research article magnetic field and gravity effects on peristaltic...

Research ArticleMagnetic Field and Gravity Effects on Peristaltic Transport ofa Jeffrey Fluid in an Asymmetric Channel

A. M. Abd-Alla,1,2 S. M. Abo-Dahab,1,3 and Maram M. Albalawi1

1 Mathematics Department, Faculty of Science, Taif University, Taif 888, Saudi Arabia2Mathematics Department, Faculty of Science, Sohag University, Sohag, Egypt3Mathematics Department, Faculty of Science, SVU, Qena 83523, Egypt

Correspondence should be addressed to A. M. Abd-Alla; [email protected]

Received 12 January 2014; Accepted 19 February 2014; Published 13 April 2014

Academic Editor: Juntao Sun

Copyright © 2014 A. M. Abd-Alla et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, the peristaltic flow of a Jeffrey fluid in an asymmetric channel has been investigated.Mathematical modeling is carriedout by utilizing longwavelength and lowReynolds number assumptions. Closed form expressions for the pressure gradient, pressurerise, stream function, axial velocity, and shear stress on the channel walls have been computed numerically. Effects of the Hartmannnumber, the ratio of relaxation to retardation times, time-mean flow, the phase angle and the gravity field on the pressure gradient,pressure rise, streamline, axial velocity, and shear stress are discussed in detail and shown graphically. The results indicate thatthe effect of Hartmann number, ratio of relaxation to retardation times, time-mean flow, phase angle, and gravity field are verypronounced in the peristaltic transport phenomena. Comparison was made with the results obtained in the presence and absenceof magnetic field and gravity field.

1. Introduction

Peristaltic pumping has been the object of scientific andengineering research in recent years. The word peristalticcomes from a Greek word “Peristaltikos” which means clasp-ing and compressing. The peristaltic transport is travellingcontraction wave along a tube-like structure, and it resultsphysiologically from neuron-muscular properties of anytubular smooth muscle. Peristaltic motion of blood (or otherfluid) in animal or human bodies have been considered bymany authors. It is an important mechanism for transportingblood, where the cross section of the artery is contractedor expanded periodically by the propagation of progressivewave. It plays an indispensable role in transporting manyphysiological fluids in the body in various situations such asurine transport from the kidney to the bladder through theureter, transport of spermatozoa in the ducts efferents of themale reproductive tract, and the movement of the ovum inthe flipping tubes.

A variety of complex theological fluids can easily betransported from one place to another with a special type

of pumping known as peristaltic pumping. This pump-ing principle is called peristalsis. The mechanism includesinvoluntary periodic contraction followed by relaxation orexpansion of the ducts that the fluidsmove through; this leadsto the rise of pressure gradient that eventually pushes the fluidforward.

The study of the peristaltic transport of a fluid in thepresence of an external magnetic field and rotation is of greatimportance with regard to certain problems involving themovement of conductive physiological fluids, for example,blood and saline water. Pandey and Chaube [1] investigatedan analytical study of the MHD flow of a micropolar fluidthrough a porous medium induced by sinusoidal peristalticwaves traveling down the channel wall. The magnetohydro-dynamic flow of a micropolar fluid in a circular cylindricaltube has been investigated by Wang et al. [2]. Nadeem andAkram [3] studied the analytical and numerical treatmentof peristaltic flows in viscous and non-Newtonian fluids.Vajravelu et al. [4] studied the influence of heat transfer onperistaltic transport of a Jeffrey fluid in a vertical porousstratum. Hayat et al. [5] discussed the influence of compliant

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 896121, 11 pageshttp://dx.doi.org/10.1155/2014/896121

