Dharmendra TripathiDepartment of Mathematics,

Indian Institute of Technology,

Ropar, Punjab-140001, India

e-mail: dtripathi@iitrpr.ac.in

Numerical Study on CreepingFlow of Burgers’ Fluids througha Peristaltic TubeMotivated by the objective of improving an understanding of the complex rheological fluiddynamics in fluid engineering and biomedical engineering, we consider the creeping flowof Burgers’ fluid with a fractional model through a peristaltic tube in the present article.Homotopy analysis method is used to solve the problem and obtain the approximate analyt-ical solution in terms of axial velocity, volumetric flow rate, pressure gradient, stream func-tion and mechanical efficiency under the long wavelength approximation. It is assumedthat the cross-section of the tube varies sinusoidally along the length of tube. The impactsof fractional parameters, material constants, time and amplitude on the pressure difference,frictional force across one wavelength and trapping, are depicted numerically. It is foundthat the second material constant helps the flow pattern, whereas the other three materialconstants resist it through the peristaltic tube. The effects of fractional parameters on flowpattern are found to be opposite to each other. [DOI: 10.1115/1.4005316]

Keywords: creeping flow, Burgers’ fluids, peristaltic tube, homotopy analysis method

1 Introduction

Calculus is a key branch of mathematics which has two majorsub-branches known as differential calculus (integer-order deriva-tives) and integral calculus (integrals). Fractional calculus is a col-lection of relatively little-known mathematical results concerninggeneralizations of differentiation and integration to non-integerorders. While these results have been accumulated over centuriesin various branches of mathematics, they have until recently foundlittle appreciation or application in physics and other mathemati-cally oriented sciences. This situation is beginning to change, andthere are now a growing number of research areas in physicswhich employ fractional calculus. The fractional derivative accu-rately describes natural phenomena that occur in such commonengineering problems as heat transfer, electrode=electrolytebehavior, and sub-threshold nerve propagation. By expanding therange of mathematical operations to include fractional calculus,we can develop new and potentially useful functional relation-ships for modeling complex biological systems in a direct and rig-orous manner. Fractional calculus was applied to problems innerve stimulation, dielectric relaxation, and viscoelastic materialsby extending the governing differential equations to include frac-tional order terms. Applications of fractional calculus to the solu-tion of time-dependent, viscous-diffusion fluid mechanicsproblems have been reported in Refs. [1–14]. Unsteady flows ofviscoelastic fluids with the fractional Maxwell model, fractionalgeneralized Maxwell model, fractional second grade fluid, frac-tional Oldroyed-B model, fractional Burgers’ model and fractionalgeneralized Burgers’ model have been studied. Laplace transform,Fourier transform, Weber transform, Hankel transform and dis-crete Laplace transform have been used to find the solution ofproblems. Comparing the fractional results to the existing analyti-cal results for the first and second Stokes problems, the fractionalmethodology is shown to be much simpler and more powerfulthan existing analytical techniques.

The study of fluid flow has a variety of applications in medi-cine, science and technology. In particular, the flow of non-Newtonian fluids in peristaltic tubes is encountered in various en-gineering applications. Furthermore, the industrial importance of

non-Newtonian fluids is widely known. The second and thirdgrade fluids, which form a subclass of the differential type fluids,can predict the shear thinning and thickening properties but lackother features such as stress relaxation and retardation. Maxwelland Oldroyd-B fluid models which are of the viscoelastic type canshow relaxation and retardation phenomena. Burgers’ model is aviscoelastic fluid model which is used for asphalt and asphaltmixes to characterize food products such as cheese, as well assoil, in the modeling of high temperature viscoelasticity of fine-grain polycrystalline olivine. A peristaltic tube is a tube in whichthe flow of fluids occurs due to peristalsis. Peristalsis is character-ized by alternate contraction and expansion, which pushes the flu-ids from one place to another place. It plays an important role inthe study of the applications in fluid engineering, biomechanicsand physiology. Peristalsis is very well known for biologist; how-ever, from a mechanical point of view, Latham [15] first intro-duced the theoretical and experimental concept of peristalsis tostudy the fluid flow pattern and since then, Burns and Parkes [16]and Shapiro et al. [17] have mathematically studied the peristalticflow of viscous fluid under the assumption of low Reynolds num-ber and large wavelength approximation through the chan-nel=tube. Furthermore, some workers [18,19] extended this studyfor viscoelastic fluid with Maxwell model and they discussed theeffect of relaxation time on the peristaltic flow. Hayat et al.[20,21] studied the peristaltic flow of viscoelastic fluid with theJeffrey model and they have also discussed the effect of relaxationand retardation time on the peristaltic transport. The flow of visco-elastic fluid with Burgers’ model has been reported by Hayat et al.[22] and perturbation solutions have been developed for axial ve-locity and pressure rise.

