Alexandria Engineering Journal (2016) 55, 1725–1735

HO ST E D BY

Alexandria University

Alexandria Engineering Journal

www.elsevier.com/locate/aejwww.sciencedirect.com

ORIGINAL ARTICLE

Rotation effect on peristaltic transport of a Jeffrey

fluid in an asymmetric channel with gravity field

* Corresponding author at: Math. Dept., Faculty of Science, Taif

University, 888, Saudi Arabia.E-mail addresses: mohmrr@yahoo.com (A.M. Abd-Alla), sdahb@

yahoo.com (S.M. Abo-Dahab).

Peer review under responsibility of Faculty of Engineering, Alexandria

University.

http://dx.doi.org/10.1016/j.aej.2016.03.0181110-0168 � 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

A.M. Abd-Alla a,c,*, S.M. Abo-Dahab b,c

aMath. Dept., Faculty of Science, Sohag University, EgyptbMath. Dept., Faculty of Science, SVU, Qena 83523, EgyptcMath. Dept., Faculty of Science, Taif University, 888, Saudi Arabia

Received 11 March 2014; revised 29 January 2016; accepted 15 March 2016Available online 28 April 2016

KEYWORDS

Jeffrey fluid;

Rotation;

Peristaltic;

Asymmetric channel;

Gravity field

Abstract In this paper, the peristaltic flow of a Jeffrey fluid in an asymmetric rotating channel is

studied under long wavelength and low Reynolds number assumptions are investigated. Closed

form expressions for the pressure gradient, pressure rise, stream function, axial velocity and shear

stress on the channel walls have been computed numerically. The effects of the ratio of relaxation to

retardation times, time-mean flow, rotation, 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 that the effect of the ratio of relaxation to retardation times, time-

mean flow, rotation, the phase angle and the gravitational field are very pronounced in the phenom-

ena. Comparison was made with the results obtained in the asymmetric channel and symmetric

channel.� 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an

open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

A variety of complex rheological fluids can easily be trans-ported from one place to another place with a special type of

pumping known as Peristaltic pumping. This pumping princi-ple is called peristalsis. The mechanism includes involuntaryperiodic contraction followed by relaxation or expansion of

the ducts the fluids move through. This leads to the rise in pres-sure gradient that eventually pushes the fluid forward. This

type of pumping is first observed in physiology where the foodmoves through the digestive tract, urine transports from thekidney to the bladder through ureters, semen moves through

the vas deferens, lymphatic fluids moves through lymphaticvessels, bile flows from the gall bladder into the duodenum,spermatozoa move through the ductus efferentes of the malereproductive tract and cervical canal, ovum moves through

the fallopian tube, and blood circulates in small blood vessels.Historically, however, the engineering analysis of peristalsiswas initiated much later than in physiological studies.

Applications in industrial fluid mechanics are like aggressivechemicals, high solid slurries, noxious fluid (nuclear industries)and other materials that are transported by peristaltic pumps.

Roller pumps, hose pumps, tube pumps, finger pumps, heart–lung machines, blood pump machines, and dialysis machinesare engineered on the basis of peristalsis. The study of the

1726 A.M. Abd-Alla, S.M. Abo-Dahab

peristaltic transport of a fluid in the presence of an externalmagnetic field and rotation is of great importance with regardto certain problems involving the movement of conductive

physiological fluids, e.g. blood and saline water. Abd-Allaet al. [1] investigated the effect of rotation on peristaltic flowof a micropolar fluid through a porous medium with an exter-

nal magnetic field. Akram et al. [2] studied the numerical andanalytical treatment on peristaltic flow of Williamson fluid inthe occurrence of induced magnetic. Akram and Nadeem [3]

discussed the influence of induced magnetic field and heattransfer on the peristaltic motion of a Jeffrey fluid in an asym-metric channel: Closed form solution field. Abd-Alla et al. [4]studied the effects of rotation and initial stress on peristaltic

transport of fourth grade fluid with heat transfer and inducedmagnetic field. Hayat et al. [5] discussed the influence of com-pliant wall properties and heat transfer on the peristaltic flow

of an incompressible viscous fluid in an curved channel.Bhargava et al. [6] have studied finite element study of nonlin-ear two-dimensional deoxygenated biomagnetic micropolar

flow. Ali et al. [7] discussed the peristaltic motion of a non-Newtonian fluid in a channel having compliant boundaries.Abd-Alla et al. [8] studied the effects of rotation and magnetic

field on 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 the peri-staltic flow of an incompressible Carreau fluid in an asymmet-

ric channel. Pandey et al. [10] concerned with the theoreticalstudy of two-dimensional peristaltic flow of power-law fluidsin three layers with different viscosities. Jimenez-Lozano and