2 Abstract and Applied Analysis

wall properties and heat transfer on the peristaltic flow of anincompressible viscous fluid in a curved channel. Bhargavaet al. [6] have studied finite element study of nonlinear two-dimensional deoxygenated biomagnetic micropolar flow. Aliet al. [7] discussed the peristaltic motion of a non-Newtonianfluid in a channel having compliant boundaries. Abd-Allaet al. [8] studied the effects of rotation and magnetic fieldon nonlinear peristaltic flow of second-order fluid in anasymmetric channel through a porous medium. Hayat et al.[9] analyzed the effect of an induced magnetic field on theperistaltic flow of an incompressible Carreau fluid in anasymmetric channel. Pandey et al. [10] were concerned withthe theoretical study of two-dimensional peristaltic flow ofpower-law fluids in three layers with different viscosities.Jimenez-Lozano and Sen [11] investigated the streamlinepatterns and their local and global bifurcations in a two-dimensional planar and axisymmetric peristaltic flow for anincompressible Newtonian fluid. Hayat et al. [12] analyzedthe effect of an induced magnetic field on the peristalticflow of an incompressible Carreau fluid in an asymmetricchannel. Srinivas and Kothandapani [13] investigated theeffects of heat and mass transfer on peristaltic transport ina porous space with compliant walls. Gad [14] discussedthe effect of hall currents on the interaction of pulsatileand peristaltic transport induced flows of a particle fluidsuspension. Abd-Alla et al. [15] investigated the peristalticflow in a tubewith an endoscope subjected to amagnetic field.Hayat and Noreen [16] discussed the influence of an inducedmagnetic field on the peristaltic flow of an incompressiblefourth grade fluid in a symmetric channel with heat transfer.Nadeem and Akbar [17] investigated the peristaltic flow of anincompressible MHD Newtonian fluid in a vertical annulus.Nadeem and Akbar [18] investigated the deals with theperistaltic motion of an incompressible non-Newtonian fluidin a nonuniform tube for long wavelength. Nadeem et al. [19]studied the concentrates on the heat transfer characteristicsand endoscope effects for the peristaltic flow of a third-order fluid. Srinivas et al. [20] studied the effects of bothwall slip conditions and heat transfer on the peristaltic flowof MHD Newtonian fluid in a porous channel with elasticwall properties. Abd-Alla et al. [21] investigated the effect ofthe rotation, magnetic field, and initial stress on peristalticmotion of micropolar fluid. Mahmoud et al. [22] discussedthe effect of the rotation on wave motion through cylindricalbore in a micropolar porous medium.The dynamic behaviorof a wet long bone that has been modeled as a piezoelectrichollow cylinder of crystal class 6 is investigated by Abd-Allaet al. [23]. Akram and Nadeem [24] discussed the peristalticmotion of a two-dimensional Jeffrey fluid in an asymmetricchannel under the effects of the induced magnetic field andheat transfer.

Recently, Rathod and Mahadev [25] investigated theeffect of heat transfer and magnetic field on the peristaltictransport of Jeffrey fluid in an inclined channel taking slipcondition into account. S. N. Reddy and G. V. Reddy [26]studied the effect of sleep on the peristaltic flow of a Jeffreyfluid through a porous medium in an asymmetric channelunder the assumptions of long wavelength and low Reynoldsnumber.

The aim of this paper is to study the effect of magneticfield and gravity field on peristaltic motion of Jeffrey typeand is electrically conducting in asymmetric channel. Herethe governing equations are nonlinear in nature; we usedinfinitely long wavelength assumption to obtain linearizedsystem of coupled differential equations which are thensolved analytically. Results have been discussed for pres-sure gradient, pressure rise, streamline and axial velocityto observe the Hartmann number, the ratio of relaxationto retardation times, time-mean flow, the phase angle, andthe gravity field effect. The numerical result displayed byfigures and the physical meaning is explained.The results anddiscussions presented in this study may be helpful to furtherunderstand MHD peristaltic motion for non-Newtonianfluids in an asymmetric channel.

2. Formulation of the Problem

Let us consider the peristaltic transport of an incompressibleviscous fluid in a two-dimensional channel of width 𝑑

1+ 𝑑2.