Motivated by the application of fractional calculus in visco-elastic fluid engineering, Tripathi et al. [23] have studied the peri-staltic flow of viscoelastic fluids with a fractional Maxwell modelthrough the channel. Homotopy perturbation method (HPM) andAdomian decomposition method (ADM) have been used to findthe solution. This study has further been improved for differentfractional models with different flow geometries and different nu-merical techniques in Refs. [24–28]. None of these models havestudied the flow of fractional Burgers’ fluid through peristaltictube. Considering these facts, in this paper, we study the creepingflow of complex rheological fluids with a fractional Burgers’model through the peristaltic tube under the long wavelengthapproximation. The fractional equations are solved by homotopy

Contributed by the Fluids Engineering Division of ASME for publication in theJOURNAL OF FLUIDS ENGINEERING. Manuscript received April 11, 2011; final manuscriptreceived October 13, 2011; published online December 20, 2011. Assoc. Editor: ZviRusak.

Journal of Fluids Engineering DECEMBER 2011, Vol. 133 / 121104-1Copyright VC 2011 by ASME

Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms

analysis method (HAM) which is developed by Liao [29]. Thisstudy is applicable in transportation of industrial fluids by peri-staltic pumping and peristaltic flow of physiological fluids in de-formable domains. This model helps to study the applications offluid engineering and biomedical engineering.

2 Mathematical Formulation and Analysis

The constitutive equation for fractional Burgers’ fluid is givenby

1þ ~ka1

@a

@~taþ ~ka

2

@2a

@~t 2a

� �~s ¼ l 1þ ~kb

3

@b

@~tbþ ~kb

4

@2b

@~t 2b

� �_c (1)

where ~t; ~s; _c, ~k1; ~k2; ~k3; ~k4 are the time, shear stress, rate ofshear strain, and material constants, l is viscosity, and a; b arethe fractional parameters such that 0 < a � b � 1. This modelreduces to a generalized Burgers’ model if a ¼ b ¼ 1; if it gives afractional Burgers’ model for ~k4 ¼ 0 and for ~k4 ¼ 0, a ¼ b ¼ 1,we get Burgers’ model. It reduces to fractional Oldroyd-B, frac-tional Maxwell, fractional second grade models respectivelywhen, ~k2 ¼ ~k4 ¼ 0, ~k2 ¼ ~k3 ¼ ~k4 ¼ 0, ~k1 ¼ ~k2 ¼ ~k4 ¼ 0 andwith a ¼ b ¼ 1, these models reduce to their ordinary models. Aclassical Navier-Stokes model is obtained by substituting~k1 ¼ ~k2 ¼ ~k3 ¼ ~k4 ¼ 0.

The governing equations of the motion of fractional Burgers’fluids for tubular flow are given by

q 1þ ~ka1

@a

@~taþ ~ka

2

@2a

@~t 2a

� �D~u

D~t¼ � 1þ ~ka

1

@a

@~taþ ~ka

2

@2a

@~t 2a

� �@~p

@~xþ l 1þ ~kb

3

@b

@~tbþ ~kb

4

@2b

@~t 2b

� �1

~r

@

@~r~r@~u

@~r

� �þ @

2 ~u

@~x2

� �

q 1þ ~ka1

@a

@~taþ ~ka

2

@2a

@~t 2a

� �D~v

D~t¼ � 1þ ~ka

1

@a

@~taþ ~ka

2

@2a

@~t 2a

� �@~p

@~rþ l 1þ ~kb

3

@b

@~tbþ ~kb

4

@2b

@~t 2b

� �@

@~r

1

~r

@

@~rð~r~vÞ

� �þ @

2~v

@~x2

� �@~u

@~xþ 1

r

@ð~r~vÞ@~r¼ 0

9>>>>>>>=>>>>>>>;

(2)

whereD

D~t� @

@~tþ ~u

@

@~xþ ~v

@

@~r. For carrying out further analysis, we introduce the following non-dimensional parameters:

x ¼ ~x

k; r ¼ ~r

a; t ¼ c~t

k; k1 ¼

c~k1

k; k2 ¼

c~k2

k; k3 ¼

c~k3

k; k4 ¼

c~k4

k; u ¼ ~u

c

v ¼ ~v

cd; d ¼ a

k; / ¼

~/a; h ¼

~h

a; p ¼ ~pa2

l ck; Q ¼

~Q

pa2c; Re ¼ q cad=l

9>>>=>>>;

(3)

where q, x; r; u; v; c; d; a; /; h; p, and Q are the fluid density, axial coordinate, radial coordinate, axial velocity, radial velocity, wavespeed, wave number, radiaus of tube, wave amplitude, radial displacement of the walls from center line, pressure, and volume flow raterespectively, and� represents the corresponding parameters in the dimensional form.

