An analysis of peristaltic transport for flowof a Jeffrey fluid

T. Hayat, N. Ali, and S. Asghar, Islamabad, Pakistan

Received May 9, 2006; revised December 21, 2006Published online: June 12, 2007 � Springer-Verlag 2007

Summary. The peristaltic mechanism of a Jeffrey fluid in a circular tube is investigated. The rheological

effects and compressibility of the fluid are taken into account. The modeled equations are solved using

perturbation technique when the ratio of the wave amplitude to the radius of the pore is small. In the

second order approximation, a net flow due to a travelling wave is obtained and effects of Reynolds

number, relaxation and retardation times, compressibility of the fluid and tube radius are studied. It is

noticed that for the Jeffrey fluid the back flow only occurs for large values of the relaxation time and small

values of the retardation time (less than 10 in the present analysis). Another interesting observation is that

oscillatory behavior of the net flow rate in the Jeffrey fluid is less than that of a Maxwell fluid. Several

results of other fluid models can be deduced as the limiting cases of our situation.

1 Introduction

Peristalsis [1] is a form of fluid transport induced by a wave travelling on the walls of a tube. It is

an inherent property of many biological systems. In living systems it is a distinctive pattern of

smooth muscle contractions that propel the contents of the tube, as food stuffs through oeso-

phagus and alimentary canal, in transport of urine from kidney to bladder, the movement of

chyme in the gastro-intestinal tract, vasomotion of the small blood vessels and in many other

glandular ducts. Physiologists also know this mechanism as a type of motility in which con-

traction above and relaxation below a segment is stimulated. Engineers have used the peristaltic

mechanism in developing pumps which have several industrial applications. To understand this

principle, several theoretical and experimental investigations have been made in various situa-

tions and the literature on the topic is now quite extensive. Recently, it has been shown that the

flow rate of a liquid is increased by an external sonic radiation through a porous medium (see [2]

and several references therein). In [2], Aarts and Ooms discussed the compressible flow of

Newtonian fluid induced by the traveling waves on the tube wall in porous media.

The behavior of most of the physiological fluids, oils, hydrocarbons and polymers is known

to be non-Newtonian. But most of the investigations in the literature have been carried out by

considering blood, physiological fluids, oils etc. as Newtonian fluids. This approach may

provide satisfactory understanding of peristaltic mechanism in the ureter but it definitely fails to

give a better understanding when it is involved in small blood vessels, lymphatic vessels,

intestine, ductus efferentes of male reproductive tracts, transport of spermatozoa, oils and

Acta Mechanica 193, 101–112 (2007)

DOI 10.1007/s00707-007-0468-2

Printed in The NetherlandsActa Mechanica

hydrocarbons. The literature regarding peristaltic flow of compressible non-Newtonian fluids is

scarce. Tsiklauri and Beresnev [3] extended the analysis of reference [2] from Newtonian to

Maxwell fluids.

The non-Newtonian fluids, exhibiting a non-linear relationship between the stresses and the

rate of strain, are now acknowledged as more appropriate in medical sciences and technological

applications than Newtonian fluids. The equations which govern the flow of such fluids offer

interesting and exciting challenges to engineers, mathematicians and computer scientists alike.

Engineers can design effective viscometers and other instruments to measure the non-Newtonian

fluid parameters [4]. Mathematicians can derive the proofs for existence of unique or multiple

solutions [5]–[8]. Computer scientists can design efficient algorithms for computing the flows.

But there is not a single model which exhibits all properties of non-Newtonian fluids. They

cannot be described simply as Newtonian fluids and such fluids may be classified as: (i) fluids for

which the shear stress depends on the shear rate, (ii) fluids for which the relation between the

shear stress and shear rate depends upon time, (iii) fluids which possess both elastic and viscous

properties called viscoelastic fluids. Due to the complex behavior of non-Newtonian fluids many

constitutive equations have been suggested and most of them are empirical or semi-empirical.

