research article peristaltic motion of viscoelastic...

Hindawi Publishing CorporationChinese Journal of EngineeringVolume 2013 Article ID 582390 7 pageshttpdxdoiorg1011552013582390

Research ArticlePeristaltic Motion of Viscoelastic Fluid with Fractional SecondGrade Model in Curved Channels

V K Narla1 K M Prasad1 and J V Ramanamurthy2

1 Department of Mathematics GITAM University Hyderabad-502329 Andhra Pradesh India2Department of Mathematics National Institute of Technology Warangal-506004 Andhra Pradesh India

Correspondence should be addressed to V K Narla vknarlagmailcom

Received 11 October 2013 Accepted 27 November 2013

Academic Editors H Hu and Y Zhu

Copyright copy 2013 V K Narla 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

Exact analytic solutions are obtained for the flow of a viscoelastic fluid with fractional second grade model by peristalsisthrough a curved channel The flow has been investigated under the assumptions of long wavelength and low Reynolds numberapproximation The streamlines for trapped bolus of Newtonian fluid are analyzed graphically The fractional calculus approach isused to get analytic solutions of the problem The influence of fractional parameter material constant amplitude and curvatureparameter on the pressure and friction force across one wavelength are discussed numerically with the help of graphs

1 Introduction

Peristalsis is a mechanism of fluid transport throughdeformable vessels with the aid of a progressive contrac-tionexpansion wave along the vessel This mechanismappears to be a major mechanism for fluid transport in manyphysiological systems It appears in the gastrointestine tracturine transport from kidney to bladder bile from the gallbladder into the duodenum the movement of spermatoza inthe ducts efferentes of the male reproductive tract transportof lymph in the lymphatic vessels and in the vasomotion ofsmall blood vessels such as arterioles venules and capillariesPeristaltic fluid transport is being increasingly used by mod-ern technology in cases where it is necessary to avoid contactbetween the pumped medium and the mechanical parts ofthe pump

A mathematical model to understand fluid mechanicsof this phenomenon has been developed using lubricationtheory provided that the fluid inertia effects are negligibleand the flow is of the low Reynolds number The flowof Newtonian and non-Newtonian fluids was described bymany researchers in straight vessels (Shapiro et al [1] Jaffrinand Shapiro [2] Jaffrin [3] Pozrikidis [4] Vajravelu et al[5] and Li and Brasseur [6]) In recent years it has turnedout that the mathematical models in areas like viscoelasticityand electrochemistry as well as in many fields of science

and engineering including fluid flow rheology diffusivetransport electrical networks electromagnetic theory andprobability can be formulated very successfully by fractionalcalculus In particular it has been found to be quite flexiblein describing viscoelastic behavior of fluids The startingpoint of the fractional derivative model of non-Newtonianfluids is usually a classical differential equation which ismodified by replacing the time derivative of an integer orderby the so-called Riemann-Liouville fractional operator Thefractional derivative models have been used in various situa-tions to analyze diverse rheological problems The fractionalsecond grade model is one among these fractional modelsWenchang et al [7] have investigated the unsteady flow ofviscoelastic fluid with fractionalMaxwell model between twoparallel plates Qi and Jin [8] have discussed unsteady flowsbetween coaxial cylinders while Qi and Xu [9] have studiedthe flowpropertiesHayat et al [10] have constructed periodicunidirectional flows of a viscoelastic fluid with the fractionalMaxwell model and solutions are solved by Fourier trans-form Khan et al [11] have discussed the decay of potentialvertex for viscoelastic fluid with fractional Maxwell modeland analytical solutions are obtained by Hankel transformand discrete Laplace transform Recently Tripathi et al havestudied the peristaltic flow of viscoelastic fluid with fractionalmodels [12ndash19]

2 Chinese Journal of Engineering

In all these previous studies authors have carried outthe analysis in straight channels or tubes In fact the shapeof most physiological ducts are curved The correspondingstudy of the peristaltic flow in curved channel was carriedout by Sato et al [20] Ali et al [21] discussed the peristalticmotion in a curved channel using wave frame Later Ali et al[22] extended the flow analysis by considering heat transfereffects Hayat et al [23] have examined the peristaltic flow ofviscous fluid in a curved channel with complaint walls Non-Newtonian fluid flow induced by peristaltic waves in a curvedchannel has been investigated by Ali et al [24] Hayat et al[25] have extended the problem to investigate the effect of aninduced magnetic field on peristaltic flow of non-Newtonianfluid in a curved channel Very recently Ramanamurthy et al[26] have investigated unsteady effects of peristaltic transportin curved channels

The objective of this paper is to investigate the effects offractional parameters of second grade fluid on peristaltic flowthrough a curved channel In order to study these effects weemployed long wave length and the low Reynolds numberapproximation The fractional calculus approach is usedto obtain analytical solution of the problem The obtainedexpressions are utilized to discuss the influences of variousphysical parameters

2 Preliminaries

Here we present the necessary definitions from fractionalcalculus theory which are useful in subsequent sectionsThese definitions can be found in the recent literature [7 2728]