Introducing the non-dimensional parameters and further applying the long wavelength approximation, Eqs. (2) reduce to:

@ p

@ x¼ 1þ ka

1

@a

@ taþ ka

2

@2a

@ t2a

� ��1

1þ kb3

@b

@ tbþ kb

4

@2b

@ t2b

� �@2u

@ r2þ 1

r

@ u

@ r

� �@ p

@ r¼ 0

@ u

@ xþ 1

r

@ðr vÞ@ r

¼ 0

9>>>>>>>=>>>>>>>;

(4)

The following boundary conditions are prescribed:

@u

@r¼ 0 at r ¼ 0; u¼ 0 at r ¼ h;

@p

@x

� �t¼0

¼ p0;d

dt

@p

@x

� �¼ p1

(5)

Integrating Eq. (4) with respect to r, and using the first conditionof Eq. (5), the velocity gradient is obtained as

@u

@r¼ r

21þ kb

3

@b

@ tbþ kb

4

@2b

@ t2b

� ��1

1þ ka1

@a

@ taþ ka

2

@2a

@ t2a

� �@p

@x

(6)

Further integrating Eq. (5) from 0 to r, we get the axial velocity as

u¼ 1

41þkb

3

@b

@ tbþkb

4

@2b

@ t2b

� ��1

1þka1

@a

@ taþka

2

@2a

@ t2a

� �@p

@xr2�h2� �

(7)

The volume flow rate is defined as Q ¼Ð h

02r u dr, which, by vir-

tue of Eq. (7), reduces to

Q ¼ � 1

81þ kb

3

@b

@ tbþ kb

4

@2b

@ t2b

� ��1

1þ ka1

@a

@ taþ ka

2

@2a

@ t2a

� �@p

@xh4

(8)The transformations between the wave and the laboratory frames,in the dimensionless form, are given by

X ¼ x� t; R ¼ r; U ¼ u� 1; V ¼ v; q ¼ Q� h2 (9)

where the left side parameters are in the wave frame and the rightside parameters are in the laboratory frame.

The geometry of the peristaltic tube is mathematically (in anon-dimensional wave frame) modeled (Fig. 1) as

h ¼ 1� / cos2ðp xÞ (10)

The averaged flow rate in the wave frame and that in the labora-tory frame is given by

121104-2 / Vol. 133, DECEMBER 2011 Transactions of the ASME

Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms

�Q ¼ qþ 1� /þ 3/2

8¼ Q� h2 þ 1� /þ 3/2

8(11)

Eq. (8), in view of Eq. (11) gives

@2a

@t2a

@p

@x

� �þ ka

1

ka2

@a

@ta@p

@x

� �þ 1

ka2

@p

@x

¼ � 8

ka2

1þ kb3

@b

@tbþ kb

4

@2b

@t2b

� � �Qþ h2 � 1þ /� ð3/2=8Þh4

� �(12)

From Eq. (7), and using Eqs. (8) and (11), the stream function (w)

in wave frame U ¼ 1

r

@w@r

� �is obtained as

w ¼ ��Qþ h2 � 1þ /� ð3/2=8Þ

2h4

� �ðr4 � 2r2h2Þ � r2

2(13)

It is clear from Eq. (13) that the stream function is independent offractional parameters and material constants.

3 Solution of the Problem by HAM

Eq. (12) is simplified as

@2af

@t2aþ ka

1

ka2

@af

@taþ 1

ka2

f ¼ 1þ kb3

@b

@tbþ kb

4

@2b

@t2b

� �A (14)

where

f ðx; tÞ ¼ @p

@x(15)

and

A ¼ � 8

ka2

�Qþ h2 � 1þ /� ð3/2=8Þh4

� �(16)

To solve Eq. (14) by means of HAM, we choose the initialapproximation

f0 ðx ; t Þ ¼ p0 þ p1 t (17)

and the linear operator

L ½u ðx ; t ; hÞ� ¼ @2 a u ðx ; t ; hÞ

@ t2 a; 0 < a � 1 (18)

with the property

L ½c1 þ c2 t � ¼ 0 (19)

where c1 and c2 are integral constants. Furthermore, Eq. (14) sug-gests that we define an equation of nonlinear operator as

N ½u ðx ; t ; hÞ� ¼ @2 au ðx ; t ; hÞ

@ t2 aþ ka

1

ka2

@au ðx ; t ; hÞ@ ta

þ 1

ka2

u ðx ; t ; hÞ � 1þ kb3

@b

@ tbþ kb

4

@2b

@ t2b

� �A (20)

Now, we construct the zeroth-order deformation equation

ð1� hÞL ½u ðx ; t ; hÞ � Z0ðx; tÞ� ¼ h �h N ½uðx ; t ; hÞ� (21)

Obviously, when h ¼ 0 and h ¼ 1,

u ðx; t ; 0Þ ¼ Z0ðx; tÞ ¼ f0ðx; t Þ and u ðx; t ; 1Þ ¼ f ðx; tÞ(22)

Therefore, as the embedding parameter h increases from zero tounity, u ðx; t ; hÞ varies from the initial guess Z0ðx; tÞ to the solu-tion f ðx; tÞ. Expanding u ðx; t ; hÞ in Taylor series with respect toh, one can find

u ðx; t ; hÞ ¼ Z0ðx; tÞ þX1m¼1

Zmðx; tÞ � hm

where, Zmðx; tÞ ¼ 1

m !

@m u ðx; t ; hÞ@ hm

����h¼0

.