Sen [11] investigated the streamline patterns and their localand global bifurcations in a two-dimensional planar andaxisymmetric peristaltic flow for an incompressible Newtonian

fluid. Hayat et al. [12] analyzed the effect of an induced mag-netic field on the peristaltic flow of an incompressible Carreaufluid in an asymmetric channel. Srinivas and Kothandapani

[13] investigated the effects of heat and mass transfer onperistaltic transport in a porous space with compliant walls.Abd-Alla et al. [14] discussed the Effects of an endoscopeand rotation on peristaltic flow in a tube with long wavelength.

Abd-Alla et al. [15] investigated the peristaltic flow in a tubewith an endoscope subjected to magnetic field. Hayat andNoreen [16] discussed the influence of an induced magnetic

field on the peristaltic flow of an incompressible fourth gradefluid in a symmetric channel with heat transfer. Nadeem andAkbar [17] investigated the peristaltic flow of an incompress-

ible MHD Newtonian fluid in a vertical annulus. Abd-Allaet al. [18] investigated the peristaltic flow in cylindrical tubeswith an endoscope subjected to effect of rotation and magneticfield. Nadeem et al. [19] studied the concentrates on the heat

transfer characteristics and endoscope effects for the peristalticflow of a third order fluid. Abd-Alla and Abo-Dahab [20] stud-ied the magnetic field and rotation effects on peristaltic trans-

port of a Jeffrey fluid in an asymmetric channel. Abd-Allaet al. [21] investigated the effect of the rotation, magnetic fieldand initial stress on peristaltic motion of micropolar fluid.

Mahmoud et al. [22] discussed the effect of the rotation onwave motion through cylindrical bore in a micropolar porousmedium. The dynamic behavior of a wet long bone that has

been modeled as a piezoelectric hollow cylinder of crystal class6 is investigated by Abd-Alla et al. [23]. Vajravelu et al. [27]investigated the peristaltic transport of a conducting Jeffreyfluid in an inclined asymmetric channel. The extensive

literature on the topic is now available and we can only men-tion a few recent interesting investigations in Refs. [28–38].

The aim of this paper was to study the effect of rotation and

gravity field on the peristaltic transport of Jeffrey type in asym-metric channel. Here the governing equations are nonlinear innature; we used infinitely long wavelength assumption to

obtain linear zed system of coupled differential equationswhich are then solved analytically. Results have been discussedfor pressure gradient, pressure rise, streamline and axial veloc-

ity to observe the ratio of relaxation to retardation times, time-mean flow, rotation, the phase angle and the gravity fieldeffect. The numerical result displayed by figures and thephysical meaning is explained. The results and discussions pre-

sented in this study may be helpful to further understand peri-staltic motion for non-Newtonian fluids in an asymmetricchannel and a symmetric channel.

2. Formulation of the problem

Let us consider the peristaltic transport of an incompressible

viscous fluid in a two-dimensional channel of width d1 + d2.The channel walls are inclined at angles a. The flow is inducedby sinusoidal wave trains propagating with constant speed c

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

�h1ðX; �tÞ ¼ d1 þ a1 cos2pkðX� c�tÞ

� �; at upper wall; ð1aÞ

�h2ðX; �tÞ ¼ �d2 � b1 cos2pkðX� c�tÞ þ /

� �; at lower wall;

ð1bÞ

where a1 and b1 are the amplitudes of the waves, k is the wave-length, c is the wave speed, / ð0 6 / 6 pÞ is the phase differ-

ence, / = 0 corresponds to symmetric channel with waves outof phase and / = p the waves are in phase, and further a1, b1,d1, d2 and / satisfy the condition.

a21 þ b21 þ 2a1b1 cos / ^ ðd1 þ d2Þ2: ð2ÞThe Cauchy ðTÞ and extra ðSÞ stress tensors areT ¼ ��pIþ S;

S ¼ l1þ k1

_cþ k2€c� �

; ð3Þ

where �p is the pressure, I is the identity tensor, k1 is the ratio ofrelaxation to retardation times, k2 is the retardation time, and_c is the shear rate.