The channel walls are inclined at angles 𝛼.The flow is inducedby sinusoidal wave trains propagating with constant speed 𝑐

along the channel walls.The geometry of the wall surfaces is

ℎ1(𝑋, 𝑡) = 𝑑

1+ 𝑎1cos [2𝜋

𝜆(𝑋 − 𝑐𝑡] at upper wall,

ℎ2(��, 𝑡) = −𝑑

2− 𝑏1cos [2𝜋

𝜆(�� − 𝑐 𝑡) + 𝜙] at lower wall,

(1)

where 𝑎1and 𝑏1are the types of amplitude of the waves, 𝜆 is

thewavelength, 𝑐 is thewave speed,𝜙 (0 ≤ 𝜙 ≤ 𝜋) is the phasedifference, 𝜙 = 0 corresponds to symmetric channel withwaves out of phase, and 𝜙 = 𝜋 indicates that the waves arein phase, and further 𝑎

1, 𝑏1, 𝑑1, 𝑑2, and 𝜙 satisfy the condition

𝑎2

1+ 𝑏2

1+ 2𝑎1𝑏1cos𝜙 ≼ (𝑑

1+ 𝑑2)2

. (2)

TheCauchy (𝑇) and extra (𝑆) stress tensors take the followingform

�� = −�� 𝐼 + 𝑆,

𝑆 =𝜇

1 + 𝜆1

( �� + 𝜆2��) ,

(3)

where 𝑝 is the pressure, 𝐼 is the identity tensor, 𝜆1is the ratio

of relaxation to retardation times, 𝜆2is the retardation time,

and 𝛾 is the shear rate.

Abstract and Applied Analysis 3

In laboratory frame, the following set of pertinent fieldequations governing the flow are

𝜌 [𝜕

𝜕𝑡+ ��

𝜕

𝜕��+ ��

𝜕

𝜕��] �� = −

𝜕��

𝜕��+

𝜕

𝜕��( 𝑆��

)

+𝜕

𝜕��( 𝑆��

)−𝜎𝐵2

0��+𝜌𝑔 sin𝛼,

𝜌 [𝜕

𝜕𝑡+ ��

𝜕

𝜕��+ ��

𝜕

𝜕��]𝑉 = −

𝜕��

𝜕��+

𝜕

𝜕��( 𝑆��

)

+𝜕

𝜕��( 𝑆��𝑌) − 𝜌𝑔 cos𝛼

(4)

and the continuity equation takes the form

𝜕��

𝜕��+𝜕��

𝜕��= 0, (5)

where𝑈 and𝑉 are the velocity components in the laboratoryframe (𝑋, 𝑌), 𝜌 is the density, 𝑔 is the earth gravity accel-eration, 𝐵

0is the magnetic induction, and 𝜎 is the electrical

conductivity of the fluid.The flow is inherently unsteady in the laboratory frame

(𝑋,𝑌). However, the flow becomes steady in a wave frame(𝑥, 𝑦) moving away from the laboratory frame with speed𝑐 in the direction of propagation of the wave. Taking 𝑢

and V the velocity components in 𝑥 and 𝑦 directions, thetransformation from the laboratory frame to the wave frameis given by

𝑥 = 𝑋 − 𝑐𝑡, 𝑦 = 𝑌, 𝑢 = 𝑈 − 𝑐,

V = 𝑉, 𝑃 (𝑥) = 𝑃 (𝑋, 𝑡) ,

(6)

where 𝑢 and V are the velocity components in the wave frame(𝑥, 𝑦), 𝑝 and 𝑃 are pressure in wave and fixed frame ofreferences, respectively.