If the auxiliary linear operator, the initial guess and the auxil-iary parameter �h are properly chosen, the above series is conver-gent at h ¼ 1, then one has

f ðx; t Þ ¼ Z0ðx; tÞ þX1m¼1

Zmðx; tÞ

which must be one of the solutions of the original nonlinear equa-tion, as proved by Liao [30]. Now we define the vector

~Znðx; tÞ ¼ fZ0ðx; tÞ; Z1ðx; tÞ; : : : : ; Znðx; tÞg

Then the mth-order deformation equation is

L ½Zmðx; tÞ � vm Zm�1ðx; tÞ� ¼ �h Rmð~Zm�1ðx; tÞÞ (23)

with the initial conditions

Zmðx; 0Þ ¼ 0 and@ Zmðx; 0Þ

@ t¼ 0 (24)

where

Rmð~Zm�1ðx; sÞÞ ¼ D2 at ðZm�1Þ þ

ka1

ka2

Dat ðZm�1Þ

þ 1

ka2

Zm�1 � 1þ kb3

@b

@ tbþ kb

4

@2b

@ t2b

� �A;

(25)

and vm ¼0; m � 1

1 ; m > 1

�.

Now, the solution of the mth-order deformation Eq. (23) form � 1, becomes

Zmðx; tÞ ¼ vm Zm�1ðx; tÞ þ �h J2 at ½Rmð~Zm�1ðx; tÞÞ d t� þ c1 þ c2 t

(26)

where the integration constants c1 and c2 are determined by theinitial condition [24]. We now successively obtain

Z1ðx; tÞ ¼ �h ka1p1

ka2

taþ1

Cðaþ 2Þ þ �hp0

ka2

� A

� �t2 a

Cð2 aþ 1Þ

þ �hp1

ka2

t2 aþ1

Cð2 aþ 2Þ � �h A kb3

t2 a�b

Cð2 a� bþ 1Þ ;

� �h A kb4

t2 a�2b

Cð2 a� 2bþ 1Þ (27)

Fig. 1 Geometry of peristaltic tube

Journal of Fluids Engineering DECEMBER 2011, Vol. 133 / 121104-3

Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms

Z2ðx; tÞ ¼ �h ð�hþ 1Þ ka1p1

ka2

taþ1

Cðaþ 2Þ þ �hð�hþ 1Þ p0

ka2

� ð�hþ 2ÞA� �

t2a

Cð2 aþ 1Þ

þ �h�h k 2 a

1

ka2

þ ð�hþ 1Þ� �

p1

ka2

t2 aþ1

Cð2 aþ 2Þ � �h ð�hþ 2ÞA kb3

t2 a�b

Cð2 a� bþ 1Þ � �h ð�hþ 2ÞA kb4

t2 a�2b

Cð2 a� 2bþ 1Þ

þ �h2 ka1

ka2

p0

ka2

� A

� �t3a

Cð3 aþ 1Þ þ 2 �h2 ka1

k 2 a2

p1

t3aþ1

Cð3 aþ 2Þ � �h2Aka

1 kb3

ka2

t3a�b

Cð3 a� bþ 1Þ

� �h2Aka

1 kb4

ka2

t3a�2b

Cð3 a� 2bþ 1Þ þ�h2

ka2

p0

ka2

� A

� �t4a

Cð4 aþ 1Þ þ �h2 p1

k 2 a2

t4aþ1

Cð4 aþ 2Þ

� �h2Akb

3

ka2

t4a�b

Cð4 a� bþ 1Þ � �h2Akb

4

ka2

t4a�2b

Cð4 a� 2bþ 1Þ (28)

Z3ðx; tÞ ¼ �h ð�hþ 1Þ2 ka1p1

ka2

taþ1

Cðaþ 2Þ þ �hð�hþ 1Þ2 p0

ka2

� ð�h2 þ 3 �hþ 3ÞA !

t2a

Cð2 aþ 1Þ

þ �h ð�hþ 1Þ 2 �h k 2 a1

ka2

þ ð�hþ 1Þ� �

p1

ka2

t2 aþ1

Cð2 aþ 2Þ � �h ð�h2 þ 3 �hþ 3ÞA kb3

t2 a�b

Cð2 a� bþ 1Þ

� �h ð�h2 þ 3 �hþ 3ÞA kb4

t2 a�2b

Cð2 a� 2bþ 1Þ þ �h2 ka1

ka2

2 ð�hþ 1Þp0

ka2

� A ð2�hþ 3Þ� �

t3a

Cð3 aþ 1Þ

þ �h2 ka1

k 2 a2

p1

�h k2 a1

ka2

þ 4ð�hþ 1Þ� �

t3aþ1

Cð3 aþ 2Þ � �h2ð2 �hþ 3ÞA ka1 kb

3

ka2

t3a�b

Cð3 a� bþ 1Þ

� �h2ð2 �hþ 3ÞA ka1 kb

4

ka2

t3a�2b

Cð3 a� 2bþ 1Þ þ�h2

ka2

�h k 2 a1 p0

k 2a2

þ 2 ð�hþ 1Þka

2

p0 � A �hk 2a

1

ka2

� Að2 �hþ 3Þ !