Definition 1 TheRiemann-Liouville fractional integral oper-ator of order 120572 gt 0 of a function 119891(119909) (0infin) rarr R is givenby

119869120572

119891 (119909) =

1

Γ (120572)

int

119909

0

(119909 minus 120585)120572minus1

119891 (120585) 119889120585 120572 gt 0 119909 gt 0

(1)

Definition 2 The fractional derivative of order 120572 gt 0 of acontinuous function 119891(119909) (0infin) rarr R is given by

119863120572

119891 (119909) =

1

Γ (119898 minus 120572)

(

119889

119889119909

)

119898

int

119909

0

(119909 minus 120585)119898minus120572minus1

119891 (120585) 119889120585

for 119898 minus 1 lt 120572 le 119898 119898 isin N 119909 gt 0 119891 isin C119898

minus1

(2)

where 119898 = [120572] + 1 provided that the right-hand side ispointwise defined on (0infin)

Remark 3 For example 119891(119909) = 119909120573 we quote for 120573 gt minus1 in

(2) one can get

119863120572

119909120573

=

Γ (120573 + 1)

Γ (120573 minus 120572 + 1)

119909120573minus120572

(3)

giving in particular 119863120572119909120573minus119899 = 0 119899 = 1 2 119873 where 119873 isthe smallest integer greater than or equal to 120572

3 Mathematical Model

We consider the flow of a fluidmodeled as a fractional secondgrade model given by

120591 = 120583(1 + 120582120572

1

120597120572

120597119905120572) 120574 (4)

where 1205821 120572 119905 120591 120583 and 120574 are the relaxation time fractional

parameter time shear stress viscosity and rate of shearstrain

We consider the flow of an incompressible viscoelasticfluid with fractional second grade model in a curved channelinduced by two infinite trains of sinusoidal waves that arepropagated along the flexible walls of the channel Whenundeformed the walls are separated by a distance 2119886 theconstant radius of curvature at the channel centreline is 119877We choose curvilinear coordinate system (119909 119903) in such a waythat 119909-axis lies along the center line of the curved channel and119903-axis is normal to it and is measured from central line Thescale factors are ℎ

1= (119903 + 119877)119877 ℎ

2= 1 and ℎ

3= 1 There is

no component in 119911 direction as shown in Figure 1The sinusoidal waves propagating along the channel walls

are described as

119903 = ℎ (119909 119905) = 119886 + 119887 cos [2120587 (

119909

120582

minus

119905

119879

)] (Upper wall)

119903 = minusℎ (119909 119905) = minus119886 minus 119887 cos [2120587 (

119909

120582

minus

119905

119879

)] (Lower wall) (5)

Here 119909 is the axial distance 119886 the radius of the stationarycurved channel 120601 the wave amplitude 120582 the wave length 119879the wave period and ℎ the radial displacement of the wavefrom the centerline The wavelength is large compared withthe channelrsquos width (119886120582 ≪ 1)

The governing equations for an ordinary second gradefluid through curved channel with the velocity vector 119881 =

119906119890119909+ V119890119903are given by

119877

120597119906

120597119909

+

120597

120597119903

(119903 + 119877) V = 0 (6)

120597119906

120597119905

+ (119881 sdot nabla) 119906 minus

119906V119903 + 119877

= minus

119877

120588 (119903 + 119877)

120597119901

120597119909

+ ](1 + 1205821

120597

120597119905

)

times [nabla2

119906 minus

119906

(119903 + 119877)2+

2119877

(119903 + 119877)2

120597V120597119909

]

(7)

Chinese Journal of Engineering 3

C

O

b

a

R

P(0 t)P(L t)

h(x t)

x u

r120582

Nw

Tw

Figure 1 Peristaltic wave in curved channel

120597V120597119905

+ (119881 sdot nabla) V minus

1199062

119903 + 119877

= minus

1

120588

120597119901

120597119903

+ ](1 + 1205821

120597

120597119905

)

times [nabla2V minus

V(119903 + 119877)

2minus

2119877

(119903 + 119877)2

120597119906

120597119909

]

(8)

where

(119881 sdot nabla) =

119877119906

(119903 + 119877)

120597

120597119909

+ V120597

120597119903

nabla2

= (

119877

119903 + 119877

)

2

1205972

1205971199092+

1

119903 + 119877

120597

120597119903

+

1205972

1205971199032

(9)

In the above equations 119901 is pressure 120588 is fluid density ] is thekinematic viscosity and 119906 and V are the velocity componentsin radial 119903 and axial 119909 directions respectively

The following dimensionless variables and parameter areintroduced

1199091015840

=

119909

120582

1199031015840

=

119903

119886

1199061015840

=

119906

119888

V1015840 =V120575119888

ℎ1015840

=

ℎ

119886

120601 =

119887

119886

120581 =

119877

119886

1199011015840

=

1198862

119901

120583119888120582

Re =

119888119886120575

] 120595

1015840

=

120595

119886119888

119876 =

119876

119886119888

1199051015840

=

119905

119879

(10)

where Re is Reynolds number119876 is volume flow rate 119888 = 120582119879

is the velocity of the wave 120575 = 119886120582 defines wave number 120601 isthe amplitude ratio and 120581 is the curvature parameter