The appropriate nondimensional variables for the flow aredefined as

𝑥 =2𝜋𝑥

𝜆, 𝑦 =

𝑦

𝑑1

, 𝑢 =𝑢

𝑐,

V =V𝑐𝛿

, 𝛿 =2𝜋𝑑1

𝜆,

𝑝 =2𝜋𝑑2

1𝑝

𝜇𝑐𝜆, 𝑡 =

2𝜋𝑐𝑡

𝜆, ℎ1=

ℎ1

𝑑1

,

ℎ2=

ℎ2

𝑑1

, 𝑆 =𝑑1𝑆

𝜇𝑐,

𝑑 =𝑑2

𝑑1

, 𝑎 =𝑎1

𝑎2

, 𝑏 =𝑏1

𝑏2

, Ω =Ω

𝜇𝑎2

1

(7a)

and the stream function is

𝑢 =𝜕𝜓

𝜕𝑦, V = −𝛿

𝜕𝜓

𝜕𝑥. (7b)

Using (6), (7a), and (7b) into (3)-(4) and eliminating pressureby cross differentiation, we get

𝛿Re [(𝜕𝜓

𝜕𝑦

𝜕

𝜕𝑥−𝜕𝜓

𝜕𝑥

𝜕

𝜕𝑦)∇2𝜓] = [(

𝜕2

𝜕y2− 𝛿2 𝜕2

𝜕𝑥2) 𝑠𝑥𝑦]

+ 𝛿[𝜕2

𝜕𝑥𝜕𝑦(𝑠𝑥𝑥

− 𝑠𝑥𝑦)]

+𝑀2 𝜕2𝜓

𝜕𝑦2+ Re

𝑔𝑑1

𝑐2sin𝛼

(8)

in which

𝑠𝑥𝑥

=2𝛿

1 + 𝜆1

[1 +𝛿𝜆2𝑐

𝑑1

(𝜕𝜓

𝜕𝑦

𝜕

𝜕𝑥+𝜕𝜓

𝜕𝑥

𝜕

𝜕𝑦)]

𝜕2𝜓

𝜕𝑥𝜕𝑦,

𝑠𝑥𝑦

=1

1 + 𝜆1

[1 +𝛿𝜆2𝑐

𝑑1

(𝜕𝜓

𝜕𝑦

𝜕

𝜕𝑥−𝜕𝜓

𝜕𝑥

𝜕

𝜕𝑦)]

× (𝜕2𝜓

𝜕𝑦2− 𝛿2 𝜕2𝜓

𝜕𝑥2) ,

(9)

𝑠𝑦𝑦

= −2𝛿

1 + 𝜆1

[1 +𝛿𝜆2𝑐

𝑑1

(𝜕𝜓

𝜕𝑦

𝜕

𝜕𝑥−𝜕𝜓

𝜕𝑥

𝜕

𝜕𝑦)]

𝜕2𝜓

𝜕𝑥𝜕𝑦, (10)

where

∇2= 𝛿2 𝜕2𝜓

𝜕𝑥2+𝜕2𝜓

𝜕𝑦2, 𝛿 =

2𝜋𝑑1

𝜆,

Re =𝜌𝑐𝑑1

𝜇is the Reynolds number,

𝑀 = √𝜎

𝜇𝐵0𝑑1is the Hartmann number.

(11)

Using the long wavelength approximation and neglecting thewave number 𝛿 along with low Reynolds number in ouranalysis, the field equations (8) and (10) now give

𝜕2

𝜕𝑦2[

1

1 + 𝜆1

𝜕2𝜓

𝜕𝑦2] +𝑀

2 𝜕2𝜓

𝜕𝑦2+ Re

𝑔𝑑1

𝑐2sin𝛼 = 0. (12)

The boundary conditions for the stream functions in thewaveframe are

𝜓 =𝑞

2, ℎ

1= 1 + 𝑎 cos [𝑥] ,

𝜓 = −𝑞

2at𝑦 = ℎ

2(𝑥) = −𝑑 − 𝑏 cos [2𝜋

𝜆(𝑥) + 𝜙] ,

𝜕𝜓

𝜕𝑦= −1 at𝑦 = ℎ

1, 𝑦 = ℎ

2,

(13)

where 𝑞 is the flux in the wave frame and 𝑎, 𝑏, 𝜙, and 𝑑 satisfythe relation

𝑎2+ 𝑏2+ 2𝑎𝑏 cos𝜙 ≤ (1 + 𝑑)