t4a

Cð4 aþ 1Þ

þ �h2 p1

k 2 a2

3 �hk 2a

1

ka2

þ 2ð�hþ 1Þ� �

t4aþ1

Cð4 aþ 2Þ � �h2Akb

3

ka2

�hk 2a1 þ 2 �hþ 3

� � t4a�b

Cð4 a� bþ 1Þ

� �h2Akb

4

ka2

�hk 2a1 þ 2 �hþ 3

� � t4a�2b

Cð4 a� 2bþ 1Þ þ 2 �h3 ka1

k 2 a2

p0

ka2

� A

� �t5a

Cð5 aþ 1Þ þ 3 �h3p1

ka1

k 3 a2

t5aþ1

Cð5 aþ 2Þ

� 2 �h3Aka

1 kb3

k 2 a2

t5 a�b

Cð5 a� bþ 1Þ � 2 �h3Aka

1 kb4

k 2 a2

t5 a�2b

Cð5 a� 2bþ 1Þ þ�h3

k 2 a2

p0

ka2

� A

� �t6a

Cð6 aþ 1Þ

þ �h3

k 3 a2

p1

t6aþ1

Cð6 aþ 2Þ ��h3Akb

3

k 2 a2

t6a�b

Cð6 a� bþ 1Þ ��h3Akb

4

k 2 a2

t6a�2b

Cð6 a� 2bþ 1Þ (29)

proceeding in this manner, the components Zn; n � 0; of theHAM can be completely obtained and the series solutions are thusentirely determined.

Finally, we approximate the analytical solution f ðx; tÞ by thetruncated series

f ðx; t Þ ¼ limN!1

UNðx; t Þ (30)

where UNðx; t Þ ¼PN�1

n¼0

Znðx; t Þ .

The pressure difference across one wavelength (Dp) and fric-tional force across one wavelength (F) [20,21] are given by

Dp ¼ð1

0

@p

@xdx (31)

F ¼ð1

0

�h2 @p

@x

� �dx (32)

Mechanical efficiency [17,31] is defined as the ratio between theaverage rate per wavelength at which work is done by the movingfluid against a pressure head and average at which the walls dowork on the fluid. It is derived for given wall geometry and frac-tional Burgers’ fluid as

E ¼�QDp

2/½�ð1þ /ÞDpþ I11 þ /I12�(33)

where I11 ¼Ð 1

0

@p

@xcos 2p xdx, I12 ¼

Ð 1

0

@p

@xcos4ðp xÞ dx.

4 Numerical Results and Discussion

The numerical features of creeping flow of complex rheologicalfluids with a fractional Burgers’ model through the peristaltic tubeare discussed in this section with the help of illustrations. Basedon Eqs. (30) and (31), Figs. 3–19 are drawn to show the effect ofphysical parameters such as first fractional parameter (a), secondfractional parameter ( b), material constants (k1; k2; k3; k4), time(t) and amplitude (/) on variations of pressure difference acrossthe one wavelength (Dp) and frictional force across the one wave-length (F) against averaged flow rate ( �Q). Numerical results areobtained by using Mathematica Software. The initial pressure gra-dient and rate of pressure gradient are taken zero (i.e. p0 ¼ 0 &p1 ¼ 0). HAM method is used to find the approximate analyticaland numerical solution of problem. It is pointed out that the con-vergence and rate of approximation [29] for the HAM solutionstrongly depends on the values of auxiliary parameter �h. To findout the convergence range for �h, we plot a graph (see Fig. 2)between Dp and �h for different values of a. It is called �h -curve. Itis found that the convergence range for the values of �h is

121104-4 / Vol. 133, DECEMBER 2011 Transactions of the ASME

Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms

Fig. 3 Pressure difference versus averaged flow rate for vari-ous values of a at / ¼ 0:5, t ¼ 0:4, b ¼ 4=5, k1 ¼ 4; k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Fig. 4 Pressure difference versus averaged flow rate for vari-ous values of b at / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, k1 ¼ 4; k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Fig. 5 Pressure difference versus averaged flow rate for vari-ous values of k1 at / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, b ¼ 4=5, k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Fig. 6 Pressure difference versus averaged flow rate for vari-ous values of k2 at / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, b ¼ 4=5,k1 ¼ 4; k3 ¼ 1; k4 ¼ 1

Fig. 7 Pressure difference versus averaged flow rate for vari-ous values of k3 at / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, b ¼ 4=5, k1 ¼ 4;k2 ¼ 1; k4 ¼ 1

Fig. 8 Pressure difference versus averaged flow rate for vari-ous values of k4 at / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, b ¼ 4=5, k1 ¼ 4;k2 ¼ 1; k3 ¼ 1

Fig. 9 Pressure difference versus averaged flow rate for differ-ent fractional models (a) / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, b ¼ 4=5,k1 ¼ 4; k2 ! 0; k3 ¼ 1; k4 ¼ 0, (b) / ¼ 0:5, t ¼ 0:4, a ¼ 1=5,b ¼ 4=5, k1 ¼ 4; k2 ! 0; k3 ¼ 0; k4 ¼ 0, (c) / ¼ 0:5, t ¼ 0:4,a ¼ 1=5, b ¼ 4=5, k1 ¼ 4; k2 ¼ 1; k3 ¼ 1; k4 ¼ 1