In laboratory frame, the following set of pertinent fieldequations governing the flow is as follows:

q@

@tþU

@

@Xþ V

@

@Y

� �U� 2qXV

¼ � @P

@Xþ @

@XSXX

� �þ @

@YSXY

� �þ q�g sin a; ð4Þ

q@

@tþU

@

@Xþ V

@

@Y

� �Vþ 2qXU

¼ � @P

@Yþ @

@XSXY

� �þ @

@YSYY

� �� q�g cos a ð5Þ

Rotation effects on peristaltic transport of a Jeffrey fluid 1727

@U

@Xþ @V

@Y¼ 0; ð6Þ

P ¼ �p� 1

2qX2R2: ð6aÞ

in which U and V are the velocity components in the X and Y –

directions, respectively, X!¼ Xk, k is the unit vector,

X!¼ ð0; 0;XÞ is the rotation vector, R is given by

R2 ¼ X2 þ Y2, P is the modified pressure, and the equationsof motion in the rotating frame have two additional terms

qðX!^ ðX!^ R!Þ is the centrifugal force, 2qðX!^ V

!Þ is the

Coriolis force, R!¼ ðX;Y; 0Þ, V!¼ ðU;V; 0Þ is the velocity vec-

tor and q is the density.The flow is inherently unsteady in the laboratory frame

X;Y� �

. However, the flow becomes steady in a wave frame

�x; �yð Þ moving away from the laboratory frame with speed cin the direction of propagation of the wave. Taking u and v

the velocity component in x and y-directions, the transforma-tion from the laboratory frame to the wave frame is given by

�x ¼ X� c�t, �y ¼ Y, �u ¼ U� c;

�v ¼ V; �pð�xÞ ¼ PðX; �tÞ; ð7Þwhere �u and �v are the velocity components in the wave frame

�x; �yð Þ, �p and P are pressure in wave and fixed frame of refer-ences, respectively.

The appropriate non-dimensional variables for the flow are

defined as follows:

x ¼ 2p�xk

; y ¼ �y

d1; u ¼ �u

c; v ¼ �v

cd; d ¼ 2pd1

k;

p ¼ 2pd21�plck

; t ¼ 2pc�tk

; h1 ¼�h1d1

; h2 ¼�h2d1

; s ¼ d1lc

�s;

d ¼ d2d1

; a ¼ a1d1

; b ¼ b1d1

; X ¼ Xa21l

:

ð8aÞIntroducing the dimensionless stream function w(x, y) such

that,

u ¼ @w@y

; v ¼ �d@w@x

: ð8bÞ

Using Eqs. (7)–(8b) into Eqs. (3)–(5) and eliminating pres-sure by cross differentiation, we get

dRe@w@y

@

@x� @w

@x

@

@y

� �r2w

� �¼ @2

@y2� d2

@2

@x2

� �sxy

� �

þ d@2

@x@ysxx � sxy� �� �

þ qlXa21

@2w@y2

þ gc2

d1sin a; ð9Þ

in which

sxx ¼ 2d1þ k1

1þ dk2cd1

@w@y

@

@xþ @w

@x

@

@y

� �� �@2w@x@y

; ð10Þ

sxy ¼ 1

1þ k11þ dk2c

d1

@w@y

@

@x� @w

@x

@

@y

� �� �@2w@y2

� d2@2w@x2

� �;

ð11Þ

syy ¼ � 2d1þ k1

1þ dk2cd1

@w@y

@

@x� @w

@x

@

@y

� �� �@2w@x@y

; ð12Þ

r2 ¼ d2@2w@x2

þ @2w@y2

; d ¼ 2pd1k

;

where

Re ¼ qcd1l

is the Reynolds number: ð13Þ

Under the long wavelength approximation i:e:k � að Þ, wehave d? 0 and along with low-Reynolds number in ouranalysis.