2. (14)


0

2

4

6

8

10

12

14

0

5

10

15

20

0 1 2 3 4 5 6 0 1 2 3 4 5 6

AsymmetricAsymmetric

M = 0.1, 0.5, 1, 1

x x

𝜆1 = 0.2, 0.8, 2, 4

dp/d

x

dp/d

x

Figure 1: Variation of 𝑑𝑝/𝑑𝑥 with influence of𝑀 and 𝜆1respect to 𝑥.

0

6

12

18

0

5

10

15

20

25

0 1 2 3 4 5 6 0 1 2 3 4 5 6


b = 0.3, 0.5, 0.7, 0.9

xx

dp/d

x

dp/d

x

𝜙 = 0, 𝜋/6, 𝜋/3, 𝜋/2

Figure 2: Variation of 𝑑𝑝/𝑑𝑥 with influence of 𝑏 and 𝜙 respect to 𝑥.

3. Solution of the Problem

The solution of (12) satisfying the corresponding boundaryconditions (13) is

𝜓 =(ℎ1+ ℎ2) [𝑁𝑞 + 2 tanh [𝑁 (ℎ

1− ℎ2) /2]]

2𝑁 (ℎ2− ℎ1) + 4 tanh [𝑁 (ℎ

1− ℎ2) /2]

+𝑁𝑞 + 2 tanh [𝑁 (ℎ

1− ℎ2) /2]

𝑁 (ℎ1− ℎ2) − 2 tanh [𝑁 (ℎ

1− ℎ2) /2]

𝑦

+ ((𝑞 + ℎ1− ℎ2) sech[

𝑁 (ℎ1− ℎ2)

2]

× sinh [𝑁 (ℎ1− ℎ2)

2])

× (𝑁(ℎ1− ℎ2) − 2 tanh[

𝑁 (ℎ1− ℎ2)

2])

−1

× cosh (𝑁𝑦)


0

3

6

9

12

15

0

3

6

9


Θ = −4, −3, −2, −1

g = 1, 5, 10, 20

xx

0 1 2 3 4 5 60 1 2 3 4 5 6

dp/d

x

dp/d

x

Figure 3: Variation of 𝑑𝑝/𝑑𝑥 with influence of Θ and 𝑔 respect to 𝑥.

+ ((𝑞 + ℎ1− ℎ2) sech[

𝑁 (ℎ1− ℎ2)

2]

× cosh [𝑁 (ℎ1− ℎ2)

2])

× (𝑁(ℎ1− ℎ2) + 2 tanh[

𝑁 (ℎ1− ℎ2)

2])

−1

× sinh (𝑁𝑦) ,(15)

where𝑁2 = 𝑀2(1 + 𝜆

1) + Re(𝑔𝑑

1/𝑐2) sin𝛼.

The flux at any axial station in the fixed frame is

𝑄 = ∫

ℎ1

ℎ2

(𝜕𝜓

𝜕𝑦+ 1)𝑑𝑦 = ℎ

1− ℎ2+ 𝑞. (16)

The time-mean flow over a period 𝑇 at a fixed position 𝑋 isdefined as

𝜃 =1

𝑇∫

𝑇

0

𝑄𝑑𝑡 =1

𝑇∫

𝑇

0

(𝑞 + ℎ1− ℎ2) 𝑑𝑡

= 𝑞 + 1 + 𝑑.

(17)

The pressure gradient is obtained from the dimensionlessmomentum equation for the axial velocity as

𝑑𝑝

𝑑𝑥=

1

1 + 𝜆1

[𝑁3(𝑞 + ℎ

1− ℎ2)

𝑁 (ℎ2− ℎ1) + 2 tanh [𝑁 (ℎ

1− ℎ2) /2]

]

+ Re𝑔𝑑1

𝑐2sin𝛼.

