Research ArticleFlow Characteristics of Distinctly ViscousMultilayered Intestinal Fluid Motion

S K Pandey1 M K Chaube2 and Dharmendra Tripathi3

1Department of Mathematical Sciences Indian Institute of Technology Banaras Hindu University Varanasi 221005 India2Department of Mathematics Echelon Institute of Technology Faridabad Haryana 121101 India3Department of Mechanical Engineering Manipal University Jaipur Rajasthan 303007 India

Correspondence should be addressed to M K Chaube mkchaubegmailcom

Received 7 July 2014 Accepted 10 March 2015

Academic Editor Andrea Cereatti

Copyright copy 2015 S K Pandey et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The goal of this investigation is to study the three layered (core layer intermediate layer and peripheral layer) tubular flow ofpower law fluids with variable viscosity by peristalsis in order to investigate the strength of the role played by an artificiallygenerated intermediate layer to ease constipationThe solution is carried out under the long wavelength and low Reynolds numberapproximations in the wave frame of reference as the flow is creeping one The stream functions for each layer such as core layerintermediate layer and peripheral layer are determinedThe expressions for axial pressure gradient interfaces trapping and refluxlimits are obtained The effects of power law index and viscosities on pressure across one wavelength mechanical efficiency andtrapping are discussed numerically It is found that the pressure required to restrain flow rates and themechanical efficiency increasewith the viscosities of the intermediate and peripheral layers as well as with the flow behaviour index It is observed that theaxisymmetric flow in intestines is less prone to constipation than two-dimensional flow and may be more easily overcome withintroducing a viscous intermediate layer

1 Introduction

Chyme is a semidigested food mixed with intestinal secrantsand is generally non-Newtonian in nature Its inherentphysical behavior is very close to the concept of power-lawfluids A viscous mucuos layer is present as a peripheral layerin the small intestine that acts as a lubricant to safeguardthe inner surface of the intestine from the rough contents ofthe chyme passing through it and secretes enzymes to helpdigestion The flow of chyme in intestines is due to peristalsiswhich is a series of rhythmic muscular contractions of theintestinal wall This is coordinated in such a way that wavesappear to start at the point of commencement of oesophagusat the neck and propagate up to the end to culminate afterpushing the contents into the stomach through the cardiacsphincter Further the flow inside an intestine may be two-dimensional or axisymmetric depending on the state of thecontent

In view of huge applications of peristaltic steady andunsteady flow of Newtonian and non-Newtonian fluids

many theoretical studies have been reported in literaturetime to time Ng and Ma [1] introduced the Lagrangianapproach for peristaltic pumping Takagi and Balmforth [23] discussed the peristaltic pumping with rigid object andviscous fluids in elastic tube Dudchenko and Guria [4]studied the self-sustained peristaltic waves Though theserecently published articles are significant contributions in thearea of peristalsis none of them are directly applicable tointestinal flowThus it looks better to concentrate on themostrelevant papers

Brasseur et al [5] developed two layers peristaltic flowmodels and studied the effect of peripheral layer viscosityon peristaltic transport through a channel and presented amodified result that a more viscous peripheral layer improvespumping performance while a less viscous peripheral layerdegrades the pumping performance Later on Rao and Usha[6] extended the previous analytical investigation [5] for acylindrical flow that yielded similar resultsMisra and Pandey[7 8] investigated peristaltic transport of a non-Newtonian

Hindawi Publishing CorporationApplied Bionics and BiomechanicsVolume 2015 Article ID 515241 15 pageshttpdxdoiorg1011552015515241

2 Applied Bionics and Biomechanics

fluid modeled as power-law fluid through a channel as wellas a tube in the presence of a peripheral layer and pointed outthat flow is enhanced with the peripheral layer viscosity aswell as the flow behavior index They [9] further investigatedthe peripheral layer effect on a core layer of Casson fluidmodelled for blood flow in a channel as well as a tube Apartfrom several other results they concluded that the peripherallayer viscosity favours pumping performance whether thecore layer is a power-law fluid or a Casson fluid Recentlya more general analysis of peristaltic transport with threelayers of different viscosities namely core intermediate andperipheral layers has been presented by Elshehawey andGharsseldien [10] They modeled embryo transfer (the finalstep of IVF) considering the fluid close to it as the core layerthe culture fluid as intermediate layer and the intrauterinefluid as peripheral layer However this concept of threelayers appears appropriate for investigating the role of anartificially generated intermediate layer through medicationto intervene constipation The three-layered peristaltic flowmodel [10] was further applied by Tripathi [11] to studythe axisymmetric blood flow through stenosed arteries byconsidering red blood cells as the core layer plateletswhiteblood cells as the intermediate layer and plasma as theperipheral layer In a previous attempt [12] we studiedsuch a flow for a channel and concluded that an artificiallygenerated comparatively viscous intermediate layer may helpease constipationOne of the refereeswas of the opinion that amore appropriate study will be for the tubular geometryThismotivated us to go for this finer quantitative investigation

In recent years this mechanism (peristalsis) has beenexploited in biomedical engineering industrial technologiesand toxic waste conveyance in chemical process engineeringContinuous refinements in designs require increasinglymoresophisticated models for peristaltic flows using complexworking fluids (non-Newtonian nanotechnological etc)The proposed model may be applicable and help investigatesimilar flows discussed above of biomedical and chemicalindustriesTherefore we propose to investigate axisymmetricperistaltic flow of a power-law fluid in three layers througha cylindrical tube The models [6 7 11 13] will be specialcases of this model Comparative study with previous inves-tigations is made

This paper is designed as follows Section 2 presentsthe formulation of the model wherein a dimensionless setof governing equations subject to appropriate boundaryconditions is derived Analytical wave form solutions areobtained in Section 3 Interface analysis pumping charac-teristic mechanical efficiency trapping and reflux are pre-sented in subsequent sections from 4 through 8 respectivelySection 9 provides the detailed numerical results along withinterpretationsThe final section concludes with our findings

2 Formulation of the Model

We consider the flow of a power-law fluid through a tubein three layers differing in terms of viscosity As discussedabove the three layers will be called core intermediateand peripheral layer The walls of the tube are considered

1205832

1205831

1205830 q3

q2

q1

120573

120572

H

H2

H1

Figure 1 The diagram represents the geometry of the three-layeredperistaltic tubular flow 119867 119867

2 1198671are radial displacement for

peripheral intermediate and core layers respectively 1205830 1205832 1205831and

1199021 1199022 1199023are the viscosities and flow rates in different layers

to be flexible so as to model the small intestine and alsobecause periodic transverse progressive wave trains are topropagate along the walls Details of the geometry of three-layered (peripheral layer intermediate layer and core layer)peristaltic flow region are presented in Figure 1

The following nondimensional variables are used in theanalysis

1199111015840=119911

120582 119903

1015840=119903

119886 119905

1015840=119888119905

120582 119906

1015840=119906

119888

V1015840 =V119896119888 119901

1015840=119901119886119899+1

1205830120582119888 119896 =

119886

120582

1205831015840=120583

1205830

Ψ1015840=

Ψ

1205871198862119888 ℎ

1015840=ℎ

119886

ℎ1015840

1=ℎ1

119886 ℎ

1015840

2=ℎ2

119886 Re =

119886119899+1119888120588

1205821205830

(1)

where 119888 120582 119886 119899 and 1205830represent respectively the wave-

speed the wavelength the radius of the tube the power-lawflow behavior index and the viscosity parameter of the corefluid The other parameters are 119909 119910 119905 119901 120583 120595 ℎ ℎ

1 and

ℎ2which represent respectively axial and transverse coor-

dinates time pressure viscosity stream function and thedistance from the centre line of the outer the intermediateand the core layer boundary their primed counterparts arethe corresponding quantities in the dimensionless form Reis the Reynolds number

Using the nondimensional variables and dropping theprimes the equation of the wall may be given by 119903 = 119867(119911 minus119905) which may acquire an arbitrary form Since the threefluids are mutually immiscible and properly conserve therespective masses during the flow the interfaces between theintermediate and core layers denoted by 119903 = 119867

1(119911minus119905) and the

peripheral and intermediate layers denoted by 119903 = 1198672(119911 minus 119905)

will be streamlinesUsing the nondimensional variables defined above and

considering the long wavelength and low Reynolds number

Applied Bionics and Biomechanics 3

approximations the governing equations of the flow of apower-law fluid after dropping the primes reduce to

120597119901

120597119911= (sign 120597119906

120597119903) [

1

119903

120597

120597119903119903120583

10038161003816100381610038161003816100381610038161003816

120597119906

120597119903

10038161003816100381610038161003816100381610038161003816

119899

]

120597119901

120597119903= 0

(2)

The following are the transformations from the laboratoryframe to the wave frame

119911 minus 119905 = 119909 119903 = 119903 119906 minus 1 = 119908

V = V 119901 = 119901

(3)

where the terms on the left side of the equality sign are in thelaboratory frame and those on the right side are in the waveframe explicitly119909 and119908 are the axial coordinate and the axialvelocity in the wave frame and 119906 is the axial velocity in thelaboratory frame other parameters remain invariant in bothframes Accordingly the stream function 120595 and the volumeflow rate 119902 in the wave frame are related to the correspondingterms in the laboratory fame Ψ being the stream function inthe laboratory frame as given in the following

120595 = Ψ minus1199032

2

119902 = 119876 minus (1 +1206012

2) = 119902

1+ 1199022+ 1199023

(4)

where 1199021 1199022 and 119902

3are the volume flow rates respectively

in the core intermediate and peripheral regions in the waveframe and 119876 = int1

0119876119889119905 is the total volume flow rate averaged

over a period henceforth known as the time-averaged flowrate in the laboratory frame Consider

119902lowast

1= 1198761minus int

1

0

1198671

2(119909) 119889119909 = 119902

1

119902lowast

2= 1198762minus int

1

0

1198672

2(119909) 119889119909 = 119902

1+ 1199022

(5)

where 1198761and 119876

2are the time-averaged flow rates in the

core and the intermediate layers respectively and 119902lowast1and 119902lowast2

are parameters defined by the aforementioned expressionsused to show the flow-rate relations between the wave andlaboratory frames Equations (2) may be expressed in thewave frame in terms of stream function 120595(119903 119909) as follows

120597119901

120597119909= [(sign 1

119903

120597120595

120597119903)1

119903

120597

120597119903(119903120583

10038161003816100381610038161003816100381610038161003816

120597

120597119903(1

119903

120597120595

120597119903)

10038161003816100381610038161003816100381610038161003816

119899

)]

120597119901

120597119903= 0

(6)

The variable viscosity is defined as follows

120583 =

1 0 le 119903 le 1198671

1205831

1198671le 119903 le 119867

2

1205832

1198672le 119903 le 119867

(7)

and the boundary conditions to be imposed on (6) are

120595 = 0 (1

119903120595119903)119903

= 0 at 119903 = 0

120595119903= minus 119867 at 119903 = 119867

120595 =119902

2= const at 119903 = 119867

120595 =119902lowast

1

2= const at 119903 = 119867

1

120595 =119902lowast

2

2= const at 119903 = 119867

2

(8)

The first boundary condition is that the stream functionand the velocity gradient are zero on the symmetry linethe second boundary condition is no-slip condition at theboundary119867 whereas the other three conditions denote massconservation in the three layers This further implies that119867

1

as well as1198672is a streamline in the course of the flow

3 Solution in the Wave Frame of Reference

Eliminating the pressure from the differential equations (6)we get

(sign(1119903

120597120595

120597119903))

120597

120597119903[1

119903

120597

120597119903(119903120583

10038161003816100381610038161003816100381610038161003816

120597

120597119903(1

119903

120597120595

120597119903)

10038161003816100381610038161003816100381610038161003816

119899

)] = 0 (9)

Integrating this equation and using the boundary conditions(8) we get

120595 = minus1

2[1199032minus (119902 + 119867

2)1198682

1198681

] (10)

where

1198681= int

119867

0

119903 int

119867

119903

(119904

120583)

1119899

119889119904 119889119903

1198682= int

119903

0

119904 int

119867

119904

(1199041

120583)

1119899

1198891199041119889119904

(11)

4 Applied Bionics and Biomechanics

Introducing definition (7) the stream function for the threeregions is given by

120595 = minus1

21199032+1

21199032(119902 + 119867

2) (3119899 + 1

119899 + 1)

sdot (1 minus1

1205831119899

1

)119867(119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2

+1

1205831119899

2

119867(119899+1)119899

minus (2119899

3119899 + 1) 119903(119899+1)119899

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 0 le 119903 le 1198671

(12)

120595 = minus1

21199032+1

2(119902 + 119867

2)

sdot [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

3119899 + 1

119899 + 1) 1199032

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

minus(2119899

3119899 + 1)

1

1205831119899

1

119903(119899+1)119899

]

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 1198671le 119903 le 119867

2

(13)

120595 = minus1

21199032+1

2(119902 + 119867

2)

sdot [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+ (3119899 + 1

119899 + 1)

1

1205831119899

2

1199032

times 119867(119899+1)119899

minus (2119899

3119899 + 1) 119903(119899+1)119899

]

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 1198672le 119903 le 119867

(14)

Moreover the pressure gradient in view of (12)ndash(14) is givenby

119889119901

119889119909= minus2 (

3119899 + 1

119899)

119899

sdot [

[

(119902 + 1198672)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

]

]

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus((119902 + 1198672)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

(15)

These results are valid for any arbitrary wave-shape givenby 119910 = 119867(119909) A further investigation of the problem willhowever be carried out for a sinusoidal wave form given by

119867(119909) = 1 + 120601 sin 2120587119909 (16)

Applied Bionics and Biomechanics 5

4 Interfaces Analysis

Using the boundary conditions (8) we obtain the followingequations for the two interfaces119902lowast

1+ 1198671

2

119902 + 1198672

= (1 minus (3119899 + 1

119899 + 1)

1

1205831119899

1

119867(3119899+1)119899

1

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

21198671

2

+ (3119899 + 1

119899 + 1)

1

1205831119899

2

119867(119899+1)119899

1198671

2)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

(17)

119902lowast

2+ 1198672

2

119902 + 1198672

= ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+

1

1205831119899

1

minus (3119899 + 1

119899 + 1)

1

1205831119899

2

sdot119867(3119899+1)119899

2+ (

3119899 + 1

119899 + 1)

1

1205831119899

2

119867(119899+1)119899

1198672

2)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)

sdot119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

)

minus1

(18)

Equations (17) and (18) will be simultaneously solved numer-ically to evaluate119867

1and119867

2as a function of 119909 (see Appendix

for the details of solution)

5 The Pumping Characteristic

Integrating the axial pressure gradient with respect to 119911 overone wavelength we get

Δ119901 = minus2 (3119899 + 1

119899)

119899

sdot int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

(19)

where

120574 (119909) = [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1

119867(3119899+1)119899+ (

1

1205831119899

1

minus1

1205831119899

2

)

sdot119867(3119899+1)119899

2

119867(3119899+1)119899+

1

1205831119899

2

]

minus1

(20)

Δ119901 attains its maximum value Δ1199010when 119876 = 0 Similarly 119876

reaches1198760 themaximumflow rate whenΔ119901 = 0 HenceΔ119901

0

may be explicitly given by

Δ1199010= minus2 (

3119899 + 1

119899)

119899

sdot int

1

0

120574 (119909)1198672minus 1 minus 120601

22

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)1198672minus 1 minus 120601

22

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

(21)

and 1198760may be evaluated from the implicit relation

int

1

0

120574 (119909)1198760+ 1198672minus 1 minus 120601

22

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)1198760+ 1198672minus 1 minus 120601