Fig. 2 �h -curves for the pressure difference at different valuesof a at / ¼ 0:5, �Q ¼ 0:2, t ¼ 0:4, b ¼ 4=5, k1 ¼ 4; k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Journal of Fluids Engineering DECEMBER 2011, Vol. 133 / 121104-5

Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms

Fig. 13 Frictional force versus averaged flow rate for variousvalues of b at / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, k1 ¼ 4; k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Fig. 17 Frictional force versus averaged flow rate for variousvalues of k4 at / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, b ¼ 4=5, k1 ¼ 4; k2 ¼ 1;k3 ¼ 1

Fig. 11 Pressure difference versus averaged flow rate for vari-ous values of / at t ¼ 0:4, a ¼ 1=5, b ¼ 4=5, k1 ¼ 4; k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Fig. 12 Frictional force versus averaged flow rate for variousvalues of a at / ¼ 0:5, t ¼ 0:4, b ¼ 4=5, k1 ¼ 4; k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Fig. 10 Pressure difference versus averaged flow rate for vari-ous values of t at / ¼ 0:5, a ¼ 1=5, b ¼ 4=5, k1 ¼ 4; k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Fig. 14 Frictional force versus averaged flow rate for variousvalues of k1 at / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, b ¼ 4=5, k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Fig. 15 Frictional force versus averaged flow rate for variousvalues of k2 at / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, b ¼ 4=5, k1 ¼ 4;k3 ¼ 1; k4 ¼ 1

Fig. 16 Frictional force versus averaged flow rate for variousvalues of k3 at / ¼ 0:5, t ¼ 0:4, a ¼ 1=5, b ¼ 4=5, k1 ¼ 4;k2 ¼ 1; k4 ¼ 1

121104-6 / Vol. 133, DECEMBER 2011 Transactions of the ASME

Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms

�1:0 � �h � �0:2. As such, in this study, the value of �h is takenas �1:0.

The relation between pressure difference across one wavelengthand averaged flow rate under the influence of emerging parame-

ters are shown in Figs. 3–11. It is found that the relation is linearand pressure difference inversely proportional to averaged flowrate, meaning pressure difference decreases with increasing themagnitude of the flow rate. Maximum pressure difference is

Fig. 19 Frictional force versus averaged flow rate for variousvalues of / at t ¼ 0:4, a ¼ 1=5, b ¼ 4=5, k1 ¼ 4; k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Fig. 18 Frictional force versus averaged flow rate for variousvalues of t at / ¼ 0:5, a ¼ 1=5, b ¼ 4=5, k1 ¼ 4; k2 ¼ 1;k3 ¼ 1; k4 ¼ 1

Fig. 20 Streamlines in the wave frame at �Q ¼ 0:7 for (a) / ¼ 0:5, (b) / ¼ 0:4, (c) / ¼ 0:3, (d) / ¼ 0:2

Journal of Fluids Engineering DECEMBER 2011, Vol. 133 / 121104-7

Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms

achieved at zero averaged flow rate and vice versa. Under the cer-tain range of the pressure difference, three types of regions aredefined as (i) Dp > 0 i.e. pumping region (ii) Dp ¼ 0 i.e. freepumping region (iii) Dp < 0 i.e. co-pumping region. The effect offractional parameters on pressure difference - averaged flow ratecurve is shown in Figs. 3 and 4. It is found that pressure differencereduces with increasing the magnitude of first fractional parame-ter, whereas it increases with increasing the magnitude of the sec-ond fractional parameter in the pumping region, unaltered in thefree pumping region and opposite behavior is noticed in the co-pumping region. Figures 5–8 illustrate the influence of materialconstants (k1; k2; k3; k4) on the pressure difference. These fig-ures reveal the pressure difference increases with increasing themagnitude of k1; k3; k4, whereas it diminishes by increasing themagnitude of k2 in the pumping region, unaltered in the freepumping region and opposite behavior in the co-pumping region.Hayat et al. [22] have been reported for Burgers’ fluid, which is aspecial case (k4 ! 0, a ¼ b ¼ 1) of fractional Burgers’ fluid;however, they have discussed the velocity profile and pressure dif-ference versus amplitude. The relation between pressure differ-ence and averaged flow rate has been discussed for fractionalMaxwell fluid (which is a special case of this study, i.e.,k2 ¼ k3 ¼ k4 ¼ 0) and fractional Oldroyd-B fluid (which is a spe-cial case of this study, i.e., k2 ¼ k4 ¼ 0) in Refs. [23,28], respec-tively, and a similar trend is noticed. The comparative study onpressure difference among the fractional Burgers’ model, frac-tional Oldroyd-B model (limiting case, k2 ! 0 and k4 ¼ 0), frac-tional Maxwell model (limiting case, k2 ! 0 and k3 ¼ k4 ¼ 0),fractional Oldroyd-B model (k2 ¼ k4 ¼ 0, [27]) and fractionalMaxwell model (k2 ¼ k3 ¼ k4 ¼ 0, [27]) of viscoelastic fluids isshown in Fig. 9. It is found that the pressure difference for frac-tional Oldroyd-B model is maximum and the pressure differencefor the fractional Burgers’ model is minimum. It is furtherobserved that results (limiting case) of this study are much moreclose to the results of a previous study [27]. Figures 10 and 11 areprepared to see the effect of time and amplitude on the pressuredifference - averaged flow rate relation. It is depicted that thepressure difference increases with increasing the magnitude oftime in the pumping region, unaltered in the free pumping regionand opposite behavior in the co-pumping region. Pressure differ-ence enhances with increasing the magnitude of amplitude in thepumping region, the free pumping region and also the co-pumpingregion ( �Q � 0:36).