(18)

The nondimensional expression for the pressure rise perwavelength Δ𝑝

𝜆and frictional forces on the lower (𝐹𝑙

𝜆) and

upper (𝐹𝑢𝜆) walls on the lower are defined as follows:

Δ𝑝𝜆= ∫

2𝜋

0

(𝑑𝑝

𝑑𝑥)𝑑𝑥,

𝐹𝑙

𝜆= ∫

1

0

ℎ2

2(−

𝑑𝑝

𝑑𝑥)𝑑𝑥,

𝐹𝑢

𝜆= ∫

1

0

ℎ2

1(−

𝑑𝑝

𝑑𝑥)𝑑𝑥.

(19)

The nondimensional expression of the shear stress at theupper wall of the channel is reduced to

𝑆𝑥𝑦

=1

1 + 𝜆1

𝜕2𝜓

𝜕𝑦2

=𝑁2(𝑞 + ℎ

1− ℎ2) [𝑁 (ℎ

1− ℎ2) /2]

(1 + 𝜆1) {𝑁 (ℎ

1− ℎ2) − 2 tanh [𝑁 (ℎ

1− ℎ2) /2]}

× {sinh[𝑁 (ℎ1− ℎ2)

2] cosh𝑀ℎ

1

− cosh[𝑁 (ℎ1− ℎ2)

2] sinh𝑀ℎ

1} .

(20)

4. Numerical Results and Discussion

In order to gain physical insight into the pressure gradient,pressure rise Δ𝑝

𝜆, streamline 𝜓, velocity 𝑢, and shear stress

𝑆𝑥𝑦

have been discussed by assigning numerical valuesto the parameter encountered in the problem in which


0

5

0 0.2 0.4 0.6 0.8 1

−5

−10

−15

Asymmetric

ΔP𝜆

𝜆1 = 0, 0.5, 1, 3

Θ

12

40

0 0.2 0.4 0.6 0.8 1

−16

−44

−72

−100

AsymmetricΔP𝜆

M = 1, 2, 4, 6

Θ

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

−5

−10

−15

Asymmetric

ΔP𝜆

b = 0.3, 0.5, 0.7, 0.9

Θ

0 0.2 0.4 0.6 0.8 1

0

5

10

−5

−10

−15

Asymmetric

ΔP𝜆

Θ

𝜙 = 0, 𝜋/6, 𝜋/3, 𝜋/2

Figure 4: Variation of Δ𝑃𝜆with influence of𝑀, 𝜆

1, 𝜙, and 𝑏 with respect to Θ.

the numerical results are displayed with the graphic illustra-tions. The variations are shown in Figures 1–7, respectively.

Figures 1, 2, and 3 show the variations of the axialpressure gradient 𝑑𝑝/𝑑𝑥 with respect to the axial 𝑥 whichit has oscillatory behavior in the whole range of the 𝑥-axisfor different values of the Hartmann number 𝑀, the ratioof relaxation to retardation times 𝜆

1, time-mean flow Θ,

the phase angle 𝜙, gravity field 𝑔, and the nondimensionalamplitude of wave 𝑏 in asymmetric channel. In both figures,it is clear that the pressure gradient has a nonzero value onlyin a bounded region of space. The effect of the Hartmannnumber, the ratio of relaxation to retardation times, time-mean flow, rotation, gravity field, and the nondimensional

amplitude of wave decreases and increases gradually. It isobserved that the pressure gradient increases with increasingof the Hartmann number, gravity field, the phase angle,and the nondimensional amplitude of the wave, while itdecreases with increasing of the time-mean flow and the ratioof relaxation to retardation times. It is noticed that the axialpressure gradient when compared to the case of asymmetricand symmetric channel takes a large values respect to thesmall values of 𝑥 and small values with an increasing of 𝑥.Moreover, it can be noticed that, on the one hand, in thewider part of the channel 𝑥 ∈ [0, 2] and [3, 5, 6], the pressuregradient is relatively small; that is, the flow can easily passwithout imposition of a large pressure gradient. On the other