In view of these approximations, Eqs. (9) and (11) reduce tothe following:

@2

@y21

ð1þ k1Þ@2w@y2

� �� qa2X

l

� �@2w@y2

þ gd1c2

sin a ¼ 0: ð14Þ

The boundary conditions for the stream functions in the

wave frame are

w ¼ q

2; h1 ¼ 1þ a cos½x�; ð15Þ

w ¼ � q

2at y ¼ h2ðxÞ ¼ �d� b cos

2pkðxÞ þ /

� �; ð16Þ

@w@y

¼ �1 at y ¼ h1 and y ¼ h2; ð17Þ

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

a2 þ b2 þ 2ab cos / 6 ð1þ dÞ2: ð18Þ

3. Solution of the problem

The solutions of Eq. (14) subject to the boundary conditions(15)–(17) are given as

w¼ðh1þh2Þ Nqþ2tanh Nðh1�h2Þ

2

h ih i

2Nðh2�h1Þþ4tanh Nðh1�h2Þ2

h i þNqþ2tanh Nðh1�h2Þ

2

h i

Nðh1�h2Þ�2tanh Nðh1�h2Þ2

h iy

þðqþh1�h2Þsech Nðh1�h2Þ

2

h isinh Nðh1�h2Þ

2

h i

Nðh1�h2Þ�2tanh Nðh1�h2Þ2

h i �coshðNyÞ

þðqþh1�h2Þsech Nðh1�h2Þ

2

h icosh Nðh1�h2Þ

2

h i

Nðh1�h2Þþ2tanh Nðh1�h2Þ2

h i �sinhðNyÞ� gd12N2c2

y2; ð19Þ

where N2 ¼ qa2X2

l ð1þ k1Þ.The flux at any axial station in the fixed frame is

Q ¼Z h1

h2

@w@y

þ 1

� �dy ¼ h1 � h2 þ q: ð20Þ

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

H ¼ 1

T

Z T

0

Q dt ¼ 1

T

Z T

0

ðqþ h1 � h2Þdt ¼ qþ 1þ d: ð21Þ

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

dp

dx¼ 1

1þk1

N3ðqþh1�h2ÞNðh2�h1Þþ2tanh Nðh1�h2Þ

2

h i24

35þqgd21

clsina: ð22Þ

1728 A.M. Abd-Alla, S.M. Abo-Dahab

The non-dimensional expression for the pressure rise per

wavelength Dpk and frictional forces on the lower Flk

� �and

upper Fuk

� �walls on the lower is defined as follows:

Dpk ¼Z 2p

0

dp

dx

� �dx; ð23Þ

Flk ¼

Z 1

0

h22 � dp

dx

� �dx; ð24Þ

g=1, 5,10,20

x

dpdx

dd

Asymmetric _____ Symmetric …….

Figure 1 Variation of dpdx

with influe

x

dpdx

Asymmetric _____ Symmetric …….

4, 3, 2, 1Θ = − − − −

Figure 2 Variation of dpdx

with influen

Fuk ¼

Z 1

0

h21 � dp

dx

� �dx: ð25Þ

The non-dimensional shear stress (11) at the upper wall of

the channel is reduced to

Sxy¼ 1

1þk1

@2w@y2

¼N2ðqþh1�h2Þsech Nðh1�h2Þ

2

h i

ð1þk1Þ Nðh1�h2Þ�2tanh Nðh1�h2Þ2

h in o

� sinhNðh1�h2Þ

2

� �coshMh1

� : ð26Þ

Asymmetric _____ Symmetric …….

x

px

1 0.2,0.8,2,4λ =

nce of g and k1 with respect to x.

x

Asymmetric _____ Symmetric …….

dpdx

0,10,20,30Ω =

ce of H and X with respect to x.

Asymmetric

x

dpdx

dpdx

0, , ,6 3 2π π π

φ =

Asymmetric

0.3,0.5,0.7,0.9b =

x

Figure 3 Variation of dpdx

with influence of b and / for asymmetric with respect to x.

Θ Θ

Asymmetric

PλΔ 1 0,0.5,1,3λ =

Asymmetric

0.3,0.5,0.7,0.9b =

PλΔ

Asymmetric

PλΔ

0,10,20,30Ω =

Asymmetric

0, , ,6 3 2

π π πφ =

Θ Θ

PλΔ

Figure 4 Variation of DPk with influence of M; k1; /; X; b with respect to H.