For most general three-dimensional representations the method of continuum mechanics is

needed. Amongst the many models, which have been employed to describe the non-Newtonian

behavior exhibited by certain real fluids, viscoelastic fluids have acquired a special status. One

class of viscoelastic fluids is of the differential type, namely the second grade, third grade and

fourth grade. These fluid models have attracted much attention, as well as much controversy.

For a detailed discussion of all the relevant issues, we refer the readers to Dunn and Rajagopal

[9] and Aksel [10]. Also the differential type fluid models take into account normal stress dif-

ferences and shear thinning/thickening effects, but lack other features such as stress relaxation

that many fluids show. Another type of models is that of rate-type fluid models, namely Max-

well, Oldroyd etc. We refer the reader to Rajagopal and Srinivasa [11] for a complete review of

these models. The simplest subclass of rate type fluids is that of the Maxwell model [12]–[14] in

which the relaxation phenomena are taken into consideration. This model has been applied to a

large variety of viscoelastic flows, usually with the small dimensionless relaxation time. The

models in which besides time derivatives of the stress tensor also a derivative of the rate of strain

tensor are included show relaxation as well as retardation behavior. An important example of

this kind is the Jeffrey’s version of the Oldroyd model [15]–[19]. This fluid model includes elastic

and memory effects exhibited by dilute polymer solutions and biological liquids. In view of this,

we therefore propose to study the peristaltic flow in a circular flexible tube through porous

media. The pore is filled with a compressible Jeffrey fluid being at rest in the absence of the

travelling wave. The problem is first non-dimensionlized and then perturbation analysis is

performed for small values of e ¼ a=R (a is the wave amplitude and R is the radius of the pore).

Moreover, a net flow due to the travelling wave is obtained in the second order approximation.

The Reynolds number and material constant are taken arbitrary. The variations of several

parameters of interest on the net flow rate are seen and discussed in detail.

2 Mathematical model

We choose cylindrical coordinates r; h; zð Þ in such a way that the z-axis is directed along the

axis of the tube (pore) with length L. An elastic sinusoidal wave induces a travelling wave on the

wall. The shape of the pore is given by

102 T. Hayat et al.

h z; tð Þ ¼ Rþ a cos2pk

z� ctð Þ� �

; ð2:1Þ

where k is the wavelength and c is the speed of the wave.

The laws which govern the flow are of mass and momentum given by

@q@tþ $ � qVð Þ ¼ 0; ð2:2Þ

q@

@tþ V � $ð Þ

� �V ¼ div T; ð2:3Þ

where q is the fluid density, t is the time and V is the velocity.

The constitutive equations for the Jeffrey fluid are [4]

T ¼ �pIþ S; ð2:4Þ

1þ k1@

@t

� �S ¼ �l 1þ k2

@

@t

� �c�; ð2:5Þ

c� ¼ $V þ $Vð Þy� 2

3$ � V; ð2:6Þ

where T and S are the Cauchy stress tensor and extra stress tensor, respectively, p is the

pressure, I is the identity tensor, k1 is the relaxation time, k2 is the retardation time, y denotesthe transpose and l is the dynamic viscosity.

We further assume that the equation

1

qdqdp¼ k ð2:7Þ

also holds. The solution of the above equation when density is a function of pressure is

q ¼ q0 exp k p� q0ð Þ½ �: ð2:8Þ

Here k is the fluid compressibility.

The no-slip boundary condition is

u h; z; tð Þ ¼ @h

@t; w h; z; tð Þ ¼ 0: ð2:9Þ

Upon making use of Eqs. (2.5) and (2.6) into Eq. (2.3) one can write

1þ k1@

@t

� �q@V

@tþ q V � $ð ÞV

� �

¼ � 1þ k1@

@t

� �$pþ l 1þ k2

@

@t

� �$V þ $Vy � 2

3$ � V

� �: ð2:10Þ

In order to obtain dimensionless equations and boundary conditions, the length has been

scaled by R and time by R=c. Defining eh ¼ h=R; eq ¼ q=q0; eu ¼ u=c; ew ¼ w=c; ep ¼ p=q0c2;