The stream function 120595 can be defined by using dimen-sionless variables and parameters in (10) which satisfiescontinuity equation (6) as

119906 = minus

120597120595

120597119903

V =

120581

119903 + 120581

120597120595

120597119909

(11)

The dimensionless equations are obtained (neglecting theprimes for clarity) by applying longwavelength and negligibleReynolds number approximation using (4) (7) and (8)as

120597119901

120597119909

= (1 + 120582120572

1

120597120572

120597119905120572)(

1

120581

120597

120597119903

(119903 + 120581)

1205972

120595

1205972119903

minus

1

120581 (119903 + 120581)

120597120595

120597119903

)

(12)

120597119901

120597119903

= 0 (13)

The dimensionless boundary conditions are given by

120595 = minus

119876 (119909 119905)

2

120597120595

120597119903

= 0 at 119903 = ℎ (119909 119905)

120595 =

119876 (119909 119905)

2

120597120595

120597119903

= 0 at 119903 = minusℎ (119909 119905)

(14)

where 119876(119909 119905) is volume flow rate and is defined as 119876(119909 119905) =

int

ℎ

minusℎ

119906(119909 119903 119905)119889119903

4 Solution of the Problem

The transformations between the wave frame and the labora-tory frame in dimensionless form are defined as

119883 = 119909 minus 119888119905 119884 = 119903 119880 = 119906 minus 1

119881 = V 119902 = 119876 minus 2ℎ Ψ = 120595 minus 119903

(15)

where the parameters on the left side are in the wave frameand thoseon the right side are in the laboratory frameThe solutions of (12) and (13) with the boundary condi-tions equation (14) in wave frame of reference are obtainedas

Ψ = 1198621+ 1198622log (119884 + 120581) + 119862

3(119884 + 119896)

2

+ 1198624(119884 + 120581)

2 log (119884 + 120581) + (119884 + 120581)

1198621= minus

1

2

[2119867 + 119902 + 2120581

+ ((119867 + 119902)2

(2119867 + 119902)

times (2119867120581 + (119867 minus 120581)2

times (1 minus 2 log (119867 + 120581))

times log ((119867 + 120581) (119867 minus 120581)) ) )


1205821 = 1 2 3 4

1210

86420

minus2minus4minus6minus8minus10minus12minus14

00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

120572 = 05

ΔP

(a)

00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

ΔP

10

5

0

minus5

minus10

120572 = 025 05 075 10

1205821 = 10

(b)

00 05 10 15 20QT

120572 = 05

120581 = 20

t = 10

1205821 = 10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

ΔP

120601 = 03 04 05 06

(c)

Curved channelStraight channel

00 05 10 15 20QT

ΔP

120601 = 04

120572 = 05

t = 10

1205821 = 10

2

0

minus2

minus4

minus6

(d)

Figure 2 Pressure difference across one wavelength against time-averaged flow rate

times (minus41198672

1205812

+ (1198672

minus 1205812

)

2

times (log ((120581 minus 119867) (120581 + 119867)))2

)

minus1

]

1198622=

(2119867 + 119902) (1198672

minus 1205812

)

2

log ((120581 minus 119867) (120581 + 119867))

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

1198623= minus ( (2119867 + 119902)

times [minus2119867120581 + (119867 minus 120581)2 log (120581 minus 119867)

minus(119867 + 120581)2 log (119867 + 120581) ])

times (2 [minus41198672

1205812

+ (1198672

minus 1205812

)

2

times(log ((120581 minus 119867) (120581 + 119867)))2

] )

minus1

1198624= minus

2 (2119867 + 119902)119867120581

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

(16)

where the channelwall takes the form119867(119883) = 1+120601 cos(2120587119883)The axial pressure gradient in terms of time-averaged flowcan be obtained by substituting (16) in (12) as

120597119901

120597119883

(119883 119905)

= (1 + 120582120572

1

120597120572

120597119905120572)

times

8 (2119867 + 119902)119867

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

(17)Using Definition 2 in (17) we get the pressure gradient

120597119901

120597119883

(119883 119905)

=

8 (2119867 + 119902)119867

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

times (1 + 120582120572

1

119905minus120572

Γ (1 minus 120572)

)

(18)


00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

8

4

6

2

0

minus2

minus4

120572 = 025 05 075 10

1205821 = 10

F

(a)

120572 = 05

00 05 10 15 20QT

8

4

6

2

0

minus2

minus4

120601 = 06

120581 = 20

t = 10

1205821 = 04 06 08 10

F

(b)

F

00 05 10 15 20QT

120572 = 05

120581 = 20

t = 10

1205821 = 10

6

5

4

3

2

1

0

minus2

minus1

minus4

minus3

120601 = 03 04 05 06

(c)

Curved channel

00 05 10 15 20QT

F

120601 = 04

120572 = 05

t = 10

1205821 = 10

2

0

1

4

5

3

minus1

Straight channel

(d)