0

2

4

6

8

0 1 2 3 4 5 6

−2

−4

Asymmetric

𝜓

x

M = 0, 0.5, 1, 1.5

20

60

100

0 1 2 3 4 5 6−20

Asymmetric

𝜓

x

𝜆1 = 0.2, 0.8, 2, 4

0

1

0 1 2 3 4 5 6

−2

−1

−4

−3Asymmetric

𝜓

x

b = 0.3, 0.5, 0.7, 0.9

5

15

25

0 1 2 3 4 5 6−5

Asymmetric

𝜓

x

0

1

2

0 1 2 3 4 5 6

−2

−4

−1

−3

Asymmetric

𝜓

x

Θ = −4, −3, −2, −1

𝜙 = 0, 𝜋/6, 𝜋/3, 𝜋/2

Figure 5: Variation of 𝜓 with influence of𝑀, 𝜆1, 𝑏, 𝜙, and Θ with respect to 𝑥.


−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−2 −1 0 1

Asymmetric

𝜆1 = 0.2, 0.8, 2, 4u

y

−1

−0.75

−0.5

−0.25

0

−1 0 1

Asymmetric

M = 0, 1, 2, 3

u

y

−1.5 −1 −0.5 0 0.5 1−1

−0.75

−0.5

−0.25

0

Asymmetric

Θ = −4, −3, −2, −1

u

y

−2 −1 0 1−1

−0.8

−0.6

−0.4

−0.2

0

Asymmetric

u

y

𝜙 = 𝜋/4, 𝜋/3, 𝜋/2, 3𝜋/5

Figure 6: Variation of 𝑢 with influence of𝑀, 𝜆1, Θ, and 𝜙 with respect to 𝑦.

hand, in a narrow part of the channel 𝑥 ∈ [2, 3.5], a muchpressure gradient is required to maintain the flux to pass itespecially near 𝑥 = 2.7.

Figure 4 shows the variations of the pressure riseΔ𝑝𝜆with

respect to the time-mean flow Θ for different values of theHartmann number 𝑀, the ratio of relaxation to retardationtimes 𝜆

1, the phase difference 𝜙, and the nondimensional

amplitude of wave 𝑏. In both figures, it is clear that thepressure rise has a nonzero value only in a bounded regionof space. It is observed that the pressure rise increases withincreasing of theHartmann number and the nondimensionalamplitude of the wave, while it decreases with increasing

the time-mean flow, the ratio of relaxation to retardationtimes, and the phase difference. The graph is sectored so thatthe upper right-hand quadrant (I) denotes the region of theperistaltic pumping (Θ > 0, Δ𝑝

𝜆> 0). Quadrant (II) is

designated as an augmented flow when Θ > 0, Δ𝑝𝜆

< 0.Quadrant (IV) such that Θ < 0, Δ𝑝

𝜆> 0 is called retrograde

or backward pumping.Figure 5 shows the variations of the streamlines 𝜓 with

respect to the axial 𝑥 which has oscillatory behavior inthe whole range of the 𝑥-axis for different values of theHartmann number 𝑀, the ratio of relaxation to retardationtimes 𝜆

1, time-mean flow Θ, the phase difference 𝜙, and


0

2

4

6

8

0 1 2 3 4 5 6

S xy

x

0

1

2

3

4

5

0 1 2 3 4 5 6

S xy

x

0

2

4

6

8

0 1 2 3 4 5 6

S xy

x

0

2

4

6

8

10

0 1 2 3 4 5 6

S xy

x

0

2

4

6

8

0 1 2 3 4 5 6−2

S xy

x

Asymmetric Asymmetric

Asymmetric Asymmetric

Asymmetric

M = 0, 0.5, 1, 1.5𝜆1 = 0.2, 0.8, 2, 4

g = 0, 5, 10, 20

Θ = −4, −3, −2, −1

𝜙 = 𝜋/4, 𝜋/3, 𝜋/2, 3𝜋/5

Figure 7: Variation of 𝑆𝑥𝑦

with influence of𝑀, 𝜆1, Θ, 𝑔, and 𝜙 with respect to 𝑥.