22

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909 = 0

(22)

6 Mechanical Efficiency of Pumping

Themechanical efficiency is defined as the ratio between theaverage rate per wavelength at which work is done by themoving fluid against a pressure head and the average rate atwhich the wall does work on the fluid (cf Shapiro et al [13])For a power-law fluid it will be given by

119864 = 119876(int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

sdot (int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

(1198672minus 1) 119889119909)

minus1

)

(23)

7 Trapping Limits

Trapping is a phenomenon of peristalsis in which an inter-nally circulating bolus of fluid is formed by closed streamlinesand this trapped bolus is pushed ahead along with theperistaltic wave (cf Shapiro et al [13]) It takes place at highflow rates and large occlusions Trapping limit is given bythe value of 119876 where 120595 = 0 for 119910 gt 0 For the given waveinformation there is a range of 119876 for trapping to occur The

6 Applied Bionics and Biomechanics

shape of the trapped region is obtained by setting 120595 = 0 for119910 = 0 in (12) as follows

119903(119899+1)119899

=1

2119899 (119902 + 1198672)

sdot [(1 minus1

1205831119899

1

)119867(119899+1)119899

1

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

1

2

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2

2

+1

1205831119899

2

119867(119899+1)119899

(3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2]

(24)

For the occurrence of trapping we must have 119903(119899+1)119899 gt 0 forsome 119909 So for a real and positive 119903(119899+1)119899 it is required thatboth the numerator and denominator be of the same signHere the denominator has its maximum value at 119909 = 14

and theminimum value at 119909 = 34Thus in this case we havetwo conditions imposed on 119876 for the existence of trappingConsider

119876minus

lt 119876 lt 119876+

(25)

where

119876minus

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1max + (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1max

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(119899+1)119899

])

minus1

) minus 2120601 minus1206012

2

119876+

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(119899+1)119899

])

minus1

) + 2120601 minus1206012

2

(26)

1198671max 1198672max 1198671min and 119867

2min are the values of theinterfaces at 119909 = 14 and 119909 = 34 respectively For 119899 = 1 theresults reduce to the results of Tripathi [11] for three-layeredperistaltic viscous flow For 120583

1= 1 the results reduce to those

reported by Misra and Pandey [8] for two-layered power-lawflow which for 119899 = 1 further reduce to those presented byRao andUsha [6] for two-layeredNewtonian flowThe resultsof Shapiro et al [13] model can be obtained by substituting1205831= 1 120583

2= 1 and 119899 = 1 in presented model

8 Reflux Limit

Reflux is another phenomenon concerning peristalsis inwhich some fluid particles on the average move in adirection opposite to the net flow (cf Shapiro et al [13])The volume flow rate 119876

120595(119909) corresponding to a particle at

a position 119911 and time 119905 in the laboratory frame and on thestreamline 120595 in the wave frame is given as

119876120595(119909) = 2120595 + 119903

2(120595 119909) (27)

On averaging over one cycle this becomes

119876120595(119909) = 2120595 + int

1

0

1199032(120595 119909) 119889119909 (28)

For evaluating the reflux limit we expand 119876120595(119909) in a power

series in terms of a small parameter 120576 about the wall where

120576 = 120595 minus119902

2 (29)

and use the reflux condition119876120595

119876

gt 1 as 120576 997888rarr 0 (30)

The coefficients of the first two terms in the expansion of1199032(120595 119909) that is

1199032= 1198672+ 1198861120576 + 11988621205762+ sdot sdot sdot (31)

Applied Bionics and Biomechanics 7

0

01

02

03

04

0 02 04 06 08Q

Δp

(a)

0

1

2

3

4

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

0 02 04 06 08Q

Δp

minus2

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 2 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205831at 120601 = 05 120572 = 07

120573 = 09 and 1205832= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

are obtained by using (14) and given by

1198861= minus2

1198862= minus

4 (119902 + 1198672)

11986741205831119899

2

120574 (119909)

(32)

Performing the integration in (28) to the second orderand using condition (30) we obtain the condition for theoccurrence of reflux as follows

int

1

0

((119876 minus 1 minus1206012

2+ 1198672)119867(1minus119899)119899

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1minus 1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)119889119909

lt 0

(33)

9 Numerical Results and Discussion

We have carried out numerical calculations and plottedgraphs to observe the effects of the viscosities of the interme-diate and peripheral layers and the flow behaviour index onpumping mechanical efficiency trapping limits and refluxlimit

Figures 2(a)ndash2(c) are based on (15) and are plottedbetween the pressure difference across one wavelength (Δ119901)

8 Applied Bionics and Biomechanics

0

01

02

03

04

05

06

07

0 02 04 06 08Q

Δp

minus01

(a)

0

1

2

3

4

5

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

14

0 02 04 06 08Q

Δp

minus2

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 3 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205832at 120601 = 05 120572 = 07

120573 = 09 and 1205831= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

and the time-averaged flow rate (119876) to observe the impact ofvariation of the intermediate layer viscosity 120583

1 The relation

is linear and the maximum flow rate 1198760 which is dependent

on 1205831 is achieved when Δ119901 is zero However for a fixed

1205832 increase of many folds in 120583

1brings about only a slight

cut in 1198760irrespective of the fluid being pseudoplastic (119899 =

05) Newtonian (119899 = 10) or dilatant (119899 = 15) (seeFigures 2(a)ndash2(c)) But a more interesting observation is thatthe pressure required to restrain flow rates has to increaseby a large magnitude if 120583

1is increased even without any

variation in 1205832 In Figures 2(a)ndash2(c) the curves for 120583

1=

10 show the results for two-layered peristaltic power-lawfluid flow which are similar to results obtained by Misraand Pandey [8] The results in Figure 2(c) are verified withthe results (three-layered peristaltic viscous flow) of Tripathi

[11] Figures 3(a)ndash3(c) are plotted to examine the influence ofthe outer (peripheral) layer viscosity on the pressure versustime-averaged flow rate relation It is noticed that for all thevalues of flow behavior index unlike 120583

1an increase in the

viscosity 1205832of the outer layer causes a slight addition to

the maximum flow rates However on the pressure-flow raterelation 120583

1and 120583

2both have similar effects The influence of

the flow behaviour index 119899 on the pressure-flow rate relationis qualitatively similar to that of intermediate layer viscosity1205831(cf Figure 4)To learn the effects of viscosities in different layers and

power-law fluid index on the mechanical efficiency graphs(Figures 5ndash7) between mechanical efficiency and the ratioof averaged flow rate and maximum averaged flow rate areplotted based on (23) The relation between them is found

Applied Bionics and Biomechanics 9

0

1

2

3

4

5

6

7

8

0 02 04 06 08Q

Δp

minus2

minus1

n = 15

n = 10

n = 05

Figure 4 Pressure difference across one wavelength versus averaged volume flow rate for different values of 119899 at 120601 = 05 120572 = 07 120573 = 091205831= 01 and 120583

2= 01

0

005

01

015

02

025

03

035

04

045

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

E

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1QQ0

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 5 Mechanical efficiency versus ratio of the time-averaged volume flow rate and the maximum time-averaged volume flow rate fordifferent 120583

1at 120601 = 05 120572 = 07 120573 = 09 and 120583

2= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

10 Applied Bionics and Biomechanics

0

01

02

03

04

05

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

0

01

02

03

04

05

06

07

08

09

0 02 04 06 08 1minus01

E

QQ0

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 6 Mechanical efficiency versus ratio of averaged volume flow rate and maximum averaged volume flow rate with the different valuesof 1205832at 120601 = 05 120572 = 07 120573 = 09 and 120583

1= 01 for fixed values of 119899 as follows (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

to be nonlinear An observation of Figures 5(a)ndash5(c) revealsthat the pumping performance improves for all values of 119899when the viscosity 120583

1of the intermediate layer is increased

This too favours the earlier conclusion that flow rate isenhanced for larger viscosity of the intermediate layer Theouter layer viscosity 120583

2too is found to improve the pumping

performance with an increase in its magnitude (cf Figures6(a)ndash6(c)) The results of Figures 6 and 7 agree with theprevious results [8 11] The pumping efficiency is found toincrease with the flow behavior index (119899) for fixed viscositiesof the outer and the intermediate layers Thus peristalticpumping with a pseudoplastic fluid is less efficient than thatwith a fluid (cf Figure 7)

Reflux is an inherent phenomenon of peristaltic pumpingwhich is also named reversal flow To discuss the refluxphenomenon under the effects of viscosities of both layersFigures 8 and 9 are drawn between the amplitude ratio andaveraged flow rate based on (33) It is observed that reflux can

occur in a larger region if the intermediate layer viscosity isincreased (Figure 8) The outer layer viscosity has a similareffect but the change observed is insignificant (Figure 9)

We have determined the trapping limits based on (26) fordifferent viscosities of the intermediate and peripheral layersand also the flow behaviour index to investigate their effectsIt is noticed that when 120583

1varies from 001 to 10 for the fixed

values of 1205832at 01 and 119899 at 05 119876minus rises from 069 to 095

and 119876+ from 095 to 11 which means both the lower andthe upper trapping limits increase with 120583

1 the intermediate

layer viscosity By setting 1205831= 01 and 119899 = 05 if 120583

2is varied

from 001 to 10 119876minus shifts from 098 to 068 and 119876+from 11to 095 showing that both the lower and the upper trappinglimits decrease with 120583

2 When at the fixed values 120583

1= 01

and 1205832= 01 119899 varies from 05 to 15 119876minus varies from 074 to

033 and119876+ varies from 11 to 10 indicating thus that both thelower and the upper trapping limits decline with increasing

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

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2 Applied Bionics and Biomechanics

fluid modeled as power-law fluid through a channel as wellas a tube in the presence of a peripheral layer and pointed outthat flow is enhanced with the peripheral layer viscosity aswell as the flow behavior index They [9] further investigatedthe peripheral layer effect on a core layer of Casson fluidmodelled for blood flow in a channel as well as a tube Apartfrom several other results they concluded that the peripherallayer viscosity favours pumping performance whether thecore layer is a power-law fluid or a Casson fluid Recentlya more general analysis of peristaltic transport with threelayers of different viscosities namely core intermediate andperipheral layers has been presented by Elshehawey andGharsseldien [10] They modeled embryo transfer (the finalstep of IVF) considering the fluid close to it as the core layerthe culture fluid as intermediate layer and the intrauterinefluid as peripheral layer However this concept of threelayers appears appropriate for investigating the role of anartificially generated intermediate layer through medicationto intervene constipation The three-layered peristaltic flowmodel [10] was further applied by Tripathi [11] to studythe axisymmetric blood flow through stenosed arteries byconsidering red blood cells as the core layer plateletswhiteblood cells as the intermediate layer and plasma as theperipheral layer In a previous attempt [12] we studiedsuch a flow for a channel and concluded that an artificiallygenerated comparatively viscous intermediate layer may helpease constipationOne of the refereeswas of the opinion that amore appropriate study will be for the tubular geometryThismotivated us to go for this finer quantitative investigation

In recent years this mechanism (peristalsis) has beenexploited in biomedical engineering industrial technologiesand toxic waste conveyance in chemical process engineeringContinuous refinements in designs require increasinglymoresophisticated models for peristaltic flows using complexworking fluids (non-Newtonian nanotechnological etc)The proposed model may be applicable and help investigatesimilar flows discussed above of biomedical and chemicalindustriesTherefore we propose to investigate axisymmetricperistaltic flow of a power-law fluid in three layers througha cylindrical tube The models [6 7 11 13] will be specialcases of this model Comparative study with previous inves-tigations is made

This paper is designed as follows Section 2 presentsthe formulation of the model wherein a dimensionless setof governing equations subject to appropriate boundaryconditions is derived Analytical wave form solutions areobtained in Section 3 Interface analysis pumping charac-teristic mechanical efficiency trapping and reflux are pre-sented in subsequent sections from 4 through 8 respectivelySection 9 provides the detailed numerical results along withinterpretationsThe final section concludes with our findings

2 Formulation of the Model

We consider the flow of a power-law fluid through a tubein three layers differing in terms of viscosity As discussedabove the three layers will be called core intermediateand peripheral layer The walls of the tube are considered

1205832

1205831

1205830 q3

q2

q1

120573

120572

H

H2

H1

Figure 1 The diagram represents the geometry of the three-layeredperistaltic tubular flow 119867 119867

2 1198671are radial displacement for

peripheral intermediate and core layers respectively 1205830 1205832 1205831and

1199021 1199022 1199023are the viscosities and flow rates in different layers

to be flexible so as to model the small intestine and alsobecause periodic transverse progressive wave trains are topropagate along the walls Details of the geometry of three-layered (peripheral layer intermediate layer and core layer)peristaltic flow region are presented in Figure 1

The following nondimensional variables are used in theanalysis

1199111015840=119911

120582 119903

1015840=119903

119886 119905

1015840=119888119905

120582 119906

1015840=119906

119888

V1015840 =V119896119888 119901

1015840=119901119886119899+1

1205830120582119888 119896 =

119886

120582

1205831015840=120583

1205830

Ψ1015840=

Ψ

1205871198862119888 ℎ

1015840=ℎ

119886

ℎ1015840

1=ℎ1

119886 ℎ

1015840

2=ℎ2

119886 Re =

119886119899+1119888120588

1205821205830

(1)

where 119888 120582 119886 119899 and 1205830represent respectively the wave-

speed the wavelength the radius of the tube the power-lawflow behavior index and the viscosity parameter of the corefluid The other parameters are 119909 119910 119905 119901 120583 120595 ℎ ℎ

1 and

ℎ2which represent respectively axial and transverse coor-

dinates time pressure viscosity stream function and thedistance from the centre line of the outer the intermediateand the core layer boundary their primed counterparts arethe corresponding quantities in the dimensionless form Reis the Reynolds number

Using the nondimensional variables and dropping theprimes the equation of the wall may be given by 119903 = 119867(119911 minus119905) which may acquire an arbitrary form Since the threefluids are mutually immiscible and properly conserve therespective masses during the flow the interfaces between theintermediate and core layers denoted by 119903 = 119867

1(119911minus119905) and the

peripheral and intermediate layers denoted by 119903 = 1198672(119911 minus 119905)

will be streamlinesUsing the nondimensional variables defined above and

considering the long wavelength and low Reynolds number

Applied Bionics and Biomechanics 3

approximations the governing equations of the flow of apower-law fluid after dropping the primes reduce to

120597119901

120597119911= (sign 120597119906

120597119903) [

1

119903

120597

120597119903119903120583

10038161003816100381610038161003816100381610038161003816

120597119906

120597119903

10038161003816100381610038161003816100381610038161003816

119899

]

120597119901

120597119903= 0

(2)

The following are the transformations from the laboratoryframe to the wave frame

119911 minus 119905 = 119909 119903 = 119903 119906 minus 1 = 119908

V = V 119901 = 119901

(3)

where the terms on the left side of the equality sign are in thelaboratory frame and those on the right side are in the waveframe explicitly119909 and119908 are the axial coordinate and the axialvelocity in the wave frame and 119906 is the axial velocity in thelaboratory frame other parameters remain invariant in bothframes Accordingly the stream function 120595 and the volumeflow rate 119902 in the wave frame are related to the correspondingterms in the laboratory fame Ψ being the stream function inthe laboratory frame as given in the following

120595 = Ψ minus1199032

2

119902 = 119876 minus (1 +1206012

2) = 119902

1+ 1199022+ 1199023

(4)

where 1199021 1199022 and 119902

3are the volume flow rates respectively

in the core intermediate and peripheral regions in the waveframe and 119876 = int1