Friction is the force resisting the relative motion of solid surfa-ces, fluid layers, and=or material elements sliding against eachother. There are several types of friction including: (i) Dry fric-tion, (ii) Fluid friction, (iii) Lubricated friction, (iv) Skin friction,and (v) Internal friction. In peristaltic flow, the disturbances in thefluid is due to compression and expansion of walls; we say thatthe walls are producing sinusoidal movement and that is why theflow is produced. As such, when flow is produced then frictionforce will be developed automatically. In this study, frictionalforce [20,21] is defined as the integration (over one wavelength)of the product of pressure gradient and tube area, and the negativesign denotes the opposite to the direction of flow. The relationbetween frictional force and averaged flow rate under the influ-ence of said parameters are presented in Figs. 12–19. The effectsof fractional parameters, material constants, time and amplitudeon frictional force are similar to that for the pressure difference ina magnitude sense but opposite in direction, which is similar tothe results for frictional force in Ref. [23].

A region of closed streamlines for certain values of / and �Q,forms a recirculation and moves with a mean speed equal to thatof the wave speed, is termed as trapping. This phenomenon is aninherent characteristic of peristaltic pumping. Based on Eq. (13),Figs. 20(a–d) are drawn for stream lines for different values of theamplitude (/ ¼ 0:2� 0:5) at �Q ¼ 0:7. It is observed that the num-ber of trapped bolus increases when the magnitude of / increases.From Eq. (13), it is clear that trapping is unaltered with otherparameters.

5 Conclusions

This paper examines the numerical study of the viscoelastic fea-ture, fractional feature on creeping flow through the peristaltic tubeunder the long wavelength approximation. It is inferred that thepressure difference -averaged flow rate curve is linear under theinfluence of all pertinent parameters. A revelation is that the pres-sure difference reduces with increasing the magnitude of the firstfractional parameter in the pumping region, whereas the secondfractional parameter has the opposite effect. It is concluded that theviscoelastic features determine that the pressure difference enhan-ces with increasing the magnitude of first, third and fourth parame-ters, whereas it diminishes with increasing the magnitude of thesecond parameter in the pumping region. It is physically interpretedthat the second parameter of viscoelastic material requires less pres-sure difference to propel the fluids while the other three parametersrequire more pressure difference. The effects of time and amplitudeon pressure difference are found to be similar to that for the secondfractional parameter in the pumping region. It is also revealed thatthe influence of all physical parameters on frictional force is similarto that for the pressure difference but opposite in direction. It isfinally concluded that the trapping is unaltered with the effect offractional parameters and viscoelastic nature while it diminisheswith decreasing the amplitude of the peristaltic wave. The presentanalysis is of a general nature, and results for Oldroyd-B fluid,Maxwell fluid with fractional and ordinary models and Newtonianfluid can be obtained as the special cases.

References[1] Hayat, T., Nadeem, S., and Asghar, S., 2004, “Periodic Unidirectional Flows of

a Viscoelastic Fluid With the Fractional Maxwell Model,” Appl. Math. Com-put., 151, pp. 153–161.

[2] Tan, W., Pan, W., and Xu, M., 2003, “A Note on Unsteady Flows of a Visco-elastic Fluid With the Fractional Maxwell Model Between Two Parallel Plates,”Int. J. Non-Linear Mech., 38, pp. 645–650.

[3] Tan, W., and Xu, M., 2002 “Plane Surface Suddenly Set in Motion in a Visco-elastic Fluid With Fractional Maxwell Model,” Acta Mech. Sin., 18, pp.342–349.

[4] Friedrich, C., 1991, “Relaxation and Retardation Functions of the Maxwellmodel With Fractional Derivatives,” Rheol. Acta, 30, pp. 151–158.

[5] Qi, H., and Jin, H., 2006, “Unsteady Rotating Flows of a Viscoelastic FluidWith the Fractional Maxwell Model Between Coaxial Cylinders,” Acta Mech.Sin., 22, pp. 301–305.

[6] Qi, H., and Xu, M., 2007, “Unsteady Flow of Viscoelastic Fluid With FractionalMaxwell Model in Channel,” Mech. Res. Commun., 34, pp. 210–212.

[7] Khan, M., Ali, S. H., Fetecau, C., and Qi, H., 2009, “Decay of Potential Vortexfor a Viscoelastic Fluid with Fractional Maxwell Model,” Appl. Math. Model.,33, pp. 2526–2533.

[8] Vieru, D., Fetecau, C., and Fetecau, C., 2008, “Flow of a Viscoelastic FluidWith the Fractional Maxwell Model Between Two Side Walls Perpendicular toa Plate,” Appl. Math. Comput., 200, pp. 459–464.