Rotation effects on peristaltic transport of a Jeffrey fluid 1729

b=0.3,0.5,0.7,0.9 Asymmetric

x

ψ

0, , ,6 3 2π π π

φ =

Asymmetric _____

Symmetric ………….

ψ

x

Asymmetric _____

Symmetric ……..….

x

ψ

1 0.2,0.8,2, 4λ =

Asymmetric _____

Symmetric ………….

ψ

x

1,10,20,30Ω =

Asymmetric _____

Symmetric ………….

ψ

x

4, 3, 2, 1Θ = − − − −

Figure 5 Variation of w with influence of X; k1; b; /; H with respect to x.

1730 A.M. Abd-Alla, S.M. Abo-Dahab

Asymmetric

y

u 1 0.2,0.8,2, 4λ =

Asymmetric

u

y

0.3,0.5,0.7,0.9b =

Asymmetric Asymmetric

u u

yy

4, 3, 2, 1Θ = − − − −

0,10,20,30Ω =

Asymmetric

u u

yy

0.01,0.1,3,5ρ =

3, , ,4 3 2 5π π π π

φ =

Figure 6 Variation of u with influence of b; k1; H; X; q; / with respect to y.

Rotation effects on peristaltic transport of a Jeffrey fluid 1731

Expression for wave shape: we can deduce the asymmetric

channel into symmetric channel by taking a ¼ b; d ¼1 and / ¼ 0.

4. Numerical results and discussion

In order to gain physical insight the pressure gradient dpdx, pres-

sure rise Dpk, streamline w, velocity u, and shear stress Sxy have

Asymmetric

x

xyS

0,10,29,30Ω =

Asymmetric

xyS

x

4, 3, 2, 1Θ = − − − −

xyS

x

3, , ,4 3 2 5π π π πφ =

b=0.3,0.5,0.7,0.9

Asymmetric

x

xyS

1 0.2,0.8, 2,4λ =Asymmetric

xyS

x

Figure 7 Variation of Sxy with influence of b; k1; H; X; / with respect to x.

1732 A.M. Abd-Alla, S.M. Abo-Dahab

Rotation effects on peristaltic transport of a Jeffrey fluid 1733

been discussed by assigning numerical values to the parameterencountered in the problem in which the numerical results aredisplayed with the graphic illustrations. The variations are

shown in Figs. 1–7 respectively.Figs. 1–3 show the variations of the axial pressure gradient

dpdxwith respect to the axial x in which it has oscillatory behav-

ior in the whole range of the x-axis for different values of thegravity field, the ratio of relaxation to retardation times k1;,time-mean flow H, rotation X, the non-dimensional amplitude

of wave b in asymmetric and the phase angle /. In both figures,it is clear that the pressure gradient has a nonzero value only ina bounded region of space. The effect of gravity field, the ratioof relaxation to retardation times, time-mean flow, rotation,

the non-dimensional amplitude of wave and the phase angledecrease and increase gradually. It is observed that the pres-sure gradient increases with increase in gravity field, rotation

and the non-dimensional amplitude of wave, while it decreaseswith increase in the time-mean flow, the ratio of relaxation toretardation times and the phase angle. It is noticed the axial

pressure gradient when compared to the case of asymmetricand symmetric channel. The pressure gradient in case of sym-metric channel exceeds in magnitude when compared withasymmetric channel. Moreover, it can be noticed that on the

one hand, in the wider part of channel x e [0, 2] and [3, 5, 6],the pressure gradient is relatively small, i.e., the flow can easilypass without imposition of a large pressure gradient. On the

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

Fig. 4 shows that the variations of the pressure rise Dpk withrespect to the time mean flowH for different values of the non-dimensional amplitude of wave b, the ratio of relaxation to

retardation times k1, the phase difference / and rotation X.In both figures, it is clear that the pressure rise has a nonzerovalue only in a bounded region of space. It is observed that thepressure rise increases with increase in the non-dimensional

amplitude of wave and rotation, while it decreases withincreasing the ratio of relaxation to retardation times andthe phase difference. The graph is sectored so that the upper

right-hand quadrant (I) denotes the region of the peristalticpumping H > 0; Dpk > 0ð Þ. Quadrant (II) is designated asaugmented flow when H > 0; Dpk < 0. Quadrant (IV) such

that H < 0; Dpk > 0 is called retrograde or backwardpumping.