0119876119889119905 is the total volume flow rate averaged

over a period henceforth known as the time-averaged flowrate in the laboratory frame Consider

119902lowast

1= 1198761minus int

1

0

1198671

2(119909) 119889119909 = 119902

1

119902lowast

2= 1198762minus int

1

0

1198672

2(119909) 119889119909 = 119902

1+ 1199022

(5)

where 1198761and 119876

2are the time-averaged flow rates in the

core and the intermediate layers respectively and 119902lowast1and 119902lowast2

are parameters defined by the aforementioned expressionsused to show the flow-rate relations between the wave andlaboratory frames Equations (2) may be expressed in thewave frame in terms of stream function 120595(119903 119909) as follows

120597119901

120597119909= [(sign 1

119903

120597120595

120597119903)1

119903

120597

120597119903(119903120583

10038161003816100381610038161003816100381610038161003816

120597

120597119903(1

119903

120597120595

120597119903)

10038161003816100381610038161003816100381610038161003816

119899

)]

120597119901

120597119903= 0

(6)

The variable viscosity is defined as follows

120583 =

1 0 le 119903 le 1198671

1205831

1198671le 119903 le 119867

2

1205832

1198672le 119903 le 119867

(7)

and the boundary conditions to be imposed on (6) are

120595 = 0 (1

119903120595119903)119903

= 0 at 119903 = 0

120595119903= minus 119867 at 119903 = 119867

120595 =119902

2= const at 119903 = 119867

120595 =119902lowast

1

2= const at 119903 = 119867

1

120595 =119902lowast

2

2= const at 119903 = 119867

2

(8)

The first boundary condition is that the stream functionand the velocity gradient are zero on the symmetry linethe second boundary condition is no-slip condition at theboundary119867 whereas the other three conditions denote massconservation in the three layers This further implies that119867

1

as well as1198672is a streamline in the course of the flow

3 Solution in the Wave Frame of Reference

Eliminating the pressure from the differential equations (6)we get

(sign(1119903

120597120595

120597119903))

120597

120597119903[1

119903

120597

120597119903(119903120583

10038161003816100381610038161003816100381610038161003816

120597

120597119903(1

119903

120597120595

120597119903)

10038161003816100381610038161003816100381610038161003816

119899

)] = 0 (9)

Integrating this equation and using the boundary conditions(8) we get

120595 = minus1

2[1199032minus (119902 + 119867

2)1198682

1198681

] (10)

where

1198681= int

119867

0

119903 int

119867

119903

(119904

120583)

1119899

119889119904 119889119903

1198682= int

119903

0

119904 int

119867

119904

(1199041

120583)

1119899

1198891199041119889119904

(11)

4 Applied Bionics and Biomechanics

Introducing definition (7) the stream function for the threeregions is given by

120595 = minus1

21199032+1

21199032(119902 + 119867

2) (3119899 + 1

119899 + 1)

sdot (1 minus1

1205831119899

1

)119867(119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2

+1

1205831119899

2

119867(119899+1)119899

minus (2119899

3119899 + 1) 119903(119899+1)119899

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 0 le 119903 le 1198671

(12)

120595 = minus1

21199032+1

2(119902 + 119867

2)

sdot [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

3119899 + 1

119899 + 1) 1199032

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

minus(2119899

3119899 + 1)

1

1205831119899

1

119903(119899+1)119899

]

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 1198671le 119903 le 119867

2

(13)

120595 = minus1

21199032+1

2(119902 + 119867

2)

sdot [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+ (3119899 + 1

119899 + 1)

1

1205831119899

2

1199032

times 119867(119899+1)119899

minus (2119899

3119899 + 1) 119903(119899+1)119899

]

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 1198672le 119903 le 119867

(14)

Moreover the pressure gradient in view of (12)ndash(14) is givenby

119889119901

119889119909= minus2 (

3119899 + 1

119899)

119899

sdot [

[

(119902 + 1198672)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

]

]

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus((119902 + 1198672)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

(15)

These results are valid for any arbitrary wave-shape givenby 119910 = 119867(119909) A further investigation of the problem willhowever be carried out for a sinusoidal wave form given by

119867(119909) = 1 + 120601 sin 2120587119909 (16)

Applied Bionics and Biomechanics 5

4 Interfaces Analysis

Using the boundary conditions (8) we obtain the followingequations for the two interfaces119902lowast

1+ 1198671

2

119902 + 1198672

= (1 minus (3119899 + 1

119899 + 1)

1

1205831119899

1

119867(3119899+1)119899

1

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

21198671

2

+ (3119899 + 1

119899 + 1)

1

1205831119899

2

119867(119899+1)119899

1198671

2)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

(17)

119902lowast

2+ 1198672

2

119902 + 1198672

= ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+

1

1205831119899

1

minus (3119899 + 1

119899 + 1)

1

1205831119899

2

sdot119867(3119899+1)119899

2+ (

3119899 + 1

119899 + 1)

1

1205831119899

2

119867(119899+1)119899

1198672

2)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)

sdot119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

)

minus1

(18)

Equations (17) and (18) will be simultaneously solved numer-ically to evaluate119867

1and119867

2as a function of 119909 (see Appendix

for the details of solution)

5 The Pumping Characteristic

Integrating the axial pressure gradient with respect to 119911 overone wavelength we get

Δ119901 = minus2 (3119899 + 1

119899)

119899

sdot int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

(19)

where

120574 (119909) = [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1

119867(3119899+1)119899+ (

1

1205831119899

1

minus1

1205831119899

2

)

sdot119867(3119899+1)119899

2

119867(3119899+1)119899+

1

1205831119899

2

]

minus1

(20)

Δ119901 attains its maximum value Δ1199010when 119876 = 0 Similarly 119876

reaches1198760 themaximumflow rate whenΔ119901 = 0 HenceΔ119901

0

may be explicitly given by

Δ1199010= minus2 (

3119899 + 1

119899)

119899

sdot int

1

0

120574 (119909)1198672minus 1 minus 120601

22

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)1198672minus 1 minus 120601

22

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

(21)

and 1198760may be evaluated from the implicit relation

int

1

0

120574 (119909)1198760+ 1198672minus 1 minus 120601

22

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)1198760+ 1198672minus 1 minus 120601

22

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909 = 0

(22)

6 Mechanical Efficiency of Pumping

Themechanical efficiency is defined as the ratio between theaverage rate per wavelength at which work is done by themoving fluid against a pressure head and the average rate atwhich the wall does work on the fluid (cf Shapiro et al [13])For a power-law fluid it will be given by

119864 = 119876(int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

sdot (int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

(1198672minus 1) 119889119909)

minus1

)

(23)

7 Trapping Limits

Trapping is a phenomenon of peristalsis in which an inter-nally circulating bolus of fluid is formed by closed streamlinesand this trapped bolus is pushed ahead along with theperistaltic wave (cf Shapiro et al [13]) It takes place at highflow rates and large occlusions Trapping limit is given bythe value of 119876 where 120595 = 0 for 119910 gt 0 For the given waveinformation there is a range of 119876 for trapping to occur The

6 Applied Bionics and Biomechanics

shape of the trapped region is obtained by setting 120595 = 0 for119910 = 0 in (12) as follows

119903(119899+1)119899

=1

2119899 (119902 + 1198672)

sdot [(1 minus1

1205831119899

1

)119867(119899+1)119899

1

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

1

2

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2

2

+1

1205831119899

2

119867(119899+1)119899

(3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2]

(24)

For the occurrence of trapping we must have 119903(119899+1)119899 gt 0 forsome 119909 So for a real and positive 119903(119899+1)119899 it is required thatboth the numerator and denominator be of the same signHere the denominator has its maximum value at 119909 = 14

and theminimum value at 119909 = 34Thus in this case we havetwo conditions imposed on 119876 for the existence of trappingConsider

119876minus

lt 119876 lt 119876+

(25)

where

119876minus

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1max + (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1max

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(119899+1)119899

])

minus1

) minus 2120601 minus1206012

2

119876+

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(119899+1)119899

])

minus1

) + 2120601 minus1206012

2

(26)

1198671max 1198672max 1198671min and 119867

2min are the values of theinterfaces at 119909 = 14 and 119909 = 34 respectively For 119899 = 1 theresults reduce to the results of Tripathi [11] for three-layeredperistaltic viscous flow For 120583

1= 1 the results reduce to those

reported by Misra and Pandey [8] for two-layered power-lawflow which for 119899 = 1 further reduce to those presented byRao andUsha [6] for two-layeredNewtonian flowThe resultsof Shapiro et al [13] model can be obtained by substituting1205831= 1 120583

2= 1 and 119899 = 1 in presented model

8 Reflux Limit

Reflux is another phenomenon concerning peristalsis inwhich some fluid particles on the average move in adirection opposite to the net flow (cf Shapiro et al [13])The volume flow rate 119876

120595(119909) corresponding to a particle at

a position 119911 and time 119905 in the laboratory frame and on thestreamline 120595 in the wave frame is given as

119876120595(119909) = 2120595 + 119903

2(120595 119909) (27)

On averaging over one cycle this becomes

119876120595(119909) = 2120595 + int

1

0

1199032(120595 119909) 119889119909 (28)

For evaluating the reflux limit we expand 119876120595(119909) in a power

series in terms of a small parameter 120576 about the wall where

120576 = 120595 minus119902

2 (29)

and use the reflux condition119876120595

119876

gt 1 as 120576 997888rarr 0 (30)

The coefficients of the first two terms in the expansion of1199032(120595 119909) that is

1199032= 1198672+ 1198861120576 + 11988621205762+ sdot sdot sdot (31)

Applied Bionics and Biomechanics 7

0

01

02

03

04

0 02 04 06 08Q

Δp

(a)

0

1

2

3

4

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

0 02 04 06 08Q

Δp

minus2

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 2 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205831at 120601 = 05 120572 = 07

120573 = 09 and 1205832= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

are obtained by using (14) and given by

1198861= minus2

1198862= minus

4 (119902 + 1198672)

11986741205831119899

2

120574 (119909)

(32)

Performing the integration in (28) to the second orderand using condition (30) we obtain the condition for theoccurrence of reflux as follows

int

1

0

((119876 minus 1 minus1206012

2+ 1198672)119867(1minus119899)119899

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1minus 1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)119889119909

lt 0

(33)

9 Numerical Results and Discussion

We have carried out numerical calculations and plottedgraphs to observe the effects of the viscosities of the interme-diate and peripheral layers and the flow behaviour index onpumping mechanical efficiency trapping limits and refluxlimit

Figures 2(a)ndash2(c) are based on (15) and are plottedbetween the pressure difference across one wavelength (Δ119901)

8 Applied Bionics and Biomechanics

0

01

02

03

04

05

06

07

0 02 04 06 08Q

Δp

minus01

(a)

0

1

2

3

4

5

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

14

0 02 04 06 08Q

Δp

minus2

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 3 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205832at 120601 = 05 120572 = 07

120573 = 09 and 1205831= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

and the time-averaged flow rate (119876) to observe the impact ofvariation of the intermediate layer viscosity 120583

1 The relation

is linear and the maximum flow rate 1198760 which is dependent

on 1205831 is achieved when Δ119901 is zero However for a fixed

1205832 increase of many folds in 120583

1brings about only a slight

cut in 1198760irrespective of the fluid being pseudoplastic (119899 =

05) Newtonian (119899 = 10) or dilatant (119899 = 15) (seeFigures 2(a)ndash2(c)) But a more interesting observation is thatthe pressure required to restrain flow rates has to increaseby a large magnitude if 120583

1is increased even without any

variation in 1205832 In Figures 2(a)ndash2(c) the curves for 120583

1=

10 show the results for two-layered peristaltic power-lawfluid flow which are similar to results obtained by Misraand Pandey [8] The results in Figure 2(c) are verified withthe results (three-layered peristaltic viscous flow) of Tripathi

[11] Figures 3(a)ndash3(c) are plotted to examine the influence ofthe outer (peripheral) layer viscosity on the pressure versustime-averaged flow rate relation It is noticed that for all thevalues of flow behavior index unlike 120583

1an increase in the

viscosity 1205832of the outer layer causes a slight addition to

the maximum flow rates However on the pressure-flow raterelation 120583

1and 120583

2both have similar effects The influence of

the flow behaviour index 119899 on the pressure-flow rate relationis qualitatively similar to that of intermediate layer viscosity1205831(cf Figure 4)To learn the effects of viscosities in different layers and

power-law fluid index on the mechanical efficiency graphs(Figures 5ndash7) between mechanical efficiency and the ratioof averaged flow rate and maximum averaged flow rate areplotted based on (23) The relation between them is found

Applied Bionics and Biomechanics 9

0

1

2

3

4

5

6

7

8

0 02 04 06 08Q

Δp

minus2

minus1

n = 15

n = 10

n = 05

Figure 4 Pressure difference across one wavelength versus averaged volume flow rate for different values of 119899 at 120601 = 05 120572 = 07 120573 = 091205831= 01 and 120583

2= 01

0

005

01

015

02

025

03

035

04

045

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

E

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1QQ0

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 5 Mechanical efficiency versus ratio of the time-averaged volume flow rate and the maximum time-averaged volume flow rate fordifferent 120583

1at 120601 = 05 120572 = 07 120573 = 09 and 120583

2= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

10 Applied Bionics and Biomechanics

0

01

02

03

04

05

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

0

01

02

03

04

05

06

07

08

09

0 02 04 06 08 1minus01

E

QQ0

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 6 Mechanical efficiency versus ratio of averaged volume flow rate and maximum averaged volume flow rate with the different valuesof 1205832at 120601 = 05 120572 = 07 120573 = 09 and 120583

1= 01 for fixed values of 119899 as follows (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

to be nonlinear An observation of Figures 5(a)ndash5(c) revealsthat the pumping performance improves for all values of 119899when the viscosity 120583

1of the intermediate layer is increased

This too favours the earlier conclusion that flow rate isenhanced for larger viscosity of the intermediate layer Theouter layer viscosity 120583

2too is found to improve the pumping

performance with an increase in its magnitude (cf Figures6(a)ndash6(c)) The results of Figures 6 and 7 agree with theprevious results [8 11] The pumping efficiency is found toincrease with the flow behavior index (119899) for fixed viscositiesof the outer and the intermediate layers Thus peristalticpumping with a pseudoplastic fluid is less efficient than thatwith a fluid (cf Figure 7)

Reflux is an inherent phenomenon of peristaltic pumpingwhich is also named reversal flow To discuss the refluxphenomenon under the effects of viscosities of both layersFigures 8 and 9 are drawn between the amplitude ratio andaveraged flow rate based on (33) It is observed that reflux can

occur in a larger region if the intermediate layer viscosity isincreased (Figure 8) The outer layer viscosity has a similareffect but the change observed is insignificant (Figure 9)

We have determined the trapping limits based on (26) fordifferent viscosities of the intermediate and peripheral layersand also the flow behaviour index to investigate their effectsIt is noticed that when 120583

1varies from 001 to 10 for the fixed

values of 1205832at 01 and 119899 at 05 119876minus rises from 069 to 095

and 119876+ from 095 to 11 which means both the lower andthe upper trapping limits increase with 120583

1 the intermediate

layer viscosity By setting 1205831= 01 and 119899 = 05 if 120583

2is varied

from 001 to 10 119876minus shifts from 098 to 068 and 119876+from 11to 095 showing that both the lower and the upper trappinglimits decrease with 120583

2 When at the fixed values 120583

1= 01

and 1205832= 01 119899 varies from 05 to 15 119876minus varies from 074 to

033 and119876+ varies from 11 to 10 indicating thus that both thelower and the upper trapping limits decline with increasing