Figure 3 Friction force across one wavelength against time-averaged flow rate

The pressure difference Δ119901 and friction force 119865 across onewavelength are given by

Δ119901 = int

1

0

120597119901

120597119909

119889119909

119865 = int

1

0

minusℎ

120597119901

120597119909

119889119909

(19)

5 Results and Discussion

In this paper we analyze the peristaltic motion of fractionalsecond grade fluid through curved channel Exact solutionfor stream function is obtained Based on this exact solutionwe discuss the effects of various pertinent parameters suchas fractional parameter (120572) material constant (120582

1) occlusion

parameter (120601) and curvature parameter (120581) on pressuredifference across one wavelength (Δ119901) and friction forceacross the one wavelength (119865)

The pressure difference across one wavelength is plottedagainst averaged flow rate in Figures 2(a)ndash2(d) for differentvalues of the fractional parameters (120572 = 025 05 075 10)

channel curvature relaxation time (1205821= 04 06 08 10)

and amplitude (120601 = 03 04 05 06) respectively One mayobserve from these figures that the pressure-flow relation islinearThe present study examined three regions of peristalticmotion with positive pumping (119876

119879gt 0) in the entire pump-

ing region (Δ119901 gt 0) in the free pumping region (Δ119901 = 0)and in the copumping region (Δ119901 lt 0) It may be noted fromFigure 2(a) that the volumetric flow rate can be graduallyincreased in the pumping region and gradually decreased inthe copumping region by increasing the value of relaxationtime 120582

1 Figure 2(b) shows that in the entire pumping region

the volumetric flow rate decreases with the increase infractional parameter 120572 whereas in the copumping regiona reverse trend is noticed It is observed from Figure 2(c)that in the range of values of pressure gradient Δ119901 gt minus32the volumetric flow rate increases with the increase in theamplitude ratio 120601 However the trend reverses as soon asthe pressure gradient drops below minus32 Figure 2(d) revealsthat the flow rate slightly increases with increasing channelcurvature in the pumping region as well as copumpingregion


(a) 120581 = 3 (b) 120581 = 5

(c) 120581 = 12 (d) 120581 rarr infin

Figure 4 Streamline patterns at different channel curvatures

Figures 3(a)ndash3(d) show the variations of friction force 119865

with the averaged flow rate 119876119879under the influence of all

parameters 120572 120581 1205821 and 120601 It is observed from the figures that

friction forces have opposite behavior in comparison withpressure

Trapping is an important phenomenon described as abolus of fluid that moves as a whole with the wave [1]Trapping may be observed in curved channel by plotting thestreamlines in a wave frame [4] The streamline patterns inthe wave frame for Newtonian fluid with 119876

119879= 15 and 120601 =

05 for different values of curvature parameter 120581 are shownin Figure 4 It is observed that for small values of 120581 only onetrapped bolus is formedThere exists two asymmetric bolusesas one moving from curved channel to straight channel It isalso observed that the bolus near the upper wall increases butthe bolus near lower wall disappears eventually as the channelcurvature is increased However the result agrees almost wellfor straight channel (120581 rarr infin) as the bolus splits with almostequal magnitudes [1 4]

6 Concluding Remarks

This paper analyzes a viscoelastic fluid flow with fractionalsecond grade model for peristaltic motion in two-dimen-sional curved channels The effects of fractional parameter

curvature of the channel and viscoelastic behaviors onperistalticmotion in curved channel are studiedWe obtainedthe analytical solution for stream function at low inertialeffect when the wavelength is moderately large compared tothe channel width An approximate analytical solution forpressure gradient is obtained by fractional calculus theoryThe following conclusions can be summarized

(1) The relation between pressure and flow is found to belinear

(2) The pressure-flow function decreases with increasingvalues of fractional parameter 120572 curvature parameter120581 and amplitude ratio120601 and it increases with increas-ing values of relaxation time 120582

1

(3) The variations of friction force against flow rate showopposite behavior to that of pressure

(4) The streamlines in wave frame contain two asymmet-rical parts the bolus near the outer wall grows big andthe bolus at inner wall diminishes as curvature of thechannel increases

References

[1] A H Shapiro M Y Jaffrin and S L Weinberg ldquoPeristalticpumping with long wave length at Low Reynolds NumberrdquoJournal of Fluid Mechanics vol 37 no 4 pp 799ndash825 1969


[2] M Y Jaffrin and A H Shapiro ldquoPeristaltic pumpingrdquo AnnualReview of Fluid Mechanics vol 3 pp 13ndash37 1971

[3] M Y Jaffrin ldquoInertia and streamline curvature effects on peri-staltic pumpingrdquo International Journal of Engineering Sciencevol 11 no 6 pp 681ndash699 1973

[4] C Pozrikidis ldquoA study of peristaltic flowrdquo Journal of FluidMechanics vol 180 pp 515ndash527 1987