Fig. 5 shows the variations of the streamlines w with respect

to the axial x in which it has oscillatory behavior in the wholerange of the x-axis for different values of the rotation X, theratio of relaxation to retardation times k1, the non-dimensional amplitude of wave b, the phase angle / and

time-mean flow H in asymmetric and symmetric channel. Inboth figures, it is clear that the streamlines have a nonzerovalue only in a bounded region of space. The effect of rotation,

the ratio of relaxation to retardation times, the non-dimensional amplitude of wave, the phase difference andtime-mean flow decrease and increase gradually.

It is observed that the streamline increases with increase inthe rotation, the non-dimensional amplitude of wave, phaseangle and time-mean flow, while it decreases with increase inthe ratio of relaxation to retardation times. It is noticed the

streamlines when compared to the case of asymmetric andsymmetric channel. The streamlines in case of symmetric

channel exceed in magnitude when compared with asymmetricchannel. The streamlines near the channel walls do nearlystrictly follow the wall waves, which are mainly engendered

by the relative movement of the walls.Fig. 6 shows the variations of the axial velocity u with

respect to the axial y in which it has oscillatory behavior in

the whole range of the y-axis for different values of the non-dimensional amplitude of wave b, ratio of relaxation to retar-dation times k1, time-mean flow H, rotation X, density q and

the phase angle / in asymmetric channel. In both figures, itis clear that the axial velocity have a nonzero value only in abounded region of space. The effect of the non-dimensionalamplitude of wave, the ratio of relaxation to retardation times,

time-mean flow, rotation, the density and the phase angledecrease and increase gradually. It is observed that the axialvelocity increases with increase in the non-dimensional ampli-

tude of wave, ratio of relaxation to retardation times, the time-mean flow and rotation, while it decreases with increase in thedensity and the phase angle.

Fig. 7 displays the variations of the value of axial shearstress Sxy with respect to the axial x in which it has oscillatorybehavior which may be due to peristalsis in the whole range of

the x-axis for different values of the non-dimensional ampli-tude of wave b, the ratio of relaxation to retardation timesk1, time-mean flow H, rotation X and the phase angle / inasymmetric channel. In both figures, it is clear that the value

of shear stress has a nonzero value only in a bounded regionof space. The effect of the non-dimensional amplitude of wave,the ratio of relaxation to retardation times, time-mean flow,

rotation, and the phase angle decrease and increase gradually.It is observed that the shear stress increases with increase in therotation, while it decreases with increase in the non-

dimensional amplitude of wave, the ratio of relaxation toretardation times, time-mean flow and phase angle. Moreoverthe values of shear stress are larger in case of a Jeffery fluid

when compared with Newtonian fluid.

5. Conclusion

Due to the complicated nature of the governing equations forthe pertinent field equations governing the peristaltic trans-port of Jeffery fluid, the work done in this field is unfortu-nately limited in number. The method used in this study

provides a quite successful in dealing with such problems.This method gives exact solutions in the peristaltic transportwithout any assumed restrictions on the actual physical quan-

tities that appear in the governing equations of the problemconsidered. Important phenomena are observed in all thesecomputations:

� If k1 ¼ 0; X ¼ 0; g ¼ 0, our results are in agreement withRefs. ([24–26]) and Mishra and Rao [28].

� It was found that for large values of the non-dimensional

amplitude of wave b, the ratio of relaxation to retardationtimes k1, time-mean flow H, rotation X, the phase angle /and the gravity field g in asymmetric and symmetric chan-

nel. The solution is obtained in the context of the peristaltictransport of fluid.

� By comparing Figs. 1–7 for the peristaltic transport of fluid

with figures without gravity field and rotation, it was foundthe same behavior in the same field.

1734 A.M. Abd-Alla, S.M. Abo-Dahab

� The magnitude of pressure gradient at the upper wall in a

symmetric channel is greater than in an asymmetricchannel.

� The results presented in this paper should be proved useful

for researchers in scientific and engineering, as well as forthose working on the development of fluid mechanics.Study of the phenomenon of the non-dimensional ampli-tude of wave, the ratio of relaxation to retardation times,

time-mean flow, rotation, the phase angle and the gravityfield in asymmetric and symmetric channel influence andoperations are also used to improve the conditions of peri-

staltic motion.

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