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

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Applied Bionics and Biomechanics 3

approximations the governing equations of the flow of apower-law fluid after dropping the primes reduce to

120597119901

120597119911= (sign 120597119906

120597119903) [

1

119903

120597

120597119903119903120583

10038161003816100381610038161003816100381610038161003816

120597119906

120597119903

10038161003816100381610038161003816100381610038161003816

119899

]

120597119901

120597119903= 0

(2)

The following are the transformations from the laboratoryframe to the wave frame

119911 minus 119905 = 119909 119903 = 119903 119906 minus 1 = 119908

V = V 119901 = 119901

(3)

where the terms on the left side of the equality sign are in thelaboratory frame and those on the right side are in the waveframe explicitly119909 and119908 are the axial coordinate and the axialvelocity in the wave frame and 119906 is the axial velocity in thelaboratory frame other parameters remain invariant in bothframes Accordingly the stream function 120595 and the volumeflow rate 119902 in the wave frame are related to the correspondingterms in the laboratory fame Ψ being the stream function inthe laboratory frame as given in the following

120595 = Ψ minus1199032

2

119902 = 119876 minus (1 +1206012

2) = 119902

1+ 1199022+ 1199023

(4)

where 1199021 1199022 and 119902

3are the volume flow rates respectively

in the core intermediate and peripheral regions in the waveframe and 119876 = int1

0119876119889119905 is the total volume flow rate averaged

over a period henceforth known as the time-averaged flowrate in the laboratory frame Consider

119902lowast

1= 1198761minus int

1

0

1198671

2(119909) 119889119909 = 119902

1

119902lowast

2= 1198762minus int

1

0

1198672

2(119909) 119889119909 = 119902

1+ 1199022

(5)

where 1198761and 119876

2are the time-averaged flow rates in the

core and the intermediate layers respectively and 119902lowast1and 119902lowast2

are parameters defined by the aforementioned expressionsused to show the flow-rate relations between the wave andlaboratory frames Equations (2) may be expressed in thewave frame in terms of stream function 120595(119903 119909) as follows

120597119901

120597119909= [(sign 1

119903

120597120595

120597119903)1

119903

120597

120597119903(119903120583

10038161003816100381610038161003816100381610038161003816

120597

120597119903(1

119903

120597120595

120597119903)

10038161003816100381610038161003816100381610038161003816

119899

)]

120597119901

120597119903= 0

(6)

The variable viscosity is defined as follows

120583 =

1 0 le 119903 le 1198671

1205831

1198671le 119903 le 119867

2

1205832

1198672le 119903 le 119867

(7)

and the boundary conditions to be imposed on (6) are

120595 = 0 (1

119903120595119903)119903

= 0 at 119903 = 0

120595119903= minus 119867 at 119903 = 119867

120595 =119902

2= const at 119903 = 119867

120595 =119902lowast

1

2= const at 119903 = 119867

1

120595 =119902lowast

2

2= const at 119903 = 119867

2

(8)

The first boundary condition is that the stream functionand the velocity gradient are zero on the symmetry linethe second boundary condition is no-slip condition at theboundary119867 whereas the other three conditions denote massconservation in the three layers This further implies that119867

1

as well as1198672is a streamline in the course of the flow

3 Solution in the Wave Frame of Reference

Eliminating the pressure from the differential equations (6)we get

(sign(1119903

120597120595

120597119903))

120597

120597119903[1

119903

120597

120597119903(119903120583

10038161003816100381610038161003816100381610038161003816

120597

120597119903(1

119903

120597120595

120597119903)

10038161003816100381610038161003816100381610038161003816

119899

)] = 0 (9)

Integrating this equation and using the boundary conditions(8) we get

120595 = minus1

2[1199032minus (119902 + 119867

2)1198682

1198681

] (10)

where

1198681= int

119867

0

119903 int

119867

119903

(119904

120583)

1119899

119889119904 119889119903

1198682= int

119903

0

119904 int

119867

119904

(1199041

120583)

1119899

1198891199041119889119904

(11)

4 Applied Bionics and Biomechanics

Introducing definition (7) the stream function for the threeregions is given by

120595 = minus1

21199032+1

21199032(119902 + 119867

2) (3119899 + 1

119899 + 1)

sdot (1 minus1

1205831119899

1

)119867(119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2

+1

1205831119899

2

119867(119899+1)119899

minus (2119899

3119899 + 1) 119903(119899+1)119899

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 0 le 119903 le 1198671

(12)

120595 = minus1

21199032+1

2(119902 + 119867

2)

sdot [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

3119899 + 1

119899 + 1) 1199032

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

minus(2119899

3119899 + 1)

1

1205831119899

1

119903(119899+1)119899

]

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 1198671le 119903 le 119867

2

(13)

120595 = minus1

21199032+1

2(119902 + 119867

2)

sdot [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+ (3119899 + 1

119899 + 1)

1

1205831119899

2

1199032

times 119867(119899+1)119899

minus (2119899

3119899 + 1) 119903(119899+1)119899

]

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 1198672le 119903 le 119867

(14)

Moreover the pressure gradient in view of (12)ndash(14) is givenby

119889119901

119889119909= minus2 (

3119899 + 1

119899)

119899

sdot [

[

(119902 + 1198672)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

]

]

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus((119902 + 1198672)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

(15)

These results are valid for any arbitrary wave-shape givenby 119910 = 119867(119909) A further investigation of the problem willhowever be carried out for a sinusoidal wave form given by

119867(119909) = 1 + 120601 sin 2120587119909 (16)

Applied Bionics and Biomechanics 5

4 Interfaces Analysis

Using the boundary conditions (8) we obtain the followingequations for the two interfaces119902lowast

1+ 1198671

2

119902 + 1198672

= (1 minus (3119899 + 1

119899 + 1)

1

1205831119899

1

119867(3119899+1)119899

1

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

21198671

2

+ (3119899 + 1

119899 + 1)

1

1205831119899

2

119867(119899+1)119899

1198671

2)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

(17)

119902lowast

2+ 1198672

2

119902 + 1198672

= ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+

1

1205831119899

1

minus (3119899 + 1

119899 + 1)

1

1205831119899

2

sdot119867(3119899+1)119899

2+ (

3119899 + 1

119899 + 1)

1

1205831119899

2

119867(119899+1)119899

1198672

2)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)

sdot119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

)

minus1

(18)

Equations (17) and (18) will be simultaneously solved numer-ically to evaluate119867

1and119867

2as a function of 119909 (see Appendix

for the details of solution)

5 The Pumping Characteristic

Integrating the axial pressure gradient with respect to 119911 overone wavelength we get

Δ119901 = minus2 (3119899 + 1

119899)

119899

sdot int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

(19)

where

120574 (119909) = [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1

119867(3119899+1)119899+ (

1

1205831119899

1

minus1

1205831119899

2

)

sdot119867(3119899+1)119899

2

119867(3119899+1)119899+

1

1205831119899

2

]

minus1

(20)

Δ119901 attains its maximum value Δ1199010when 119876 = 0 Similarly 119876

reaches1198760 themaximumflow rate whenΔ119901 = 0 HenceΔ119901

0

may be explicitly given by

Δ1199010= minus2 (

3119899 + 1

119899)

119899

sdot int

1

0

120574 (119909)1198672minus 1 minus 120601

22

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)1198672minus 1 minus 120601

22

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

(21)

and 1198760may be evaluated from the implicit relation

int

1

0

120574 (119909)1198760+ 1198672minus 1 minus 120601

22

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)1198760+ 1198672minus 1 minus 120601

22

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909 = 0

(22)

6 Mechanical Efficiency of Pumping

Themechanical efficiency is defined as the ratio between theaverage rate per wavelength at which work is done by themoving fluid against a pressure head and the average rate atwhich the wall does work on the fluid (cf Shapiro et al [13])For a power-law fluid it will be given by

119864 = 119876(int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

sdot (int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

(1198672minus 1) 119889119909)

minus1

)

(23)

7 Trapping Limits

Trapping is a phenomenon of peristalsis in which an inter-nally circulating bolus of fluid is formed by closed streamlinesand this trapped bolus is pushed ahead along with theperistaltic wave (cf Shapiro et al [13]) It takes place at highflow rates and large occlusions Trapping limit is given bythe value of 119876 where 120595 = 0 for 119910 gt 0 For the given waveinformation there is a range of 119876 for trapping to occur The

6 Applied Bionics and Biomechanics

shape of the trapped region is obtained by setting 120595 = 0 for119910 = 0 in (12) as follows

119903(119899+1)119899

=1

2119899 (119902 + 1198672)

sdot [(1 minus1

1205831119899

1

)119867(119899+1)119899

1

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

1

2

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2

2

+1

1205831119899

2

119867(119899+1)119899

(3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2]

(24)

For the occurrence of trapping we must have 119903(119899+1)119899 gt 0 forsome 119909 So for a real and positive 119903(119899+1)119899 it is required thatboth the numerator and denominator be of the same signHere the denominator has its maximum value at 119909 = 14

and theminimum value at 119909 = 34Thus in this case we havetwo conditions imposed on 119876 for the existence of trappingConsider

119876minus

lt 119876 lt 119876+

(25)

where

119876minus

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1max + (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1max

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(119899+1)119899

])

minus1

) minus 2120601 minus1206012

2

119876+

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(119899+1)119899

])

minus1

) + 2120601 minus1206012

2

(26)

1198671max 1198672max 1198671min and 119867

2min are the values of theinterfaces at 119909 = 14 and 119909 = 34 respectively For 119899 = 1 theresults reduce to the results of Tripathi [11] for three-layeredperistaltic viscous flow For 120583

1= 1 the results reduce to those

reported by Misra and Pandey [8] for two-layered power-lawflow which for 119899 = 1 further reduce to those presented byRao andUsha [6] for two-layeredNewtonian flowThe resultsof Shapiro et al [13] model can be obtained by substituting1205831= 1 120583

2= 1 and 119899 = 1 in presented model

8 Reflux Limit

Reflux is another phenomenon concerning peristalsis inwhich some fluid particles on the average move in adirection opposite to the net flow (cf Shapiro et al [13])The volume flow rate 119876

120595(119909) corresponding to a particle at

a position 119911 and time 119905 in the laboratory frame and on thestreamline 120595 in the wave frame is given as

119876120595(119909) = 2120595 + 119903

2(120595 119909) (27)

On averaging over one cycle this becomes

119876120595(119909) = 2120595 + int

1

0

1199032(120595 119909) 119889119909 (28)

For evaluating the reflux limit we expand 119876120595(119909) in a power

series in terms of a small parameter 120576 about the wall where

120576 = 120595 minus119902

2 (29)

and use the reflux condition119876120595

119876

gt 1 as 120576 997888rarr 0 (30)

The coefficients of the first two terms in the expansion of1199032(120595 119909) that is

1199032= 1198672+ 1198861120576 + 11988621205762+ sdot sdot sdot (31)

Applied Bionics and Biomechanics 7

0

01

02

03

04

0 02 04 06 08Q

Δp

(a)

0

1

2

3

4

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

0 02 04 06 08Q

Δp

minus2

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 2 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205831at 120601 = 05 120572 = 07

120573 = 09 and 1205832= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

are obtained by using (14) and given by

1198861= minus2

1198862= minus

4 (119902 + 1198672)

11986741205831119899

2

120574 (119909)

(32)

Performing the integration in (28) to the second orderand using condition (30) we obtain the condition for theoccurrence of reflux as follows

int

1

0

((119876 minus 1 minus1206012

2+ 1198672)119867(1minus119899)119899

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1minus 1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)119889119909

lt 0

(33)

9 Numerical Results and Discussion

We have carried out numerical calculations and plottedgraphs to observe the effects of the viscosities of the interme-diate and peripheral layers and the flow behaviour index onpumping mechanical efficiency trapping limits and refluxlimit

Figures 2(a)ndash2(c) are based on (15) and are plottedbetween the pressure difference across one wavelength (Δ119901)

8 Applied Bionics and Biomechanics

0

01

02

03

04

05

06

07

0 02 04 06 08Q

Δp

minus01

(a)

0

1

2

3

4

5

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

14

0 02 04 06 08Q

Δp

minus2

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 3 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205832at 120601 = 05 120572 = 07

120573 = 09 and 1205831= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

and the time-averaged flow rate (119876) to observe the impact ofvariation of the intermediate layer viscosity 120583

1 The relation

is linear and the maximum flow rate 1198760 which is dependent

on 1205831 is achieved when Δ119901 is zero However for a fixed

1205832 increase of many folds in 120583

1brings about only a slight

cut in 1198760irrespective of the fluid being pseudoplastic (119899 =

05) Newtonian (119899 = 10) or dilatant (119899 = 15) (seeFigures 2(a)ndash2(c)) But a more interesting observation is thatthe pressure required to restrain flow rates has to increaseby a large magnitude if 120583

1is increased even without any

variation in 1205832 In Figures 2(a)ndash2(c) the curves for 120583

1=

10 show the results for two-layered peristaltic power-lawfluid flow which are similar to results obtained by Misraand Pandey [8] The results in Figure 2(c) are verified withthe results (three-layered peristaltic viscous flow) of Tripathi

[11] Figures 3(a)ndash3(c) are plotted to examine the influence ofthe outer (peripheral) layer viscosity on the pressure versustime-averaged flow rate relation It is noticed that for all thevalues of flow behavior index unlike 120583

1an increase in the

viscosity 1205832of the outer layer causes a slight addition to

the maximum flow rates However on the pressure-flow raterelation 120583

1and 120583

2both have similar effects The influence of

the flow behaviour index 119899 on the pressure-flow rate relationis qualitatively similar to that of intermediate layer viscosity1205831(cf Figure 4)To learn the effects of viscosities in different layers and

power-law fluid index on the mechanical efficiency graphs(Figures 5ndash7) between mechanical efficiency and the ratioof averaged flow rate and maximum averaged flow rate areplotted based on (23) The relation between them is found

Applied Bionics and Biomechanics 9

0

1

2

3

4

5

6

7

8

0 02 04 06 08Q

Δp

minus2

minus1

n = 15

n = 10

n = 05

Figure 4 Pressure difference across one wavelength versus averaged volume flow rate for different values of 119899 at 120601 = 05 120572 = 07 120573 = 091205831= 01 and 120583

2= 01

0

005

01

015

02

025

03

035

04

045

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

E

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1QQ0

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 5 Mechanical efficiency versus ratio of the time-averaged volume flow rate and the maximum time-averaged volume flow rate fordifferent 120583

1at 120601 = 05 120572 = 07 120573 = 09 and 120583

2= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

10 Applied Bionics and Biomechanics

0

01

02

03

04

05

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

0

01

02

03

04

05

06

07

08

09

0 02 04 06 08 1minus01

E

QQ0

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 6 Mechanical efficiency versus ratio of averaged volume flow rate and maximum averaged volume flow rate with the different valuesof 1205832at 120601 = 05 120572 = 07 120573 = 09 and 120583

1= 01 for fixed values of 119899 as follows (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

to be nonlinear An observation of Figures 5(a)ndash5(c) revealsthat the pumping performance improves for all values of 119899when the viscosity 120583

1of the intermediate layer is increased

This too favours the earlier conclusion that flow rate isenhanced for larger viscosity of the intermediate layer Theouter layer viscosity 120583