[5] K Vajravelu G Radhakrishnamacharya and V Radhakrishna-murty ldquoPeristaltic flow and heat transfer in a vertical porousannulus with long wave approximationrdquo International Journalof Non-Linear Mechanics vol 42 no 5 pp 754ndash759 2007

[6] M Li and J G Brasseur ldquoNon-steady peristaltic transport infinite-length tubesrdquo Journal of Fluid Mechanics vol 248 pp129ndash151 1993

[7] T Wenchang P Wenxiao and X Mingyu ldquoA note on unsteadyflows of a viscoelastic fluid with the fractional Maxwell modelbetween two parallel platesrdquo International Journal of Non-LinearMechanics vol 38 no 5 pp 645ndash650 2003

[8] H Qi and H Jin ldquoUnsteady rotating flows of a viscoelastic fluidwith the fractional Maxwell model between coaxial cylindersrdquoActa Mechanica Sinica vol 22 no 4 pp 301ndash305 2006

[9] H Qi and M Xu ldquoUnsteady flow of viscoelastic fluid withfractional Maxwell model in a channelrdquo Mechanics ResearchCommunications vol 34 no 2 pp 210ndash212 2007

[10] T Hayat S Nadeem and S Asghar ldquoPeriodic unidirectionalflows of a viscoelastic fluid with the fractional Maxwell modelrdquoApplied Mathematics and Computation vol 151 no 1 pp 153ndash161 2004

[11] M Khan S Hyder Ali C Fetecau and H Qi ldquoDecay ofpotential vortex for a viscoelastic fluid with fractional Maxwellmodelrdquo Applied Mathematical Modelling vol 33 no 5 pp2526ndash2533 2009

[12] D Tripathi S K Pandey and S K Das ldquoPeristaltic flow ofviscoelastic fluid with fractional Maxwell model through achannelrdquo Applied Mathematics and Computation vol 215 no10 pp 3645ndash3654 2010

[13] D Tripathi S K Pandey and S K Das ldquoPeristaltic transportof a generalized Burgersrsquo fluid application to the movement ofchyme in small intestinerdquo Acta Astronautica vol 69 no 1-2 pp30ndash38 2011

[14] D Tripathi ldquoPeristaltic transport of a viscoelastic fluid in achannelrdquoActa Astronautica vol 68 no 7-8 pp 1379ndash1385 2011

[15] D Tripathi ldquoNumerical study on peristaltic flow of general-ized burgersrsquo fluids in uniform tubes in the presence of anendoscoperdquo International Journal for Numerical Methods inBiomedical Engineering vol 27 no 11 pp 1812ndash1828 2011

[16] D Tripathi ldquoPeristaltic flow of a fractional second grade fluidthrough a cylindrical tuberdquo Thermal Science vol 15 pp 5167ndash5173 2011

[17] D Tripathi ldquoNumerical and analytical simulation of peristalticflows of generalized Oldroyd-B fluidsrdquo International Journal forNumerical Methods in Fluids vol 67 no 12 pp 1932ndash1943 2011

[18] D Tripathi ldquoA mathematical model for the peristaltic flow ofchyme movement in small intestinerdquoMathematical Biosciencesvol 233 no 2 pp 90ndash97 2011

[19] D Tripathi ldquoPeristaltic transport of fractional Maxwell fluidsin uniform tubes applications in endoscopyrdquo Computers andMathematics with Applications vol 62 no 3 pp 1116ndash1126 2011

[20] H Sato T Kawai T Fujita and M Okabe ldquoTwo-dimensionalperistaltic flow in curved channelsrdquo Transactions of the JapanSociety of Mechanical Engineers B vol 66 no 643 pp 679ndash6852000

[21] N Ali M Sajid and T Hayat ldquoLong wavelength flow analysisin a curved channelrdquo Zeitschrift fur Naturforschung A vol 65no 3 pp 191ndash196 2010

[22] N Ali M Sajid T Javed and Z Abbas ldquoHeat transfer analysisof peristaltic flow in a curved channelrdquo International Journal ofHeat and Mass Transfer vol 53 no 15-16 pp 3319ndash3325 2010

[23] T Hayat M Javed and A A Hendi ldquoPeristaltic transportof viscous fluid in a curved channel with compliant wallsrdquoInternational Journal of Heat andMass Transfer vol 54 no 7-8pp 1615ndash1621 2011

[24] N Ali M Sajid Z Abbas and T Javed ldquoNon-Newtonian fluidflow induced by peristalticwaves in a curved channelrdquoEuropeanJournal of Mechanics B vol 29 no 5 pp 387ndash394 2010

[25] T Hayat S Noreen and A Alsaedi ldquoEffect of an inducedmagnetic field on peristaltic flow of non-Newtonian fluid in acurved channelrdquo Journal of Mechanics in Medicine and Biologyvol 12 Article ID 125005 26 pages 2012

[26] J V Ramanamurthy K M Prasad and V K Narla ldquoUnsteadyperistaltic transport in curved channelsrdquo Physics of Fluids vol25 Article ID 091903 20 pages 2013