a ¼ 2pR=k; Re ¼ q0cR=l; ek1 ¼ k1c=R; ek2 ¼ k2c=R and v ¼ kq0c2 ¼ c2=c20; c0 ¼ 1=

ffiffiffiffiffiffiffiffikq0

p(where

q0 is the constant density at the reference pressure p0) the governing equations and boundary

conditions after dropping the tilde sign for simplicity can be written as

@q@tþ u

@q@rþw

@q@zþ q

@u

@rþ u

rþ @w

@z

� �¼ 0; ð2:11Þ

An analysis of peristaltic transport 103

� 1þ k1@

@t

� �@p

@rþ lRe

1þ k2@

@t

� � @2u@r2 þ 1

3@u@r� u

r2 þ @2u@z2

þ 13

@@r

@u@rþ u

rþ @w

@z

� �� �24

35

¼ 1þ k1@

@t

� �q@u

@tþ q u

@u

@rþw

@u

@z

� �� �; ð2:12Þ

� 1þ k1@

@t

� �@p

@zþ lRe

1þ k2@

@t

� � 1r@@r

r @w@r

� �þ @2w

@z2

þ 13

@@z

@u@rþ u

rþ @w

@z

� �� �24

35

¼ 1þ k1@

@t

� �q@w

@tþ q u

@w

@rþw

@w

@z

� �� �; ð2:13Þ

u ¼ @h

@t; w ¼ 0; at r ¼ h ¼ 1þ g z; tð Þ; ð2:14Þ

where g z; tð Þ ¼ e cos a z� tð Þ and v < 1 since the wave speed c is less than the speed of sound c0.

Equation (2.8) in dimensionless form becomes

q ¼ q0 exp v p� q0ð Þ½ �; ð2:15Þ

where v is a compressibility parameter. It relates the wave speed to the speed of sound in a

medium and thus can be interpreted as the square of a Mach number. The mathematical

statement of the problem is the solution of Eqs. (2.11) to (2.13) subject to the conditions in Eq.

(2.14). We also note with interest that if k1 ¼ k2 ¼ 0, Eqs. (2.12) and (2.13) reduce to that of

viscous fluid [2] and if k2 ¼ 0, we recover the governing equations for the Maxwell fluid [3].

3 Solution of the problem

To facilitate the solution of equations in the previous Section we express pressure, velocity

components and density as [2]

p ¼ p0 þ ep1 r; z; tð Þ þ e2p2 r; z; tð Þ þ . . . ;

u ¼ eu1 r; z; tð Þ þ e2u2 r; z; tð Þ þ . . . ;

w ¼ ew1 r; z; tð Þ þ e2w2 r; z; tð Þ þ . . . ;

q ¼ 1þ eq1 r; z; tð Þ þ e2q2 r; z; tð Þ þ . . . :

ð3:1Þ

Substituting Eq. (3.1) into Eqs. (2.11) to (2.14) and writing

u1 r; z; tð Þ ¼ U1 rð Þeia z�tð Þ þ �U1 rð Þe�ia z�tð Þ;

w1 r; z; tð Þ ¼ W1 rð Þeia z�tð Þ þ �W1 rð Þe�ia z�tð Þ;

p1 r; z; tð Þ ¼ P1 rð Þeia z�tð Þ þ �P1 rð Þe�ia z�tð Þ;

q1 r; z; tð Þ ¼ vP1 rð Þeia z�tð Þ þ v �P1 rð Þe�ia z�tð Þ;

ð3:2Þ

we obtain the following set of equations from the o eð Þ system:

� 1� iak1ð ÞP01 þ1� iak2ð Þ

ReU001 þ

U01r� U1

r2� a2U1

� �

104 T. Hayat et al.

þ 1� iak2ð Þ3Re

d

drU01 þ

U1

rþ iaW1

� �¼ �ia 1� iak1ð ÞU1; ð3:3Þ

� 1� iak1ð ÞP1 þ1� iak2ð Þ

ReW 00

1 þW 0

1

r� a2W1

� �

þ ia 1� iak2ð Þ3Re

U01 þU1

rþ iaW1

� �¼ �ia 1� iak1ð ÞW1; ð3:4Þ

U01 þU1

rþ iaW1 ¼ iavP1; ð3:5Þ

U1 1ð Þ ¼ � ia2; W1 1ð Þ ¼ 0; ð3:6Þ

where a bar over the quantities denotes the complex conjugate.