2too is found to improve the pumping

performance with an increase in its magnitude (cf Figures6(a)ndash6(c)) The results of Figures 6 and 7 agree with theprevious results [8 11] The pumping efficiency is found toincrease with the flow behavior index (119899) for fixed viscositiesof the outer and the intermediate layers Thus peristalticpumping with a pseudoplastic fluid is less efficient than thatwith a fluid (cf Figure 7)

Reflux is an inherent phenomenon of peristaltic pumpingwhich is also named reversal flow To discuss the refluxphenomenon under the effects of viscosities of both layersFigures 8 and 9 are drawn between the amplitude ratio andaveraged flow rate based on (33) It is observed that reflux can

occur in a larger region if the intermediate layer viscosity isincreased (Figure 8) The outer layer viscosity has a similareffect but the change observed is insignificant (Figure 9)

We have determined the trapping limits based on (26) fordifferent viscosities of the intermediate and peripheral layersand also the flow behaviour index to investigate their effectsIt is noticed that when 120583

1varies from 001 to 10 for the fixed

values of 1205832at 01 and 119899 at 05 119876minus rises from 069 to 095

and 119876+ from 095 to 11 which means both the lower andthe upper trapping limits increase with 120583

1 the intermediate

layer viscosity By setting 1205831= 01 and 119899 = 05 if 120583

2is varied

from 001 to 10 119876minus shifts from 098 to 068 and 119876+from 11to 095 showing that both the lower and the upper trappinglimits decrease with 120583

2 When at the fixed values 120583

1= 01

and 1205832= 01 119899 varies from 05 to 15 119876minus varies from 074 to

033 and119876+ varies from 11 to 10 indicating thus that both thelower and the upper trapping limits decline with increasing

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

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4 Applied Bionics and Biomechanics

Introducing definition (7) the stream function for the threeregions is given by

120595 = minus1

21199032+1

21199032(119902 + 119867

2) (3119899 + 1

119899 + 1)

sdot (1 minus1

1205831119899

1

)119867(119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2

+1

1205831119899

2

119867(119899+1)119899

minus (2119899

3119899 + 1) 119903(119899+1)119899

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 0 le 119903 le 1198671

(12)

120595 = minus1

21199032+1

2(119902 + 119867

2)

sdot [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

3119899 + 1

119899 + 1) 1199032

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

minus(2119899

3119899 + 1)

1

1205831119899

1

119903(119899+1)119899

]

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 1198671le 119903 le 119867

2

(13)

120595 = minus1

21199032+1

2(119902 + 119867

2)

sdot [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+ (3119899 + 1

119899 + 1)

1

1205831119899

2

1199032

times 119867(119899+1)119899

minus (2119899

3119899 + 1) 119903(119899+1)119899

]

times (1 sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

for 1198672le 119903 le 119867

(14)

Moreover the pressure gradient in view of (12)ndash(14) is givenby

119889119901

119889119909= minus2 (

3119899 + 1

119899)

119899

sdot [

[

(119902 + 1198672)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

]

]

times

10038161003816100381610038161003816100381610038161003816100381610038161003816

minus((119902 + 1198672)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

(15)

These results are valid for any arbitrary wave-shape givenby 119910 = 119867(119909) A further investigation of the problem willhowever be carried out for a sinusoidal wave form given by

119867(119909) = 1 + 120601 sin 2120587119909 (16)

Applied Bionics and Biomechanics 5

4 Interfaces Analysis

Using the boundary conditions (8) we obtain the followingequations for the two interfaces119902lowast

1+ 1198671

2

119902 + 1198672

= (1 minus (3119899 + 1

119899 + 1)

1

1205831119899

1

119867(3119899+1)119899

1

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

21198671

2

+ (3119899 + 1

119899 + 1)

1

1205831119899

2

119867(119899+1)119899

1198671

2)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

(17)

119902lowast

2+ 1198672

2

119902 + 1198672

= ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+

1

1205831119899

1

minus (3119899 + 1

119899 + 1)

1

1205831119899

2

sdot119867(3119899+1)119899

2+ (

3119899 + 1

119899 + 1)

1

1205831119899

2

119867(119899+1)119899

1198672

2)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)

sdot119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

)

minus1

(18)

Equations (17) and (18) will be simultaneously solved numer-ically to evaluate119867

1and119867

2as a function of 119909 (see Appendix

for the details of solution)

5 The Pumping Characteristic

Integrating the axial pressure gradient with respect to 119911 overone wavelength we get

Δ119901 = minus2 (3119899 + 1

119899)

119899

sdot int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

(19)

where

120574 (119909) = [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1

119867(3119899+1)119899+ (

1

1205831119899

1

minus1

1205831119899

2

)

sdot119867(3119899+1)119899

2

119867(3119899+1)119899+

1

1205831119899

2

]

minus1

(20)

Δ119901 attains its maximum value Δ1199010when 119876 = 0 Similarly 119876

reaches1198760 themaximumflow rate whenΔ119901 = 0 HenceΔ119901

0

may be explicitly given by

Δ1199010= minus2 (

3119899 + 1

119899)

119899

sdot int

1

0

120574 (119909)1198672minus 1 minus 120601

22

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)1198672minus 1 minus 120601

22

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

(21)

and 1198760may be evaluated from the implicit relation

int

1

0

120574 (119909)1198760+ 1198672minus 1 minus 120601

22

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)1198760+ 1198672minus 1 minus 120601

22

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909 = 0

(22)

6 Mechanical Efficiency of Pumping

Themechanical efficiency is defined as the ratio between theaverage rate per wavelength at which work is done by themoving fluid against a pressure head and the average rate atwhich the wall does work on the fluid (cf Shapiro et al [13])For a power-law fluid it will be given by

119864 = 119876(int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

sdot (int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

(1198672minus 1) 119889119909)

minus1

)

(23)

7 Trapping Limits

Trapping is a phenomenon of peristalsis in which an inter-nally circulating bolus of fluid is formed by closed streamlinesand this trapped bolus is pushed ahead along with theperistaltic wave (cf Shapiro et al [13]) It takes place at highflow rates and large occlusions Trapping limit is given bythe value of 119876 where 120595 = 0 for 119910 gt 0 For the given waveinformation there is a range of 119876 for trapping to occur The

6 Applied Bionics and Biomechanics

shape of the trapped region is obtained by setting 120595 = 0 for119910 = 0 in (12) as follows

119903(119899+1)119899

=1

2119899 (119902 + 1198672)

sdot [(1 minus1

1205831119899

1

)119867(119899+1)119899

1

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

1

2

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2

2

+1

1205831119899

2

119867(119899+1)119899

(3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2]

(24)

For the occurrence of trapping we must have 119903(119899+1)119899 gt 0 forsome 119909 So for a real and positive 119903(119899+1)119899 it is required thatboth the numerator and denominator be of the same signHere the denominator has its maximum value at 119909 = 14

and theminimum value at 119909 = 34Thus in this case we havetwo conditions imposed on 119876 for the existence of trappingConsider

119876minus

lt 119876 lt 119876+

(25)

where

119876minus

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1max + (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1max

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(119899+1)119899

])

minus1

) minus 2120601 minus1206012

2

119876+

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(119899+1)119899

])

minus1

) + 2120601 minus1206012

2

(26)

1198671max 1198672max 1198671min and 119867

2min are the values of theinterfaces at 119909 = 14 and 119909 = 34 respectively For 119899 = 1 theresults reduce to the results of Tripathi [11] for three-layeredperistaltic viscous flow For 120583

1= 1 the results reduce to those

reported by Misra and Pandey [8] for two-layered power-lawflow which for 119899 = 1 further reduce to those presented byRao andUsha [6] for two-layeredNewtonian flowThe resultsof Shapiro et al [13] model can be obtained by substituting1205831= 1 120583

2= 1 and 119899 = 1 in presented model

8 Reflux Limit

Reflux is another phenomenon concerning peristalsis inwhich some fluid particles on the average move in adirection opposite to the net flow (cf Shapiro et al [13])The volume flow rate 119876

120595(119909) corresponding to a particle at

a position 119911 and time 119905 in the laboratory frame and on thestreamline 120595 in the wave frame is given as

119876120595(119909) = 2120595 + 119903

2(120595 119909) (27)

On averaging over one cycle this becomes

119876120595(119909) = 2120595 + int

1

0

1199032(120595 119909) 119889119909 (28)

For evaluating the reflux limit we expand 119876120595(119909) in a power

series in terms of a small parameter 120576 about the wall where

120576 = 120595 minus119902

2 (29)

and use the reflux condition119876120595

119876

gt 1 as 120576 997888rarr 0 (30)

The coefficients of the first two terms in the expansion of1199032(120595 119909) that is

1199032= 1198672+ 1198861120576 + 11988621205762+ sdot sdot sdot (31)

Applied Bionics and Biomechanics 7

0

01

02

03

04

0 02 04 06 08Q

Δp

(a)

0

1

2

3

4

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

0 02 04 06 08Q

Δp

minus2

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 2 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205831at 120601 = 05 120572 = 07

120573 = 09 and 1205832= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

are obtained by using (14) and given by

1198861= minus2

1198862= minus

4 (119902 + 1198672)

11986741205831119899

2

120574 (119909)

(32)

Performing the integration in (28) to the second orderand using condition (30) we obtain the condition for theoccurrence of reflux as follows

int

1

0

((119876 minus 1 minus1206012

2+ 1198672)119867(1minus119899)119899

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1minus 1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)119889119909

lt 0

(33)

9 Numerical Results and Discussion

We have carried out numerical calculations and plottedgraphs to observe the effects of the viscosities of the interme-diate and peripheral layers and the flow behaviour index onpumping mechanical efficiency trapping limits and refluxlimit

Figures 2(a)ndash2(c) are based on (15) and are plottedbetween the pressure difference across one wavelength (Δ119901)

8 Applied Bionics and Biomechanics

0

01

02

03

04

05

06

07

0 02 04 06 08Q

Δp

minus01

(a)

0

1

2

3

4

5

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

14

0 02 04 06 08Q

Δp

minus2

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 3 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205832at 120601 = 05 120572 = 07

120573 = 09 and 1205831= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

and the time-averaged flow rate (119876) to observe the impact ofvariation of the intermediate layer viscosity 120583

1 The relation

is linear and the maximum flow rate 1198760 which is dependent

on 1205831 is achieved when Δ119901 is zero However for a fixed

1205832 increase of many folds in 120583

1brings about only a slight

cut in 1198760irrespective of the fluid being pseudoplastic (119899 =

05) Newtonian (119899 = 10) or dilatant (119899 = 15) (seeFigures 2(a)ndash2(c)) But a more interesting observation is thatthe pressure required to restrain flow rates has to increaseby a large magnitude if 120583

1is increased even without any

variation in 1205832 In Figures 2(a)ndash2(c) the curves for 120583

1=

10 show the results for two-layered peristaltic power-lawfluid flow which are similar to results obtained by Misraand Pandey [8] The results in Figure 2(c) are verified withthe results (three-layered peristaltic viscous flow) of Tripathi

[11] Figures 3(a)ndash3(c) are plotted to examine the influence ofthe outer (peripheral) layer viscosity on the pressure versustime-averaged flow rate relation It is noticed that for all thevalues of flow behavior index unlike 120583

1an increase in the

viscosity 1205832of the outer layer causes a slight addition to

the maximum flow rates However on the pressure-flow raterelation 120583

1and 120583

2both have similar effects The influence of

the flow behaviour index 119899 on the pressure-flow rate relationis qualitatively similar to that of intermediate layer viscosity1205831(cf Figure 4)To learn the effects of viscosities in different layers and

power-law fluid index on the mechanical efficiency graphs(Figures 5ndash7) between mechanical efficiency and the ratioof averaged flow rate and maximum averaged flow rate areplotted based on (23) The relation between them is found

Applied Bionics and Biomechanics 9

0

1

2

3

4

5

6

7

8

0 02 04 06 08Q

Δp

minus2

minus1

n = 15

n = 10

n = 05

Figure 4 Pressure difference across one wavelength versus averaged volume flow rate for different values of 119899 at 120601 = 05 120572 = 07 120573 = 091205831= 01 and 120583

2= 01

0

005

01

015

02

025

03

035

04

045

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

E

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1QQ0

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 5 Mechanical efficiency versus ratio of the time-averaged volume flow rate and the maximum time-averaged volume flow rate fordifferent 120583

1at 120601 = 05 120572 = 07 120573 = 09 and 120583

2= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

10 Applied Bionics and Biomechanics

0

01

02

03

04

05

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

0

01

02

03

04

05

06

07

08

09

0 02 04 06 08 1minus01

E

QQ0

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 6 Mechanical efficiency versus ratio of averaged volume flow rate and maximum averaged volume flow rate with the different valuesof 1205832at 120601 = 05 120572 = 07 120573 = 09 and 120583

1= 01 for fixed values of 119899 as follows (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

to be nonlinear An observation of Figures 5(a)ndash5(c) revealsthat the pumping performance improves for all values of 119899when the viscosity 120583

1of the intermediate layer is increased

This too favours the earlier conclusion that flow rate isenhanced for larger viscosity of the intermediate layer Theouter layer viscosity 120583

2too is found to improve the pumping

performance with an increase in its magnitude (cf Figures6(a)ndash6(c)) The results of Figures 6 and 7 agree with theprevious results [8 11] The pumping efficiency is found toincrease with the flow behavior index (119899) for fixed viscositiesof the outer and the intermediate layers Thus peristalticpumping with a pseudoplastic fluid is less efficient than thatwith a fluid (cf Figure 7)

Reflux is an inherent phenomenon of peristaltic pumpingwhich is also named reversal flow To discuss the refluxphenomenon under the effects of viscosities of both layersFigures 8 and 9 are drawn between the amplitude ratio andaveraged flow rate based on (33) It is observed that reflux can

occur in a larger region if the intermediate layer viscosity isincreased (Figure 8) The outer layer viscosity has a similareffect but the change observed is insignificant (Figure 9)

We have determined the trapping limits based on (26) fordifferent viscosities of the intermediate and peripheral layersand also the flow behaviour index to investigate their effectsIt is noticed that when 120583

1varies from 001 to 10 for the fixed

values of 1205832at 01 and 119899 at 05 119876minus rises from 069 to 095

and 119876+ from 095 to 11 which means both the lower andthe upper trapping limits increase with 120583

1 the intermediate

layer viscosity By setting 1205831= 01 and 119899 = 05 if 120583

2is varied

from 001 to 10 119876minus shifts from 098 to 068 and 119876+from 11to 095 showing that both the lower and the upper trappinglimits decrease with 120583

2 When at the fixed values 120583

1= 01

and 1205832= 01 119899 varies from 05 to 15 119876minus varies from 074 to

033 and119876+ varies from 11 to 10 indicating thus that both thelower and the upper trapping limits decline with increasing

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

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International Journal of

Applied Bionics and Biomechanics 5

4 Interfaces Analysis

Using the boundary conditions (8) we obtain the followingequations for the two interfaces119902lowast

1+ 1198671

2

119902 + 1198672

= (1 minus (3119899 + 1

119899 + 1)

1

1205831119899

1

119867(3119899+1)119899

1

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

21198671

2

+ (3119899 + 1

119899 + 1)

1

1205831119899

2

119867(119899+1)119899

1198671

2)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

(17)

119902lowast

2+ 1198672

2

119902 + 1198672

= ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+

1

1205831119899

1

minus (3119899 + 1

119899 + 1)

1

1205831119899

2

sdot119867(3119899+1)119899

2+ (

3119899 + 1

119899 + 1)

1

1205831119899

2

119867(119899+1)119899

1198672

2)

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1+ (

1

1205831119899

1

minus1

1205831119899

2

)

sdot119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

)

minus1

(18)