[27] G Jumarie ldquoTable of some basic fractional calculus formulaederived from amodified Riemann-Liouville derivative for non-differentiable functionsrdquo Applied Mathematics Letters vol 22no 3 pp 378ndash385 2009

[28] C Friedrich ldquoRelaxation and retardation functions of theMaxwell model with fractional derivativesrdquo Rheologica Actavol 30 no 2 pp 151ndash158 1991

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In all these previous studies authors have carried outthe analysis in straight channels or tubes In fact the shapeof most physiological ducts are curved The correspondingstudy of the peristaltic flow in curved channel was carriedout by Sato et al [20] Ali et al [21] discussed the peristalticmotion in a curved channel using wave frame Later Ali et al[22] extended the flow analysis by considering heat transfereffects Hayat et al [23] have examined the peristaltic flow ofviscous fluid in a curved channel with complaint walls Non-Newtonian fluid flow induced by peristaltic waves in a curvedchannel has been investigated by Ali et al [24] Hayat et al[25] have extended the problem to investigate the effect of aninduced magnetic field on peristaltic flow of non-Newtonianfluid in a curved channel Very recently Ramanamurthy et al[26] have investigated unsteady effects of peristaltic transportin curved channels

The objective of this paper is to investigate the effects offractional parameters of second grade fluid on peristaltic flowthrough a curved channel In order to study these effects weemployed long wave length and the low Reynolds numberapproximation The fractional calculus approach is usedto obtain analytical solution of the problem The obtainedexpressions are utilized to discuss the influences of variousphysical parameters

2 Preliminaries

Here we present the necessary definitions from fractionalcalculus theory which are useful in subsequent sectionsThese definitions can be found in the recent literature [7 2728]

Definition 1 TheRiemann-Liouville fractional integral oper-ator of order 120572 gt 0 of a function 119891(119909) (0infin) rarr R is givenby

119869120572

119891 (119909) =

1

Γ (120572)

int

119909

0

(119909 minus 120585)120572minus1

119891 (120585) 119889120585 120572 gt 0 119909 gt 0

(1)

Definition 2 The fractional derivative of order 120572 gt 0 of acontinuous function 119891(119909) (0infin) rarr R is given by

119863120572

119891 (119909) =

1

Γ (119898 minus 120572)

(

119889

119889119909

)

119898

int

119909

0

(119909 minus 120585)119898minus120572minus1

119891 (120585) 119889120585

for 119898 minus 1 lt 120572 le 119898 119898 isin N 119909 gt 0 119891 isin C119898

minus1

(2)

where 119898 = [120572] + 1 provided that the right-hand side ispointwise defined on (0infin)

Remark 3 For example 119891(119909) = 119909120573 we quote for 120573 gt minus1 in

(2) one can get

119863120572

119909120573

=

Γ (120573 + 1)

Γ (120573 minus 120572 + 1)

119909120573minus120572

(3)

giving in particular 119863120572119909120573minus119899 = 0 119899 = 1 2 119873 where 119873 isthe smallest integer greater than or equal to 120572

3 Mathematical Model

We consider the flow of a fluidmodeled as a fractional secondgrade model given by

120591 = 120583(1 + 120582120572

1

120597120572

120597119905120572) 120574 (4)

where 1205821 120572 119905 120591 120583 and 120574 are the relaxation time fractional

parameter time shear stress viscosity and rate of shearstrain

We consider the flow of an incompressible viscoelasticfluid with fractional second grade model in a curved channelinduced by two infinite trains of sinusoidal waves that arepropagated along the flexible walls of the channel Whenundeformed the walls are separated by a distance 2119886 theconstant radius of curvature at the channel centreline is 119877We choose curvilinear coordinate system (119909 119903) in such a waythat 119909-axis lies along the center line of the curved channel and119903-axis is normal to it and is measured from central line Thescale factors are ℎ

1= (119903 + 119877)119877 ℎ

2= 1 and ℎ

3= 1 There is

no component in 119911 direction as shown in Figure 1The sinusoidal waves propagating along the channel walls

are described as

119903 = ℎ (119909 119905) = 119886 + 119887 cos [2120587 (

119909

120582

minus

119905

119879

)] (Upper wall)

119903 = minusℎ (119909 119905) = minus119886 minus 119887 cos [2120587 (

119909

120582

minus

119905

119879

)] (Lower wall) (5)

Here 119909 is the axial distance 119886 the radius of the stationarycurved channel 120601 the wave amplitude 120582 the wave length 119879the wave period and ℎ the radial displacement of the wavefrom the centerline The wavelength is large compared withthe channelrsquos width (119886120582 ≪ 1)

The governing equations for an ordinary second gradefluid through curved channel with the velocity vector 119881 =

119906119890119909+ V119890119903are given by

119877

120597119906

120597119909

+

120597

120597119903

(119903 + 119877) V = 0 (6)

120597119906

120597119905

+ (119881 sdot nabla) 119906 minus

119906V119903 + 119877

= minus

119877

120588 (119903 + 119877)