For the o e2� �

system we will use

u2 r; z; tð Þ ¼ U20 rð Þ þ U2 rð Þe2ia z�tð Þ þ �U2 rð Þe�2ia z�tð Þ;

v2 r; z; tð Þ ¼ W20 rð Þ þW2 rð Þe2ia z�tð Þ þ �W2 rð Þe�2ia z�tð Þ;

p2 r; z; tð Þ ¼ P20 rð Þ þ P2 rð Þe2ia z�tð Þ þ �P2 rð Þe�2ia z�tð Þ;

q2 r; z; tð Þ ¼ D20 rð Þ þ D2 rð Þe2ia z�tð Þ þ �D2 rð Þe�2ia z�tð Þ:

ð3:7Þ

The Equations (3.3) and (3.4) after using Eq. (3.5) become

� cP01 þ U001 þU01r� U1

r2� b2

U1

� �¼ 0; ð3:8Þ

� cP1 �i

aW 00

1 þW 0

1

r� b2

W1

� �¼ 0; ð3:9Þ

where

c ¼ 1� iak1

1� iak2

� �Re� iax

3; b2 ¼ a2 � ia

1� iak1

1� iak2

� �Re: ð3:10Þ

Without giving the details of algebra involved we can write the solutions of Eqs. (3.8) and (3.9)

as

U1 rð Þ ¼ C1I1 mrð Þ þ C2I1 brð Þ; ð3:11Þ

W1 rð Þ ¼ iaC1

mI0 mrð Þ þ ibC2

aI0 brð Þ; ð3:12Þ

where I1 is the modified Bessel function of the first kind of order 1; I0 is the modified Bessel

function of the first kind of order 0 and C1 and C2 are complex constants.

To determine the values of C1 and C2, we use conditions given in Eq. (3.6) and get

C1 ¼abmiI0 bð Þ

2 a2I0 mð ÞI1 bð Þ � bmI0 bð ÞI1 mð Þ½ � ; C2 ¼�a3iI0 mð Þ

2 a2I0 mð ÞI1 bð Þ � bmI0 bð ÞI1 mð Þ½ � ; ð3:13Þ

in which

m2 ¼ a21� vð Þ 1�iak1

1�iak2

Re� 4

3iav

1�iak1

1�iak2

Re� 4

3iav

: ð3:14Þ

An analysis of peristaltic transport 105

Employing the same methodology of solutions as in [3] we find that

W20 rð Þ ¼ D2 �Re

Z1

r

W1 yð Þ �U1 yð Þ þ �W1 yð ÞU1 yð Þ� �

dy; ð3:15Þ

where

D2 ¼ �iaC1

2I1 mð Þ � ib2

C2

2aI1 bð Þ þ ia �C1

2I1 �mð Þ þ i�b2 �C2

2aI1

�b� �

: ð3:16Þ

Note that the expression V20 rð Þ in [3] is incorrect. In fact, for correct expression the factor

1� iatmð Þ should be 1. The factor 1� iatmð Þ in Eq. (16) of [3] should also be 1. It has been

further pointed out that present solutions include the results of viscous fluid for k1 ¼ k2 or

k1 ¼ k2 ¼ 0 [2]. Moreover, if k2 ¼ 0, the solutions corresponding to the Maxwell fluid [3] are

obtained.