Equations (17) and (18) will be simultaneously solved numer-ically to evaluate119867

1and119867

2as a function of 119909 (see Appendix

for the details of solution)

5 The Pumping Characteristic

Integrating the axial pressure gradient with respect to 119911 overone wavelength we get

Δ119901 = minus2 (3119899 + 1

119899)

119899

sdot int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

(19)

where

120574 (119909) = [(1 minus1

1205831119899

1

)119867(3119899+1)119899

1

119867(3119899+1)119899+ (

1

1205831119899

1

minus1

1205831119899

2

)

sdot119867(3119899+1)119899

2

119867(3119899+1)119899+

1

1205831119899

2

]

minus1

(20)

Δ119901 attains its maximum value Δ1199010when 119876 = 0 Similarly 119876

reaches1198760 themaximumflow rate whenΔ119901 = 0 HenceΔ119901

0

may be explicitly given by

Δ1199010= minus2 (

3119899 + 1

119899)

119899

sdot int

1

0

120574 (119909)1198672minus 1 minus 120601

22

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)1198672minus 1 minus 120601

22

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

(21)

and 1198760may be evaluated from the implicit relation

int

1

0

120574 (119909)1198760+ 1198672minus 1 minus 120601

22

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)1198760+ 1198672minus 1 minus 120601

22

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909 = 0

(22)

6 Mechanical Efficiency of Pumping

Themechanical efficiency is defined as the ratio between theaverage rate per wavelength at which work is done by themoving fluid against a pressure head and the average rate atwhich the wall does work on the fluid (cf Shapiro et al [13])For a power-law fluid it will be given by

119864 = 119876(int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

119889119909

sdot (int

1

0

120574 (119909)119902 + 119867

2

119867(3119899+1)119899

sdot

100381610038161003816100381610038161003816100381610038161003816

minus120574 (119909)119902 + 119867

2

119867(3119899+1)119899

100381610038161003816100381610038161003816100381610038161003816

(119899minus1)

(1198672minus 1) 119889119909)

minus1

)

(23)

7 Trapping Limits

Trapping is a phenomenon of peristalsis in which an inter-nally circulating bolus of fluid is formed by closed streamlinesand this trapped bolus is pushed ahead along with theperistaltic wave (cf Shapiro et al [13]) It takes place at highflow rates and large occlusions Trapping limit is given bythe value of 119876 where 120595 = 0 for 119910 gt 0 For the given waveinformation there is a range of 119876 for trapping to occur The

6 Applied Bionics and Biomechanics

shape of the trapped region is obtained by setting 120595 = 0 for119910 = 0 in (12) as follows

119903(119899+1)119899

=1

2119899 (119902 + 1198672)

sdot [(1 minus1

1205831119899

1

)119867(119899+1)119899

1

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

1

2

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2

2

+1

1205831119899

2

119867(119899+1)119899

(3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2]

(24)

For the occurrence of trapping we must have 119903(119899+1)119899 gt 0 forsome 119909 So for a real and positive 119903(119899+1)119899 it is required thatboth the numerator and denominator be of the same signHere the denominator has its maximum value at 119909 = 14

and theminimum value at 119909 = 34Thus in this case we havetwo conditions imposed on 119876 for the existence of trappingConsider

119876minus

lt 119876 lt 119876+

(25)

where

119876minus

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1max + (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1max

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(119899+1)119899

])

minus1

) minus 2120601 minus1206012

2

119876+

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(119899+1)119899

])

minus1

) + 2120601 minus1206012

2

(26)

1198671max 1198672max 1198671min and 119867

2min are the values of theinterfaces at 119909 = 14 and 119909 = 34 respectively For 119899 = 1 theresults reduce to the results of Tripathi [11] for three-layeredperistaltic viscous flow For 120583

1= 1 the results reduce to those

reported by Misra and Pandey [8] for two-layered power-lawflow which for 119899 = 1 further reduce to those presented byRao andUsha [6] for two-layeredNewtonian flowThe resultsof Shapiro et al [13] model can be obtained by substituting1205831= 1 120583

2= 1 and 119899 = 1 in presented model

8 Reflux Limit

Reflux is another phenomenon concerning peristalsis inwhich some fluid particles on the average move in adirection opposite to the net flow (cf Shapiro et al [13])The volume flow rate 119876

120595(119909) corresponding to a particle at

a position 119911 and time 119905 in the laboratory frame and on thestreamline 120595 in the wave frame is given as

119876120595(119909) = 2120595 + 119903

2(120595 119909) (27)

On averaging over one cycle this becomes

119876120595(119909) = 2120595 + int

1

0

1199032(120595 119909) 119889119909 (28)

For evaluating the reflux limit we expand 119876120595(119909) in a power

series in terms of a small parameter 120576 about the wall where

120576 = 120595 minus119902

2 (29)

and use the reflux condition119876120595

119876

gt 1 as 120576 997888rarr 0 (30)

The coefficients of the first two terms in the expansion of1199032(120595 119909) that is

1199032= 1198672+ 1198861120576 + 11988621205762+ sdot sdot sdot (31)

Applied Bionics and Biomechanics 7

0

01

02

03

04

0 02 04 06 08Q

Δp

(a)

0

1

2

3

4

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

0 02 04 06 08Q

Δp

minus2

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 2 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205831at 120601 = 05 120572 = 07

120573 = 09 and 1205832= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

are obtained by using (14) and given by

1198861= minus2

1198862= minus

4 (119902 + 1198672)

11986741205831119899

2

120574 (119909)

(32)

Performing the integration in (28) to the second orderand using condition (30) we obtain the condition for theoccurrence of reflux as follows

int

1

0

((119876 minus 1 minus1206012

2+ 1198672)119867(1minus119899)119899

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1minus 1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)119889119909

lt 0

(33)

9 Numerical Results and Discussion

We have carried out numerical calculations and plottedgraphs to observe the effects of the viscosities of the interme-diate and peripheral layers and the flow behaviour index onpumping mechanical efficiency trapping limits and refluxlimit

Figures 2(a)ndash2(c) are based on (15) and are plottedbetween the pressure difference across one wavelength (Δ119901)

8 Applied Bionics and Biomechanics

0

01

02

03

04

05

06

07

0 02 04 06 08Q

Δp

minus01

(a)

0

1

2

3

4

5

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

14

0 02 04 06 08Q

Δp

minus2

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 3 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205832at 120601 = 05 120572 = 07

120573 = 09 and 1205831= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

and the time-averaged flow rate (119876) to observe the impact ofvariation of the intermediate layer viscosity 120583

1 The relation

is linear and the maximum flow rate 1198760 which is dependent

on 1205831 is achieved when Δ119901 is zero However for a fixed

1205832 increase of many folds in 120583

1brings about only a slight

cut in 1198760irrespective of the fluid being pseudoplastic (119899 =

05) Newtonian (119899 = 10) or dilatant (119899 = 15) (seeFigures 2(a)ndash2(c)) But a more interesting observation is thatthe pressure required to restrain flow rates has to increaseby a large magnitude if 120583

1is increased even without any

variation in 1205832 In Figures 2(a)ndash2(c) the curves for 120583

1=

10 show the results for two-layered peristaltic power-lawfluid flow which are similar to results obtained by Misraand Pandey [8] The results in Figure 2(c) are verified withthe results (three-layered peristaltic viscous flow) of Tripathi

[11] Figures 3(a)ndash3(c) are plotted to examine the influence ofthe outer (peripheral) layer viscosity on the pressure versustime-averaged flow rate relation It is noticed that for all thevalues of flow behavior index unlike 120583

1an increase in the

viscosity 1205832of the outer layer causes a slight addition to

the maximum flow rates However on the pressure-flow raterelation 120583

1and 120583

2both have similar effects The influence of

the flow behaviour index 119899 on the pressure-flow rate relationis qualitatively similar to that of intermediate layer viscosity1205831(cf Figure 4)To learn the effects of viscosities in different layers and

power-law fluid index on the mechanical efficiency graphs(Figures 5ndash7) between mechanical efficiency and the ratioof averaged flow rate and maximum averaged flow rate areplotted based on (23) The relation between them is found

Applied Bionics and Biomechanics 9

0

1

2

3

4

5

6

7

8

0 02 04 06 08Q

Δp

minus2

minus1

n = 15

n = 10

n = 05

Figure 4 Pressure difference across one wavelength versus averaged volume flow rate for different values of 119899 at 120601 = 05 120572 = 07 120573 = 091205831= 01 and 120583

2= 01

0

005

01

015

02

025

03

035

04

045

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

E

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1QQ0

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 5 Mechanical efficiency versus ratio of the time-averaged volume flow rate and the maximum time-averaged volume flow rate fordifferent 120583

1at 120601 = 05 120572 = 07 120573 = 09 and 120583

2= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

10 Applied Bionics and Biomechanics

0

01

02

03

04

05

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

0

01

02

03

04

05

06

07

08

09

0 02 04 06 08 1minus01

E

QQ0

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 6 Mechanical efficiency versus ratio of averaged volume flow rate and maximum averaged volume flow rate with the different valuesof 1205832at 120601 = 05 120572 = 07 120573 = 09 and 120583

1= 01 for fixed values of 119899 as follows (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

to be nonlinear An observation of Figures 5(a)ndash5(c) revealsthat the pumping performance improves for all values of 119899when the viscosity 120583

1of the intermediate layer is increased

This too favours the earlier conclusion that flow rate isenhanced for larger viscosity of the intermediate layer Theouter layer viscosity 120583

2too is found to improve the pumping

performance with an increase in its magnitude (cf Figures6(a)ndash6(c)) The results of Figures 6 and 7 agree with theprevious results [8 11] The pumping efficiency is found toincrease with the flow behavior index (119899) for fixed viscositiesof the outer and the intermediate layers Thus peristalticpumping with a pseudoplastic fluid is less efficient than thatwith a fluid (cf Figure 7)

Reflux is an inherent phenomenon of peristaltic pumpingwhich is also named reversal flow To discuss the refluxphenomenon under the effects of viscosities of both layersFigures 8 and 9 are drawn between the amplitude ratio andaveraged flow rate based on (33) It is observed that reflux can

occur in a larger region if the intermediate layer viscosity isincreased (Figure 8) The outer layer viscosity has a similareffect but the change observed is insignificant (Figure 9)

We have determined the trapping limits based on (26) fordifferent viscosities of the intermediate and peripheral layersand also the flow behaviour index to investigate their effectsIt is noticed that when 120583

1varies from 001 to 10 for the fixed

values of 1205832at 01 and 119899 at 05 119876minus rises from 069 to 095

and 119876+ from 095 to 11 which means both the lower andthe upper trapping limits increase with 120583

1 the intermediate

layer viscosity By setting 1205831= 01 and 119899 = 05 if 120583

2is varied

from 001 to 10 119876minus shifts from 098 to 068 and 119876+from 11to 095 showing that both the lower and the upper trappinglimits decrease with 120583

2 When at the fixed values 120583

1= 01

and 1205832= 01 119899 varies from 05 to 15 119876minus varies from 074 to

033 and119876+ varies from 11 to 10 indicating thus that both thelower and the upper trapping limits decline with increasing

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

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6 Applied Bionics and Biomechanics

shape of the trapped region is obtained by setting 120595 = 0 for119910 = 0 in (12) as follows

119903(119899+1)119899

=1

2119899 (119902 + 1198672)

sdot [(1 minus1

1205831119899

1

)119867(119899+1)119899

1

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

1

2

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2

sdot (3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2

2

+1

1205831119899

2

119867(119899+1)119899

(3119899 + 1) (119902 + 1198672) minus (119899 + 1)119867

2]

(24)

For the occurrence of trapping we must have 119903(119899+1)119899 gt 0 forsome 119909 So for a real and positive 119903(119899+1)119899 it is required thatboth the numerator and denominator be of the same signHere the denominator has its maximum value at 119909 = 14

and theminimum value at 119909 = 34Thus in this case we havetwo conditions imposed on 119876 for the existence of trappingConsider

119876minus

lt 119876 lt 119876+

(25)

where

119876minus

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1max + (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1max

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2max

+1

1205831119899

2

(1 + 120601)(119899+1)119899

])

minus1

) minus 2120601 minus1206012

2

119876+

=(119899 + 1)

(3119899 + 1)

sdot ([(1 minus1

1205831119899

1

)119867(3119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(3119899+1)119899

]

sdot ([(1 minus1

1205831119899

1

)119867(119899+1)119899

1min

+ (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2min

+1

1205831119899

2

(1 minus 120601)(119899+1)119899

])

minus1

) + 2120601 minus1206012

2

(26)

1198671max 1198672max 1198671min and 119867

2min are the values of theinterfaces at 119909 = 14 and 119909 = 34 respectively For 119899 = 1 theresults reduce to the results of Tripathi [11] for three-layeredperistaltic viscous flow For 120583

1= 1 the results reduce to those

reported by Misra and Pandey [8] for two-layered power-lawflow which for 119899 = 1 further reduce to those presented byRao andUsha [6] for two-layeredNewtonian flowThe resultsof Shapiro et al [13] model can be obtained by substituting1205831= 1 120583

2= 1 and 119899 = 1 in presented model

8 Reflux Limit

Reflux is another phenomenon concerning peristalsis inwhich some fluid particles on the average move in adirection opposite to the net flow (cf Shapiro et al [13])The volume flow rate 119876

120595(119909) corresponding to a particle at

a position 119911 and time 119905 in the laboratory frame and on thestreamline 120595 in the wave frame is given as

119876120595(119909) = 2120595 + 119903

2(120595 119909) (27)

On averaging over one cycle this becomes

119876120595(119909) = 2120595 + int

1

0

1199032(120595 119909) 119889119909 (28)

For evaluating the reflux limit we expand 119876120595(119909) in a power

series in terms of a small parameter 120576 about the wall where

120576 = 120595 minus119902

2 (29)

and use the reflux condition119876120595

119876

gt 1 as 120576 997888rarr 0 (30)

The coefficients of the first two terms in the expansion of1199032(120595 119909) that is

1199032= 1198672+ 1198861120576 + 11988621205762+ sdot sdot sdot (31)

Applied Bionics and Biomechanics 7

0

01

02

03

04

0 02 04 06 08Q

Δp

(a)

0

1

2

3

4

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

0 02 04 06 08Q

Δp

minus2

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 2 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205831at 120601 = 05 120572 = 07

120573 = 09 and 1205832= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

are obtained by using (14) and given by

1198861= minus2

1198862= minus

4 (119902 + 1198672)

11986741205831119899

2

120574 (119909)

(32)

Performing the integration in (28) to the second orderand using condition (30) we obtain the condition for theoccurrence of reflux as follows

int

1

0

((119876 minus 1 minus1206012

2+ 1198672)119867(1minus119899)119899

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1minus 1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)119889119909

lt 0

(33)

9 Numerical Results and Discussion

We have carried out numerical calculations and plottedgraphs to observe the effects of the viscosities of the interme-diate and peripheral layers and the flow behaviour index onpumping mechanical efficiency trapping limits and refluxlimit

Figures 2(a)ndash2(c) are based on (15) and are plottedbetween the pressure difference across one wavelength (Δ119901)

8 Applied Bionics and Biomechanics

0

01

02

03

04

05

06

07

0 02 04 06 08Q

Δp

minus01

(a)

0

1

2

3

4

5

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

14

0 02 04 06 08Q

Δp

minus2

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 3 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205832at 120601 = 05 120572 = 07

120573 = 09 and 1205831= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

and the time-averaged flow rate (119876) to observe the impact ofvariation of the intermediate layer viscosity 120583