120597119901

120597119909

+ ](1 + 1205821

120597

120597119905

)

times [nabla2

119906 minus

119906

(119903 + 119877)2+

2119877

(119903 + 119877)2

120597V120597119909

]

(7)


C

O

b

a

R

P(0 t)P(L t)

h(x t)

x u

r120582

Nw

Tw


120597V120597119905


1199062

119903 + 119877

= minus

1

120588

120597119901

120597119903

+ ](1 + 1205821

120597

120597119905

)


V(119903 + 119877)

2minus

2119877

(119903 + 119877)2

120597119906

120597119909

]

(8)

where


119877119906

(119903 + 119877)

120597

120597119909

+ V120597

120597119903

nabla2

= (

119877

119903 + 119877

)

2

1205972

1205971199092+

1

119903 + 119877

120597

120597119903

+

1205972

1205971199032

(9)



1199091015840

=

119909

120582

1199031015840

=

119903

119886

1199061015840

=

119906

119888

V1015840 =V120575119888

ℎ1015840

=

ℎ

119886

120601 =

119887

119886

120581 =

119877

119886

1199011015840

=

1198862

119901

120583119888120582

Re =

119888119886120575

] 120595

1015840

=

120595

119886119888

119876 =

119876

119886119888

1199051015840

=

119905

119879

(10)




119906 = minus

120597120595

120597119903

V =

120581

119903 + 120581

120597120595

120597119909

(11)


120597119901

120597119909

= (1 + 120582120572

1

120597120572

120597119905120572)(

1

120581

120597

120597119903

(119903 + 120581)

1205972

120595

1205972119903

minus

1

120581 (119903 + 120581)

120597120595

120597119903

)

(12)

120597119901

120597119903

= 0 (13)


120595 = minus

119876 (119909 119905)

2

120597120595

120597119903

= 0 at 119903 = ℎ (119909 119905)

120595 =

119876 (119909 119905)

2

120597120595

120597119903

= 0 at 119903 = minusℎ (119909 119905)

(14)


int

ℎ

minusℎ

119906(119909 119903 119905)119889119903



119883 = 119909 minus 119888119905 119884 = 119903 119880 = 119906 minus 1

119881 = V 119902 = 119876 minus 2ℎ Ψ = 120595 minus 119903

(15)


Ψ = 1198621+ 1198622log (119884 + 120581) + 119862

3(119884 + 119896)

2

+ 1198624(119884 + 120581)

2 log (119884 + 120581) + (119884 + 120581)

1198621= minus

1

2

[2119867 + 119902 + 2120581

+ ((119867 + 119902)2

(2119867 + 119902)

times (2119867120581 + (119867 minus 120581)2

times (1 minus 2 log (119867 + 120581))

times log ((119867 + 120581) (119867 minus 120581)) ) )


1205821 = 1 2 3 4

1210

86420


00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

120572 = 05

ΔP

(a)

00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

ΔP

10

5

0

minus5

minus10

120572 = 025 05 075 10

1205821 = 10

(b)

00 05 10 15 20QT

120572 = 05

120581 = 20

t = 10

1205821 = 10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

ΔP

120601 = 03 04 05 06

(c)


00 05 10 15 20QT

ΔP

120601 = 04

120572 = 05

t = 10

1205821 = 10

2

0

minus2

minus4

minus6

(d)



1205812

+ (1198672

minus 1205812

)

2

times (log ((120581 minus 119867) (120581 + 119867)))2

)

minus1

]

1198622=

(2119867 + 119902) (1198672

minus 1205812

)

2

log ((120581 minus 119867) (120581 + 119867))

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

1198623= minus ( (2119867 + 119902)


minus(119867 + 120581)2 log (119867 + 120581) ])


1205812

+ (1198672

minus 1205812

)

2

times(log ((120581 minus 119867) (120581 + 119867)))2

] )

minus1

1198624= minus

2 (2119867 + 119902)119867120581

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

(16)


120597119901

120597119883

(119883 119905)

= (1 + 120582120572

1

120597120572

120597119905120572)

times

8 (2119867 + 119902)119867

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2


120597119901

120597119883

(119883 119905)

=

8 (2119867 + 119902)119867

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

times (1 + 120582120572

1

119905minus120572

Γ (1 minus 120572)

)

(18)


00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

8

4

6

2

0

minus2

minus4

120572 = 025 05 075 10

1205821 = 10

F

(a)

120572 = 05

00 05 10 15 20QT

8

4

6

2

0

minus2

minus4

120601 = 06

120581 = 20

t = 10

1205821 = 04 06 08 10

F

(b)

F

00 05 10 15 20QT

120572 = 05

120581 = 20

t = 10

1205821 = 10

6

5

4

3

2

1

0

minus2

minus1

minus4

minus3

120601 = 03 04 05 06

(c)

Curved channel

00 05 10 15 20QT

F

120601 = 04

120572 = 05

t = 10

1205821 = 10

2

0

1

4

5

3

minus1

Straight channel

(d)



Δ119901 = int

1

0

120597119901

120597119909

119889119909

119865 = int

1

0

minusℎ

120597119901

120597119909

119889119909

(19)