The net dimensionless flow rate Q is

Q z; tð Þ ¼ 2pZ1

0

rw r; z; tð Þdr: ð3:17Þ

The net flow averaged over one period of time is given by

hQi ¼ a2p

Z2p=a

0

Q z; tð Þdt

¼ 2pe2

Z1

0

rW20 rð Þdr: ð3:18Þ

The above expression after substituting W20 rð Þ from Eq. (3.15) becomes

hQi ¼ pe2 D2 �Re

Z1

0

r2 W1 yð Þ �U1 yð Þ þ �W1 yð ÞU1 yð Þ �

dr

24

35: ð3:19Þ

4 Numerical results and discussion

Our primary interest in this model is the effect of non-Newtonian parameters, compressibility

and tube radius on the net flow rate. The present analysis shows that the net flow rate is

modified by non-Newtonian parameters k1 and k2: The influence of various parameters on the

net flow rate hQi is investigated with the help of graphs. Following [2] and [3] the value of v is

chosen between 0 and 1 in all the graphs. It is seen that for large value of k1 the system exhibits

viscoelastic behavior while for small values of k1 the conventional viscous effects dominate. The

situation is reversed in the case of k2 for a fixed value of k1:

We have prepared Figs. 1–3 to compare the behavior of Newtonian, Maxwell and Jeffrey

fluids for a fixed value of Re. The values of Re in Figs. 1 and 2 are 2000 and 5000,

respectively. Figure 1 shows that for small values of the compressibility parameter the net

flow rate in case of Maxwell fluid is less than that of Jeffrey fluid. Moreover, the net flow rate

attains its maximum value in case of a Maxwell fluid when compared with a Jeffrey fluid for

106 T. Hayat et al.

large values of the compressibility parameter. Also the maximum value of net flow rate in

case of a Jeffrey fluid is less than that of Maxwell fluid over the whole interval of a. Thenoticeable fact is that both for Newtonian and Jeffrey fluid the flow rate attains a maximum

value and then decreases as the compressibility parameter increases, but this feature is absent

in a Maxwell fluid. Here, the flow rate increases monotonically as the compressibilty para-

meter increases.

Figure 2 shows that the behavior of a Maxwell fluid is similar to that of a Jeffrey fluid i.e. for

both fluids, the flow rate attains the maximum value and then decreases. It is interesting to note

that the flow rate in case of a Maxwell fluid is negative over a certain range of v but it becomes

positive as v increases. While for the Jeffrey fluid this is not the case and the flow rate is positive

over the whole range of v. Moreover, the net flow rate decays rapidly as the compressibility

parameter increases in case of a Newtonian fluid as compared to Maxwell and Jeffrey fluids

(Figs. 1 and 2).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10−5

<Q

>

NewtonianMaxwellJeffrey

c

Fig. 1. Plot showing the dimensionless

flow rate hQi versus v; for e ¼ 0:001,a ¼ 0:001 and Re ¼ 2000: Here for

Maxwell fluid k1 ¼ 5000 and forJeffrey fluid k1 ¼ 5000; k2 ¼ 1000

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5x 10−5

NewtonianMaxwellJeffrey

<Q

>

c

Fig. 2. Plot showing the dimensionlessflow rate hQi versus v; for e ¼ 0:001,

a ¼ 0:001 and Re ¼ 5000: Here forMaxwell fluid k1 ¼ 5000 and for

Jeffrey fluid k1 ¼ 5000; k2 ¼ 1000

An analysis of peristaltic transport 107

It is clear from Figs. 1 and 2 that for fixed values of k1 and k2, if Reynolds number is small

enough, the difference between Maxwell and Jeffrey fluid is prominent but as the Reynolds

number increases the difference is not noticeable. Of course there is a quantitative difference but

qualitatively the behavior is the same. The above fact can be described in another way, i.e., for

fixed value of Re, if k1 and k2 are small the difference between Maxwell and Jeffrey fluid is less

prominent (Fig. 3). The above fact will be more clear in view of Fig. 5 and is discussed later.

In Fig. 4, the dependence of hQi on the compressibility parameter is displayed for various

values of Re: It is observed that for small Reynolds number the net flow hQi first decreases andafter a certain value of v no change is observed, i.e., it becomes constant. On the other hand for

large values of Re Re ¼ 5000; 10000ð Þ it increases up to a certain maximum and then decreases.