1 The relation

is linear and the maximum flow rate 1198760 which is dependent

on 1205831 is achieved when Δ119901 is zero However for a fixed

1205832 increase of many folds in 120583

1brings about only a slight

cut in 1198760irrespective of the fluid being pseudoplastic (119899 =

05) Newtonian (119899 = 10) or dilatant (119899 = 15) (seeFigures 2(a)ndash2(c)) But a more interesting observation is thatthe pressure required to restrain flow rates has to increaseby a large magnitude if 120583

1is increased even without any

variation in 1205832 In Figures 2(a)ndash2(c) the curves for 120583

1=

10 show the results for two-layered peristaltic power-lawfluid flow which are similar to results obtained by Misraand Pandey [8] The results in Figure 2(c) are verified withthe results (three-layered peristaltic viscous flow) of Tripathi

[11] Figures 3(a)ndash3(c) are plotted to examine the influence ofthe outer (peripheral) layer viscosity on the pressure versustime-averaged flow rate relation It is noticed that for all thevalues of flow behavior index unlike 120583

1an increase in the

viscosity 1205832of the outer layer causes a slight addition to

the maximum flow rates However on the pressure-flow raterelation 120583

1and 120583

2both have similar effects The influence of

the flow behaviour index 119899 on the pressure-flow rate relationis qualitatively similar to that of intermediate layer viscosity1205831(cf Figure 4)To learn the effects of viscosities in different layers and

power-law fluid index on the mechanical efficiency graphs(Figures 5ndash7) between mechanical efficiency and the ratioof averaged flow rate and maximum averaged flow rate areplotted based on (23) The relation between them is found

Applied Bionics and Biomechanics 9

0

1

2

3

4

5

6

7

8

0 02 04 06 08Q

Δp

minus2

minus1

n = 15

n = 10

n = 05

Figure 4 Pressure difference across one wavelength versus averaged volume flow rate for different values of 119899 at 120601 = 05 120572 = 07 120573 = 091205831= 01 and 120583

2= 01

0

005

01

015

02

025

03

035

04

045

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

E

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1QQ0

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 5 Mechanical efficiency versus ratio of the time-averaged volume flow rate and the maximum time-averaged volume flow rate fordifferent 120583

1at 120601 = 05 120572 = 07 120573 = 09 and 120583

2= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

10 Applied Bionics and Biomechanics

0

01

02

03

04

05

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

0

01

02

03

04

05

06

07

08

09

0 02 04 06 08 1minus01

E

QQ0

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 6 Mechanical efficiency versus ratio of averaged volume flow rate and maximum averaged volume flow rate with the different valuesof 1205832at 120601 = 05 120572 = 07 120573 = 09 and 120583

1= 01 for fixed values of 119899 as follows (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

to be nonlinear An observation of Figures 5(a)ndash5(c) revealsthat the pumping performance improves for all values of 119899when the viscosity 120583

1of the intermediate layer is increased

This too favours the earlier conclusion that flow rate isenhanced for larger viscosity of the intermediate layer Theouter layer viscosity 120583

2too is found to improve the pumping

performance with an increase in its magnitude (cf Figures6(a)ndash6(c)) The results of Figures 6 and 7 agree with theprevious results [8 11] The pumping efficiency is found toincrease with the flow behavior index (119899) for fixed viscositiesof the outer and the intermediate layers Thus peristalticpumping with a pseudoplastic fluid is less efficient than thatwith a fluid (cf Figure 7)

Reflux is an inherent phenomenon of peristaltic pumpingwhich is also named reversal flow To discuss the refluxphenomenon under the effects of viscosities of both layersFigures 8 and 9 are drawn between the amplitude ratio andaveraged flow rate based on (33) It is observed that reflux can

occur in a larger region if the intermediate layer viscosity isincreased (Figure 8) The outer layer viscosity has a similareffect but the change observed is insignificant (Figure 9)

We have determined the trapping limits based on (26) fordifferent viscosities of the intermediate and peripheral layersand also the flow behaviour index to investigate their effectsIt is noticed that when 120583

1varies from 001 to 10 for the fixed

values of 1205832at 01 and 119899 at 05 119876minus rises from 069 to 095

and 119876+ from 095 to 11 which means both the lower andthe upper trapping limits increase with 120583

1 the intermediate

layer viscosity By setting 1205831= 01 and 119899 = 05 if 120583

2is varied

from 001 to 10 119876minus shifts from 098 to 068 and 119876+from 11to 095 showing that both the lower and the upper trappinglimits decrease with 120583

2 When at the fixed values 120583

1= 01

and 1205832= 01 119899 varies from 05 to 15 119876minus varies from 074 to

033 and119876+ varies from 11 to 10 indicating thus that both thelower and the upper trapping limits decline with increasing

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Applied Bionics and Biomechanics 7

0

01

02

03

04

0 02 04 06 08Q

Δp

(a)

0

1

2

3

4

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

0 02 04 06 08Q

Δp

minus2

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 2 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205831at 120601 = 05 120572 = 07

120573 = 09 and 1205832= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

are obtained by using (14) and given by

1198861= minus2

1198862= minus

4 (119902 + 1198672)

11986741205831119899

2

120574 (119909)

(32)

Performing the integration in (28) to the second orderand using condition (30) we obtain the condition for theoccurrence of reflux as follows

int

1

0

((119876 minus 1 minus1206012

2+ 1198672)119867(1minus119899)119899

sdot ((1 minus1

1205831119899

1

)119867(3119899+1)119899

1

+ (1

1205831119899

1minus 1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

)

minus1

)119889119909

lt 0

(33)

9 Numerical Results and Discussion

We have carried out numerical calculations and plottedgraphs to observe the effects of the viscosities of the interme-diate and peripheral layers and the flow behaviour index onpumping mechanical efficiency trapping limits and refluxlimit

Figures 2(a)ndash2(c) are based on (15) and are plottedbetween the pressure difference across one wavelength (Δ119901)

8 Applied Bionics and Biomechanics

0

01

02

03

04

05

06

07

0 02 04 06 08Q

Δp

minus01

(a)

0

1

2

3

4

5

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

14

0 02 04 06 08Q

Δp

minus2

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 3 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205832at 120601 = 05 120572 = 07

120573 = 09 and 1205831= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

and the time-averaged flow rate (119876) to observe the impact ofvariation of the intermediate layer viscosity 120583

1 The relation

is linear and the maximum flow rate 1198760 which is dependent

on 1205831 is achieved when Δ119901 is zero However for a fixed

1205832 increase of many folds in 120583

1brings about only a slight

cut in 1198760irrespective of the fluid being pseudoplastic (119899 =

05) Newtonian (119899 = 10) or dilatant (119899 = 15) (seeFigures 2(a)ndash2(c)) But a more interesting observation is thatthe pressure required to restrain flow rates has to increaseby a large magnitude if 120583

1is increased even without any

variation in 1205832 In Figures 2(a)ndash2(c) the curves for 120583

1=

10 show the results for two-layered peristaltic power-lawfluid flow which are similar to results obtained by Misraand Pandey [8] The results in Figure 2(c) are verified withthe results (three-layered peristaltic viscous flow) of Tripathi

[11] Figures 3(a)ndash3(c) are plotted to examine the influence ofthe outer (peripheral) layer viscosity on the pressure versustime-averaged flow rate relation It is noticed that for all thevalues of flow behavior index unlike 120583

1an increase in the

viscosity 1205832of the outer layer causes a slight addition to

the maximum flow rates However on the pressure-flow raterelation 120583

1and 120583

2both have similar effects The influence of

the flow behaviour index 119899 on the pressure-flow rate relationis qualitatively similar to that of intermediate layer viscosity1205831(cf Figure 4)To learn the effects of viscosities in different layers and

power-law fluid index on the mechanical efficiency graphs(Figures 5ndash7) between mechanical efficiency and the ratioof averaged flow rate and maximum averaged flow rate areplotted based on (23) The relation between them is found

Applied Bionics and Biomechanics 9

0

1

2

3

4

5

6

7

8

0 02 04 06 08Q

Δp

minus2

minus1

n = 15

n = 10

n = 05

Figure 4 Pressure difference across one wavelength versus averaged volume flow rate for different values of 119899 at 120601 = 05 120572 = 07 120573 = 091205831= 01 and 120583

2= 01

0

005

01

015

02

025

03

035

04

045

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

E

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1QQ0

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 5 Mechanical efficiency versus ratio of the time-averaged volume flow rate and the maximum time-averaged volume flow rate fordifferent 120583

1at 120601 = 05 120572 = 07 120573 = 09 and 120583

2= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

10 Applied Bionics and Biomechanics

0

01

02

03

04

05

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

0

01

02

03

04

05

06

07

08

09

0 02 04 06 08 1minus01

E

QQ0

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 6 Mechanical efficiency versus ratio of averaged volume flow rate and maximum averaged volume flow rate with the different valuesof 1205832at 120601 = 05 120572 = 07 120573 = 09 and 120583

1= 01 for fixed values of 119899 as follows (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

to be nonlinear An observation of Figures 5(a)ndash5(c) revealsthat the pumping performance improves for all values of 119899when the viscosity 120583

1of the intermediate layer is increased

This too favours the earlier conclusion that flow rate isenhanced for larger viscosity of the intermediate layer Theouter layer viscosity 120583

2too is found to improve the pumping

performance with an increase in its magnitude (cf Figures6(a)ndash6(c)) The results of Figures 6 and 7 agree with theprevious results [8 11] The pumping efficiency is found toincrease with the flow behavior index (119899) for fixed viscositiesof the outer and the intermediate layers Thus peristalticpumping with a pseudoplastic fluid is less efficient than thatwith a fluid (cf Figure 7)

Reflux is an inherent phenomenon of peristaltic pumpingwhich is also named reversal flow To discuss the refluxphenomenon under the effects of viscosities of both layersFigures 8 and 9 are drawn between the amplitude ratio andaveraged flow rate based on (33) It is observed that reflux can

occur in a larger region if the intermediate layer viscosity isincreased (Figure 8) The outer layer viscosity has a similareffect but the change observed is insignificant (Figure 9)

We have determined the trapping limits based on (26) fordifferent viscosities of the intermediate and peripheral layersand also the flow behaviour index to investigate their effectsIt is noticed that when 120583

1varies from 001 to 10 for the fixed

values of 1205832at 01 and 119899 at 05 119876minus rises from 069 to 095

and 119876+ from 095 to 11 which means both the lower andthe upper trapping limits increase with 120583

1 the intermediate

layer viscosity By setting 1205831= 01 and 119899 = 05 if 120583

2is varied

from 001 to 10 119876minus shifts from 098 to 068 and 119876+from 11to 095 showing that both the lower and the upper trappinglimits decrease with 120583

2 When at the fixed values 120583

1= 01

and 1205832= 01 119899 varies from 05 to 15 119876minus varies from 074 to

033 and119876+ varies from 11 to 10 indicating thus that both thelower and the upper trapping limits decline with increasing

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Applied Bionics and Biomechanics

0

01

02

03

04

05

06

07

0 02 04 06 08Q

Δp

minus01

(a)

0

1

2

3

4

5

0 02 04 06 08Q

Δp

minus1

(b)

0

2

4

6

8

10

12

14

0 02 04 06 08Q

Δp

minus2

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 3 Pressure difference across one wavelength versus time-averaged volume flow rate for different values of 1205832at 120601 = 05 120572 = 07

120573 = 09 and 1205831= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

and the time-averaged flow rate (119876) to observe the impact ofvariation of the intermediate layer viscosity 120583

1 The relation

is linear and the maximum flow rate 1198760 which is dependent

on 1205831 is achieved when Δ119901 is zero However for a fixed

1205832 increase of many folds in 120583

1brings about only a slight

cut in 1198760irrespective of the fluid being pseudoplastic (119899 =

05) Newtonian (119899 = 10) or dilatant (119899 = 15) (seeFigures 2(a)ndash2(c)) But a more interesting observation is thatthe pressure required to restrain flow rates has to increaseby a large magnitude if 120583

1is increased even without any

variation in 1205832 In Figures 2(a)ndash2(c) the curves for 120583

1=

10 show the results for two-layered peristaltic power-lawfluid flow which are similar to results obtained by Misraand Pandey [8] The results in Figure 2(c) are verified withthe results (three-layered peristaltic viscous flow) of Tripathi

[11] Figures 3(a)ndash3(c) are plotted to examine the influence ofthe outer (peripheral) layer viscosity on the pressure versustime-averaged flow rate relation It is noticed that for all thevalues of flow behavior index unlike 120583

1an increase in the

viscosity 1205832of the outer layer causes a slight addition to

the maximum flow rates However on the pressure-flow raterelation 120583

1and 120583

2both have similar effects The influence of

the flow behaviour index 119899 on the pressure-flow rate relationis qualitatively similar to that of intermediate layer viscosity1205831(cf Figure 4)To learn the effects of viscosities in different layers and

power-law fluid index on the mechanical efficiency graphs(Figures 5ndash7) between mechanical efficiency and the ratioof averaged flow rate and maximum averaged flow rate areplotted based on (23) The relation between them is found

Applied Bionics and Biomechanics 9

0

1

2

3

4

5

6

7

8

0 02 04 06 08Q

Δp

minus2

minus1

n = 15

n = 10

n = 05

Figure 4 Pressure difference across one wavelength versus averaged volume flow rate for different values of 119899 at 120601 = 05 120572 = 07 120573 = 091205831= 01 and 120583

2= 01

0

005

01

015

02

025

03

035

04

045

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

E

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1QQ0

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 5 Mechanical efficiency versus ratio of the time-averaged volume flow rate and the maximum time-averaged volume flow rate fordifferent 120583

1at 120601 = 05 120572 = 07 120573 = 09 and 120583

2= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

10 Applied Bionics and Biomechanics

0

01

02

03

04

05

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

0

01

02

03

04

05

06

07

08

09

0 02 04 06 08 1minus01

E

QQ0

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 6 Mechanical efficiency versus ratio of averaged volume flow rate and maximum averaged volume flow rate with the different valuesof 1205832at 120601 = 05 120572 = 07 120573 = 09 and 120583

1= 01 for fixed values of 119899 as follows (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

to be nonlinear An observation of Figures 5(a)ndash5(c) revealsthat the pumping performance improves for all values of 119899when the viscosity 120583

1of the intermediate layer is increased

This too favours the earlier conclusion that flow rate isenhanced for larger viscosity of the intermediate layer Theouter layer viscosity 120583

2too is found to improve the pumping

performance with an increase in its magnitude (cf Figures6(a)ndash6(c)) The results of Figures 6 and 7 agree with theprevious results [8 11] The pumping efficiency is found toincrease with the flow behavior index (119899) for fixed viscositiesof the outer and the intermediate layers Thus peristalticpumping with a pseudoplastic fluid is less efficient than thatwith a fluid (cf Figure 7)

Reflux is an inherent phenomenon of peristaltic pumpingwhich is also named reversal flow To discuss the refluxphenomenon under the effects of viscosities of both layersFigures 8 and 9 are drawn between the amplitude ratio andaveraged flow rate based on (33) It is observed that reflux can

occur in a larger region if the intermediate layer viscosity isincreased (Figure 8) The outer layer viscosity has a similareffect but the change observed is insignificant (Figure 9)

We have determined the trapping limits based on (26) fordifferent viscosities of the intermediate and peripheral layersand also the flow behaviour index to investigate their effectsIt is noticed that when 120583

1varies from 001 to 10 for the fixed

values of 1205832at 01 and 119899 at 05 119876minus rises from 069 to 095

and 119876+ from 095 to 11 which means both the lower andthe upper trapping limits increase with 120583