1) occlusion










(a) 120581 = 3 (b) 120581 = 5

(c) 120581 = 12 (d) 120581 rarr infin







119879= 15 and 120601 =







1



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










C

O

b

a

R

P(0 t)P(L t)

h(x t)

x u

r120582

Nw

Tw


120597V120597119905


1199062

119903 + 119877

= minus

1

120588

120597119901

120597119903

+ ](1 + 1205821

120597

120597119905

)


V(119903 + 119877)

2minus

2119877

(119903 + 119877)2

120597119906

120597119909

]

(8)

where


119877119906

(119903 + 119877)

120597

120597119909

+ V120597

120597119903

nabla2

= (

119877

119903 + 119877

)

2

1205972

1205971199092+

1

119903 + 119877

120597

120597119903

+

1205972

1205971199032

(9)



1199091015840

=

119909

120582

1199031015840

=

119903

119886

1199061015840

=

119906

119888

V1015840 =V120575119888

ℎ1015840

=

ℎ

119886

120601 =

119887

119886

120581 =

119877

119886

1199011015840

=

1198862

119901

120583119888120582

Re =

119888119886120575

] 120595

1015840

=

120595

119886119888

119876 =

119876

119886119888

1199051015840

=

119905

119879

(10)




119906 = minus

120597120595

120597119903

V =

120581

119903 + 120581

120597120595

120597119909

(11)


120597119901

120597119909

= (1 + 120582120572

1

120597120572

120597119905120572)(

1

120581

120597

120597119903

(119903 + 120581)

1205972

120595

1205972119903

minus

1

120581 (119903 + 120581)

120597120595

120597119903

)

(12)

120597119901

120597119903

= 0 (13)


120595 = minus

119876 (119909 119905)

2

120597120595

120597119903

= 0 at 119903 = ℎ (119909 119905)

120595 =

119876 (119909 119905)

2

120597120595

120597119903

= 0 at 119903 = minusℎ (119909 119905)

(14)


int

ℎ

minusℎ

119906(119909 119903 119905)119889119903



119883 = 119909 minus 119888119905 119884 = 119903 119880 = 119906 minus 1

119881 = V 119902 = 119876 minus 2ℎ Ψ = 120595 minus 119903

(15)


Ψ = 1198621+ 1198622log (119884 + 120581) + 119862

3(119884 + 119896)

2

+ 1198624(119884 + 120581)

2 log (119884 + 120581) + (119884 + 120581)

1198621= minus

1

2

[2119867 + 119902 + 2120581

+ ((119867 + 119902)2

(2119867 + 119902)

times (2119867120581 + (119867 minus 120581)2

times (1 minus 2 log (119867 + 120581))

times log ((119867 + 120581) (119867 minus 120581)) ) )


1205821 = 1 2 3 4

1210

86420


00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

120572 = 05

ΔP

(a)

00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

ΔP

10

5

0

minus5

minus10

120572 = 025 05 075 10

1205821 = 10

(b)

00 05 10 15 20QT

120572 = 05

120581 = 20

t = 10

1205821 = 10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

ΔP

120601 = 03 04 05 06

(c)


00 05 10 15 20QT

ΔP

120601 = 04

120572 = 05

t = 10

1205821 = 10

2

0

minus2

minus4

minus6

(d)



1205812

+ (1198672

minus 1205812

)

2

times (log ((120581 minus 119867) (120581 + 119867)))2

)

minus1

]

1198622=

(2119867 + 119902) (1198672

minus 1205812

)

2

log ((120581 minus 119867) (120581 + 119867))

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

1198623= minus ( (2119867 + 119902)


minus(119867 + 120581)2 log (119867 + 120581) ])


1205812

+ (1198672

minus 1205812

)

2

times(log ((120581 minus 119867) (120581 + 119867)))2

] )

minus1

1198624= minus

2 (2119867 + 119902)119867120581

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

(16)


120597119901

120597119883

(119883 119905)

= (1 + 120582120572

1

120597120572

120597119905120572)

times

8 (2119867 + 119902)119867

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2


120597119901

120597119883

(119883 119905)

=

8 (2119867 + 119902)119867

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

times (1 + 120582120572

1

119905minus120572

Γ (1 minus 120572)

)

(18)


00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

8

4

6

2

0

minus2

minus4

120572 = 025 05 075 10

1205821 = 10

F

(a)

120572 = 05

00 05 10 15 20QT

8

4

6

2

0

minus2

minus4

120601 = 06

120581 = 20

t = 10

1205821 = 04 06 08 10

F

(b)

F

00 05 10 15 20QT

120572 = 05

120581 = 20

t = 10

1205821 = 10

6

5

4

3

2

1

0

minus2

minus1

minus4

minus3

120601 = 03 04 05 06

(c)

Curved channel

00 05 10 15 20QT

F

120601 = 04

120572 = 05

t = 10

1205821 = 10

2

0

1

4

5

3

minus1

Straight channel

(d)



Δ119901 = int

1

0

120597119901

120597119909

119889119909

119865 = int

1

0

minusