rect-angular ducts with suction and injection, Acta Mechanica 162(2003), 157.

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