1 the intermediate

layer viscosity By setting 1205831= 01 and 119899 = 05 if 120583

2is varied

from 001 to 10 119876minus shifts from 098 to 068 and 119876+from 11to 095 showing that both the lower and the upper trappinglimits decrease with 120583

2 When at the fixed values 120583

1= 01

and 1205832= 01 119899 varies from 05 to 15 119876minus varies from 074 to

033 and119876+ varies from 11 to 10 indicating thus that both thelower and the upper trapping limits decline with increasing

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

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International Journal of

Applied Bionics and Biomechanics 9

0

1

2

3

4

5

6

7

8

0 02 04 06 08Q

Δp

minus2

minus1

n = 15

n = 10

n = 05

Figure 4 Pressure difference across one wavelength versus averaged volume flow rate for different values of 119899 at 120601 = 05 120572 = 07 120573 = 091205831= 01 and 120583

2= 01

0

005

01

015

02

025

03

035

04

045

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

E

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1QQ0

1205831 = 10

1205831 = 01

1205831 = 001

(c)

Figure 5 Mechanical efficiency versus ratio of the time-averaged volume flow rate and the maximum time-averaged volume flow rate fordifferent 120583

1at 120601 = 05 120572 = 07 120573 = 09 and 120583

2= 01 for (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

10 Applied Bionics and Biomechanics

0

01

02

03

04

05

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

0

01

02

03

04

05

06

07

08

09

0 02 04 06 08 1minus01

E

QQ0

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 6 Mechanical efficiency versus ratio of averaged volume flow rate and maximum averaged volume flow rate with the different valuesof 1205832at 120601 = 05 120572 = 07 120573 = 09 and 120583

1= 01 for fixed values of 119899 as follows (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

to be nonlinear An observation of Figures 5(a)ndash5(c) revealsthat the pumping performance improves for all values of 119899when the viscosity 120583

1of the intermediate layer is increased

This too favours the earlier conclusion that flow rate isenhanced for larger viscosity of the intermediate layer Theouter layer viscosity 120583

2too is found to improve the pumping

performance with an increase in its magnitude (cf Figures6(a)ndash6(c)) The results of Figures 6 and 7 agree with theprevious results [8 11] The pumping efficiency is found toincrease with the flow behavior index (119899) for fixed viscositiesof the outer and the intermediate layers Thus peristalticpumping with a pseudoplastic fluid is less efficient than thatwith a fluid (cf Figure 7)

Reflux is an inherent phenomenon of peristaltic pumpingwhich is also named reversal flow To discuss the refluxphenomenon under the effects of viscosities of both layersFigures 8 and 9 are drawn between the amplitude ratio andaveraged flow rate based on (33) It is observed that reflux can

occur in a larger region if the intermediate layer viscosity isincreased (Figure 8) The outer layer viscosity has a similareffect but the change observed is insignificant (Figure 9)

We have determined the trapping limits based on (26) fordifferent viscosities of the intermediate and peripheral layersand also the flow behaviour index to investigate their effectsIt is noticed that when 120583

1varies from 001 to 10 for the fixed

values of 1205832at 01 and 119899 at 05 119876minus rises from 069 to 095

and 119876+ from 095 to 11 which means both the lower andthe upper trapping limits increase with 120583

1 the intermediate

layer viscosity By setting 1205831= 01 and 119899 = 05 if 120583

2is varied

from 001 to 10 119876minus shifts from 098 to 068 and 119876+from 11to 095 showing that both the lower and the upper trappinglimits decrease with 120583

2 When at the fixed values 120583

1= 01

and 1205832= 01 119899 varies from 05 to 15 119876minus varies from 074 to

033 and119876+ varies from 11 to 10 indicating thus that both thelower and the upper trapping limits decline with increasing

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Applied Bionics and Biomechanics

0

01

02

03

04

05

0 02 04 06 08 1

E

QQ0

(a)

0

01

02

03

04

05

06

0 02 04 06 08 1

E

QQ0

(b)

0

01

02

03

04

05

06

07

08

09

0 02 04 06 08 1minus01

E

QQ0

1205832 = 10

1205832 = 01

1205832 = 001

(c)

Figure 6 Mechanical efficiency versus ratio of averaged volume flow rate and maximum averaged volume flow rate with the different valuesof 1205832at 120601 = 05 120572 = 07 120573 = 09 and 120583

1= 01 for fixed values of 119899 as follows (a) 119899 = 05 (b) 119899 = 10 and (c) 119899 = 15

to be nonlinear An observation of Figures 5(a)ndash5(c) revealsthat the pumping performance improves for all values of 119899when the viscosity 120583

1of the intermediate layer is increased

This too favours the earlier conclusion that flow rate isenhanced for larger viscosity of the intermediate layer Theouter layer viscosity 120583

2too is found to improve the pumping

performance with an increase in its magnitude (cf Figures6(a)ndash6(c)) The results of Figures 6 and 7 agree with theprevious results [8 11] The pumping efficiency is found toincrease with the flow behavior index (119899) for fixed viscositiesof the outer and the intermediate layers Thus peristalticpumping with a pseudoplastic fluid is less efficient than thatwith a fluid (cf Figure 7)

Reflux is an inherent phenomenon of peristaltic pumpingwhich is also named reversal flow To discuss the refluxphenomenon under the effects of viscosities of both layersFigures 8 and 9 are drawn between the amplitude ratio andaveraged flow rate based on (33) It is observed that reflux can

occur in a larger region if the intermediate layer viscosity isincreased (Figure 8) The outer layer viscosity has a similareffect but the change observed is insignificant (Figure 9)

We have determined the trapping limits based on (26) fordifferent viscosities of the intermediate and peripheral layersand also the flow behaviour index to investigate their effectsIt is noticed that when 120583

1varies from 001 to 10 for the fixed

values of 1205832at 01 and 119899 at 05 119876minus rises from 069 to 095

and 119876+ from 095 to 11 which means both the lower andthe upper trapping limits increase with 120583

1 the intermediate

layer viscosity By setting 1205831= 01 and 119899 = 05 if 120583

2is varied

from 001 to 10 119876minus shifts from 098 to 068 and 119876+from 11to 095 showing that both the lower and the upper trappinglimits decrease with 120583

2 When at the fixed values 120583

1= 01

and 1205832= 01 119899 varies from 05 to 15 119876minus varies from 074 to

033 and119876+ varies from 11 to 10 indicating thus that both thelower and the upper trapping limits decline with increasing

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Applied Bionics and Biomechanics 11

0

01

02

03

04

05

06

07

08

0 02 04 06 08 1

E

QQ0

n = 15

n = 10

n = 05

Figure 7 Mechanical efficiency versus ratio of averaged volumeflow rate and maximum averaged volume flow rate for differentvalues of 119899 at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and 120583

2= 01

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205831 = 01

1205831 = 001

Figure 8 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

1at 120601 = 05 120572 = 07 120573 = 09 120583

2= 01 and

119899 = 05

119899 Depression of trapping limits is an indication that trappingtakes place at lower flow rates whereas elevation of the samelimits means trapping occurs at higher flow rates Moreoverthe size of the trapped bolus increases with 120583

2and 119899 while it

decreases with increasing 1205831(cf Figures 10ndash12)

If the entire flow behaviour is further compared withtwo-dimensional channel flow [12] it is observed that notonly the quantity of flow is more but also even the pressurerequired to stop the flow is almost ten times thus indicating

0

02

04

06

08

1

12

14

0 02 04 06 08 1

Reflux region

No reflux region

Q

120601

1205832 = 01

1205832 = 001

Figure 9 Averaged flow rate versus amplitude ratio with thedifferent values of 120583

2at 120601 = 05 120572 = 07 120573 = 09 120583

1= 01 and

119899 = 05

that the axisymmetric flow is less prone to constipation andmay be more easily overcome with introducing a viscousintermediate layer

10 Conclusions

It is found that the maximum flow rate diminishes whenthe intermediate layer viscosity and flow behaviour index areincreased or the peripheral layer viscosity is decreased Itis observed that the pressure required to restrain flow rateshas to increase by a large magnitude if 120583

1is increased even

without any variation in 1205832and vice versa This revelation

may lead to the physical interpretation in view of constipationthat though the mucous layer viscosity may not be varied theintroduction of a viscous intermediate layer caused by somemedicinal intervention may cure constipation

Themechanical efficiency increases with the viscosities ofboth the intermediate and the peripheral layers as well as theflow behaviour index Reflux region shrinks if the viscosity ofthe intermediate layer is increased or that of the peripherallayer is decreased Both the lower and the upper trappinglimits increase if either the viscosity of the intermediate layeris increased or that of the peripheral layer is decreased Thesaid limits decline when the flow behaviour index is raisedMoreover the size of the trapped bolus increases with theviscosity of the peripheral layer and the flow behaviour indexbut it decreases when the viscosity of the intermediate layeris increased

Finally it is observed that the axisymmetric flow intestineis less prone to constipation than two-dimensional flow andmay be more easily overcome with introducing a viscousintermediate layer

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(b)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(c)

0

02

04

06

08

1

12

14

0 02 04 06minus02

(d)

Figure 10 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205832= 002 and 119899 = 15 (a) 120583

1= 002 (b) 120583

1= 005 (c) 120583

1= 008 and (d)

1205831= 01

Appendix

Equations (17) and (18) form a system of nonlinear algebraicequations in the variables119867

1and119867

2given by

1198601119867(5119899+1)119899

1+ 1198611119867(3119899+1)119899

1+ 1198621(1198672)1198672

1+ 1198631(1198672) = 0

1198602119867(5119899+1)119899

2+ 1198612119867(3119899+1)119899

2+ 1198622(1198671)1198672

2+ 1198632(1198671) = 0

(A1)

where

1198601= (1 minus

1

1205831119899

1

)

1198602= (

1

1205831119899

1

minus1

1205831119899

2

)

1198611= [119902lowast

1(1 minus

1

1205831119899

1

) minus (119902 + 1198672)(1 minus

3119899 + 1

119899 + 1

1

1205831119899

1

)]

1198612= [119902lowast

2(

1

1205831119899

1

minus1

1205831119899

2

)

minus (119902 + 1198672)(

1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)]

1198621(1198672)

= [(1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1(119902 + 119867

2)

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Applied Bionics and Biomechanics 13

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 11 Streamlines for119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 119899 = 15 (a) 120583

2= 001 (b) 120583

2= 005 (c) 120583

2= 01 and (d) 120583

2= 015

sdot (1

1205831119899

1

minus1

1205831119899

2

)119867(119899+1)119899

2+

1

1205831119899

2

119867(119899+1)119899

]

1198622(1198671) = [(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+1

1205831119899

2

119867(3119899+1)119899

minus3119899 + 1

119899 + 1

sdot (119902 + 1198672)1

1205831119899

2

119867(119899+1)119899

]

1198631(1198672)

= 119902lowast

1[(

1

1205831119899

1

minus1

1205831119899

2

)119867(3119899+1)119899

2+

1

1205831119899

2

119867(3119899+1)119899

]

1198632(1198671) = [(119902

lowast

2minus 119902 minus 119867

2)(1 minus

1

1205831119899

1

)119867(3119899+1)119899

1

+ 119902lowast

2

1

1205831119899

2

119867(3119899+1)119899

]

(A2)

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Applied Bionics and Biomechanics

0 02 04 060

02

04

06

08

1

12

14

minus02

(a)

0 02 04 060

02

04

06

08

1

12

14

minus02

(b)

0 02 04 060

02

04

06

08

1

12

14

minus02

(c)

0 02 04 060

02

04

06

08

1

12

14

minus02

(d)

Figure 12 Streamlines for 119876 = 07 120601 = 05 120572 = 07 120573 = 09 1205831= 01 and 120583

2= 01 (a) 119899 = 14 (b) 119899 = 12 (c) 119899 = 1 and (d) 119899 = 08

The mean thicknesses of these three layers are taken at 119909 = 0and we assume that the interfaces are 119867

1(0) = 120572 119867

2(0) = 120573

(the same at 119909 = 12 1) The values of 119902lowast1 119902lowast

2are related to 120572

and 120573 in the following way

119902lowast

1= minus1205722+ (119902 + 1)

sdot (((1 minus3119899 + 1

119899 + 1

1

1205831119899

1

)120572(3119899+1)119899

+ (3119899 + 1

119899 + 1)(

1

1205831119899

1

minus1

1205831119899

2

)1205722120573(119899+1)119899

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205722)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

119902lowast

2= minus1205732+ (119902 + 1)

sdot (((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus3119899 + 1

119899 + 1

1

1205831119899

2

)120573(3119899+1)119899

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Applied Bionics and Biomechanics 15

+(3119899 + 1

119899 + 1)

1

1205831119899

2

1205732)

sdot ((1 minus1

1205831119899

1

)120572(3119899+1)119899

+ (1

1205831119899

1

minus1

1205831119899

2

)120573(3119899+1)119899

+1

1205831119899

2

)

minus1

)

(A3)

Conflict of Interests

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

References

[1] C-O Ng and Y Ma ldquoLagrangian transport induced by peri-staltic pumping in a closed channelrdquo Physical Review E vol 80no 5 Article ID 056307 2009

[2] D Takagi and N J Balmforth ldquoPeristaltic pumping of rigidobjects in an elastic tuberdquo Journal of Fluid Mechanics vol 672pp 219ndash244 2011

[3] D Takagi and N J Balmforth ldquoPeristaltic pumping of viscousfluid in an elastic tuberdquo Journal of Fluid Mechanics vol 672 pp196ndash218 2011

[4] O A Dudchenko and G T Guria ldquoSelf-sustained peristalticwaves explicit asymptotic solutionsrdquo Physical Review E vol 85no 2 Article ID 020902(R) 2012

[5] J G Brasseur S Corrsin and N Q Lu ldquoThe influence ofperipheral layer of different viscosity on peristaltic pumpingwith Newtonian fluidsrdquo Journal of Fluid Mechanics vol 174 pp495ndash519 1987

[6] A R Rao and S Usha ldquoPeristaltic transport of two immiscibleviscous fluids in a circular tuberdquo Journal of FluidMechanics vol298 pp 271ndash285 1995

[7] J C Misra and S K Pandey ldquoPeristaltic transport of a non-Newtonian fluid with a peripheral layerrdquo International Journalof Engineering Science vol 37 no 14 pp 1841ndash1858 1999

[8] J C Misra and S K Pandey ldquoPeristaltic flow of a multilayeredpower-law fluid through a cylindrical tuberdquo International Jour-nal of Engineering Science vol 39 no 4 pp 387ndash402 2001

[9] J C Misra and S K Pandey ldquoPeristaltic transport of blood insmall vessels study of a mathematical modelrdquo Computers andMathematics with Applications vol 43 no 8-9 pp 1183ndash11932002

[10] E F Elshehawey and Z M Gharsseldien ldquoPeristaltic transportof three-layered flow with variable viscosityrdquo Applied Mathe-matics and Computation vol 153 no 2 pp 417ndash432 2004

[11] D Tripathi ldquoA mathematical study on three layered oscillatoryblood flow through stenosed arteriesrdquo Journal of Bionic Engi-neering vol 9 no 1 pp 119ndash131 2012

[12] S K Pandey M K Chaube and D Tripathi ldquoPeristaltic trans-port of multilayered power-law fluids with distinct viscosities amathematical model for intestinal flowsrdquo Journal of TheoreticalBiology vol 278 pp 11ndash19 2011

[13] A H Shapiro M Y Jafferin and S L Weinberg ldquoPeristalticpumping with long wavelengths at low Reynolds numberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of