[9] Wang, S., and Xu, M., 2009, “Axial Couette Flow of Two Kinds of FractionalViscoelastic Fluids in an Annulus,” Nonlinear Anal.: Real World Appl., 10, pp.1087–1096.

[10] Khan, M., Ali, S. H., and Qi, H., 2009, “On Accelerated Flows of a ViscoelasticFluid With the Fractional Burgers’ Model,” Nonlinear Anal.: Real World Appl.,10, pp. 2286–2296.

[11] Qi, H., and Xu, M., 2009, “Some Unsteady Unidirectional Flows of a General-ized Oldroyd-B Fluid with Fractional Derivative,” Appl. Math. Model., 33, pp.4184–4191.

[12] Qi, H., and Xu, M., 2007, “Stokes’ First Problem for a Viscoelastic Fluid Withthe Generalized Oldroyd-B Model,” Acta Mech. Sin., 23, pp. 463–469.

[13] Nadeem, S., 2007, “General Periodic Flows of Fractional Oldroyd-B Fluid foran Edge,” Phys. Lett. A, 368, pp. 181–187.

[14] Hayat, T., Khan, M., and Asghar, S., 2007, “On the MHD Flow of FractionalGeneralized Burgers’ Fluid With Modified Darcy’s Law,” Acta Mech. Sin., 23,pp. 257–261.

[15] Latham, T. W., 1966, “Fluid Motion in a Peristaltic Pump,” M. S. Thesis, MIT,Cambridge.

[16] Burns, J. C., and Parkes, T., 1970, “Peristaltic Motion,” J. Fluid Mech., 29, pp.731–743.

[17] Shapiro, A. H., Jafferin, M. Y., and Weinberg, S. L., 1969, “Peristaltic PumpingWith Long Wavelengths at Low Reynolds Number,” J. Fluid Mech., 37, pp.799–825.

[18] Tsiklauri, D., and Beresnev, I., 2001, “Non-Newtonian Effects in the PeristalticFlow of a Maxwell Fluid,” Phys. Rev. E, 64, p. 036303.

[19] Ali, N., Hayat, T., and Asghar, S., 2009, “Peristaltic Flow of a Maxwell Fluid ina Channel With Compliant Walls,” Chaos, Solitons Fractals, 39, pp. 407–416.

[20] Hayat, T., Ali, N., Asghar, S., and Siddiqui, A. M., 2006, “Exact PeristalticFlow in Tubes With an Endoscope,” Appl. Math. Comput., 182, pp. 359–368.

121104-8 / Vol. 133, DECEMBER 2011 Transactions of the ASME

Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms

[21] Hayat, T., and Ali, N., 2008, “Peristaltic Motion of a Jeffrey Fluid Under theEffect of a Magnetic Field in a Tube,” Commun. Nonlinear Sci. Numer. Simul.,13, pp. 1343–1352.

[22] Hayat, T., Ali, N., and Asghar, S., 2007, “Peristaltic Motion of a Burger’s Fluidin a Planar Channel,” Appl. Math. Comput., 186, pp. 309–329.

[23] Tripathi, D., Pandey, S. K., and Das, S., 2010, “Peristaltic Flow of ViscoelasticFluid With Fractional Maxwell Model through a Channel,” Appl. Math. Com-put., 215, pp. 3645–3654.

[24] Tripathi, D., 2011, “Peristaltic Transport of a Viscoelastic Fluid in a Channel,”Acta Astron., 68, pp. 1379–1385.

[25] Tripathi, D., 2010, “Numerical and Analytical Simulation of Peristaltic Flowsof Generalized Oldroyd-B Fluids,” Int. J. Numer. Methods Fluids, to bepublished.

[26] Tripathi, D., 2011, “Peristaltic Transport of Fractional Maxwell Fluids in UniformTubes: Application of an Endoscope,” Comput. Math. App., 62, pp. 1116–1126.

[27] Tripathi, D., 2011, “A Mathematical Model for the Peristaltic Flow of ChymeMovement in Small Intestine,” Math. Biosci., 233, pp. 90–97.

[28] Tripathi, D., 2011, “Numerical Study on Peristaltic Transport of Fractional Bio-Fluids,” J. Mech. Med. Biol., to be published.

[29] Liao, S. J., 1998, “Homotopy Analysis Method: A New Analytic Method forNonlinear Problems,” Appl. Math. Mech., 19, pp. 957–962.

[30] Liao, S. J., 2003, Beyond Perturbation: Introduction to the Homotopy AnalysisMethod, CRC Press, Chapman and Hall, Boca Raton, FL.

[31] Takabatake, S., Ayukawa, K., and Mori, A., 1988, “Peristaltic Pumping in Cir-cular Cylindrical Tubes: A Numerical Study of Fluid Transport and itsEfficiency,” J. Fluid Mech., 193, pp. 267–283.

Journal of Fluids Engineering DECEMBER 2011, Vol. 133 / 121104-9

Downloaded From: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/07/2014 Terms of Use: http://asme.org/terms