Moreover, the maxima in the curve for the flow rate are shifting towards larger values of a as

the Reynolds number increases. Consequently, we can say that increasing Re has a similar effect

on hQi in both Jeffrey and Newtonian fluid. The case of Newtonian fluid is discussed in [2].

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10−5

NewtonianMaxwellJeffrey

<Q

>

c

Fig. 3. Plot showing the dimensionlessflow rate hQi versus v; for e ¼ 0:001,

a ¼ 0:001 and Re ¼ 5000: Here forMaxwell fluid k1 ¼ 1000 and for

Jeffrey fluid k1 ¼ 1000; k2 ¼ 200

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

−5

Re = 100Re = 1000Re = 5000Re = 10000

<Q

>

c

Fig. 4. Plot showing the dimensionlessflow rate hQi versus v; for k1 ¼ 1000;k2 ¼ 200; e ¼ 0:001 and a ¼ 0:001

108 T. Hayat et al.

Figure 5 shows the effects of k1 and k2 on the net flow rate hQi when plotted against v for a

fixed value of Re: Clearly k1 and k2 in the solution has pronounced effects on hQi: For small

values of k1 the behavior of hQi for Jeffrey fluid is similar to that of Newtonian and Maxwell

fluid qualitatively but differs quantitatively. For large values of k1 (i.e., in the extreme non-

Newtonian regime) the Maxwell fluid exhibits strong viscoelastic behavior, i.e., the absence of

maximum and rapid growth of hQi in the considered interval of variation of compressibility

parameter v [3] but the Jeffrey fluid shows interesting behavior. It possesses the characteristics

of both Newtonian and Maxwell fluids, i.e., the hQi for Jeffrey fluid increases rapidly and

attains a certain maximum and then decreases. This observed pattern of hQi is due to the

presence of k2:

In order to illustrate the dependence of the flow rate hQi on the parameter a (which is the

tube radius measured in wavelengths) we prepared Fig. 6. We notice that for k1 ¼ 100 the flow

rate changes in the sense that it attains lower values as a increases both for Jeffrey and Maxwell

fluids. The case k1 ¼ 1000 needs more attention for the appearance of an effect of negative flow

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2x 10−4

<Q

>

c

Fig. 5. Plot showing the dimensionlessflow rate hQi versus v; for k1 ¼ k2

(dotted line with crosses), k1 ¼ 1000;k2 ¼ 0 (dashed line with asterisks),

k1 ¼ 1000; k2 ¼ 200 (dash-dotted linewith squares), k1 ¼ 10000; k2 ¼ 0 (so-

lid line with crosses) and k1 ¼ 10000;k2 ¼ 2000 (solid line with squares).

Here, e ¼ 0:001; Re ¼ 10000 anda ¼ 0:001

1 2 3 4 5 6 7 8 9 10x 10

−3

0

0.2

0.4

0.6

0.8

1

1.2x 10

−4

<Q

>

a

Fig. 6. Plot showing the dimensionless

flow rate hQi as a function of a: Here,e ¼ 0:001; Re ¼ 10000 and v ¼ 0:6:k1 ¼ k2 correspond to the solid linewith crosses, k1 ¼ 100; k2 ¼ 0 corre-

spond to the dotted line with asterisks,k1 ¼ 100; k2 ¼ 20 correspond to the

dashed line with crosses, k1 ¼ 1000;k2 ¼ 0 correspond to the dash-dotted

line with squares and k1 ¼ 1000;k2 ¼ 150 correspond to the solid line

with squares

An analysis of peristaltic transport 109

rate when the interval of variation of a is increased both in Jeffrey and Maxwell fluids, and

therefore we have plotted this case separately in Fig. 7. We have taken the interval of a from

0.01 to 0.05. It can be seen that for the Jeffrey fluid the net flow rate is negative for very small

values of k2 k2 < 1ð Þ. However, it becomes positive and approaches the Newtonian behavior

when k2 increases. Moreover, it is quite obvious from Fig. 7 that similar to the Maxwell fluid,

back flow occurs in the present analysis for Jeffrey’s fluid. In both fluid models, the peristalsis

yields a nonlinear effect. Here, the fluid flow induced by peristaltic motion is in the opposite

direction of propagation of the travelling wave. Therefore such a situation arises due to non-

linear response of these fluids to the stress exerted by the travelling wave.