the nondimensional amplitude of wave 𝑏 in asymmetricchannel. In both figures, it is clear that the streamlines havea nonzero value only in a bounded region of space. Theeffect of the Hartmann number, the ratio of relaxation toretardation times, time-mean flow, the phase angle, and thenondimensional amplitude of wave decreases and increasesgradually. It is observed that the streamlines increase withthe increasing of the phase angle, the Hartmann number, thetime-mean flow, and the ratio of relaxation to retardationtimes. The streamlines near the channel walls do nearlystrictly follow the wall waves, which are mainly engenderedby the relative movement of the walls.

Figure 6 shows the variations of the axial velocity 𝑢 withrespect to the axial 𝑦 which has oscillatory behavior in thewhole range of the𝑦-axis for different values of theHartmannnumber 𝑀, the ratio of relaxation to retardation times 𝜆

1,

time-mean flow Θ, and the phase angle 𝜙 in asymmetricchannel. In both figures, it is clear that the axial velocityhas a nonzero value only in a bounded region of space.The effect of the Hartmann number, the ratio of relaxationto retardation times, time-mean flow, and the phase angledecreases and increases gradually. It is observed that the axialvelocity increases with increasing of the time-mean flow, theHartmann number, and the ratio of relaxation to retardation,while it decreases with increasing of the phase angle.

Figure 7 displays that the variations of the value of axialshear stress 𝑆

𝑥𝑦with respect to the axial 𝑥 has oscillatory

behavior may be due to peristalsis in the whole range ofthe 𝑥-axis for different values of the Hartmann number𝑀, the ratio of relaxation to retardation times 𝜆

1, time-

mean flow Θ, the phase angle 𝜙, and the gravity field 𝑔

in asymmetric channel. In both figures, it is clear that thevalue of shear stress has a nonzero value only in a boundedregion of space. The effect of the Hartmann number, theratio of relaxation to retardation times, time-mean flow,rotation, the phase difference, and the phase angle decreasesand increases gradually. It is observed that the shear stressincreases with increasing of the Hartmann number, gravityfield, and the phase angle, while it decreases with increasingof the ratio of relaxation to retardation times and time-meanflow. Moreover the values of shear stress are larger in case ofa Jeffery fluid when compared with Newtonian fluid.

5. Conclusion

Due to the complicated nature of the governing equationsof the pertinent field equations governing the peristaltictransport of Jeffery fluid, the work done in this field isunfortunately limited in number. The method used in thisstudy is quite successful in dealing with such problems.This method gives exact solutions in the peristaltic transportwithout any assumed restrictions on the actual physicalquantities that appear in the governing equations of theproblem considered. Important phenomena are observed inall these computations.

(i) It was found that, for large values of the Hartmannnumber𝑀, the ratio of relaxation to retardation times𝜆1, time-mean flow Θ, the phase angle 𝜙, and the

gravity field𝑔 in asymmetric channel, the solution hasbeen obtained in the context of the peristaltic trans-port of fluid.

(ii) By comparing Figures 1–7 for the peristaltic transportof fluid with figures without magnetic field andgravity field, it was found that it has the same behaviorin the same field.

(iii) The results presented in this paper should proveuseful for researchers in science and engineering,as well as for those working on the developmentof fluid mechanics. The study of the phenomenonof the Hartmann number, the ratio of relaxation toretardation times, time-mean flow, the phase angle,and the gravity field in asymmetric channel influenceand operations is also used to improve the conditionsof peristaltic motion.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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