In Fig. 8 we have fixed k1 ¼ 10000 and showed the effects of k2 on the net flow rate hQi:We

observed that for Maxwell fluid hQi is negative and has highly oscillating behavior which is in

agreement with the results of [3]. On the other hand for Jeffrey fluid, the behavior is less

oscillatory when compared with Maxwell fluid even for a very small value of k2 ¼ 0:1. It is

further observed that in the present analysis of Jeffrey fluid the oscillatory behavior vanishes

and net flow rate becomes positive when k2 � 10: This provides a clear-cut indication to the

0 0.01 0.02 0.03 0.04 0.05−6

−4

−2

0

2

4

6

8x 10

−5

2 = 0

2 = 0.1

2 = 1

2 = 800

<Q

>

a

llll

Fig. 7. Plot showing the dimensionlessflow rate hQi versus a; for k1 ¼ 1000;e ¼ 0:001; Re ¼ 10000 and v ¼ 0:6

0 0.01 0.02 0.03 0.04 0.05−4

−3

−2

−1

0

1

2

3x 10

−3

2 = 0

2 = 0.1

2 = 10

<Q

>

a

lll

Fig. 8. Plot showing the dimensionlessflow rate hQi versus a; for k1 ¼ 10000;e ¼ 0:001; Re ¼ 10000 and v ¼ 0:6

110 T. Hayat et al.

fact that oscillation decay is highly dependent upon the retardation time occuring in the con-

stitutive equation of the Jeffrey fluid. Such kind of decay is not possible in the Maxwell fluid

because there is no retardation time in the corresponding constitutive equation which is also

clear from the existing study [3], [20].

5 Concluding remarks

In this work, the peristaltic flow of a Jeffrey fluid is studied. The governing equations for a

compressible Jeffrey fluid are modelled and then used for flow in a tube. The considered

problem is important from the rheological point of view and has applications in various

branches of science including stimulation of fluid flow. Special emphasis has been given to the

effects of non-Newtonian parameters k1 and k2, Reynolds number Re, fluid compressibility vand tube radius a on the net flow rate. The main findings of the presented analysis are sum-

marized as:

� The net flow rate increases for large values of the Reynolds number. The difference among

the behavior of net flow rate for Newtonian, Maxwell and Jeffrey fluid is strongly dependent

on the values of Re, k1 and k2, i.e., one can find the values of Re; k1 and k2 where the

difference is prominent.

� The back flow occurs for both Maxwell and Jeffrey fluid models. Such flow is not possible for

the Newtonian fluid model. However, it is important to note that for the Jeffrey fluid the

back flow occurs for large values of relaxation time and small values of retardation time. It is

further observed that for large values of retardation time the behavior of net flow rate in

Newtonian and Jeffrey fluid models is similar.

� The presence of retardation time k2 makes the net flow rate hQi less oscillatory in extreme

non-Newtonian regime, and this behavior vanishes for large values of k2 (k2 > 10 in the

present analysis).

� The present analysis is more general and results for Newtonian [2] and Maxwell [3] fluids can

be obtained by taking k1 ¼ k2 (or k1 ¼ k2 ¼ 0) and k2 ¼ 0, respectively. This provides good

agreement with the previous studies.

Acknowledgements

We are grateful to the reviewers for their constructive comments regarding an earlier version of thismanuscript. We are further thankful to the Higher Education Commission (HEC) for the financial

support.

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Authors’ addresses: T. Hayat and N. Ali, Department of Mathematics, Quaid-i-Azam University 45320,Islamabad-44000 (E-mail: nasirali_qau@yahoo.com); S. Asghar, Department of Mathematical

Sciences, COMSATS Institute of Information Technology, Islamabad, Pakistan

112 T. Hayat et al.: An analysis of peristaltic transport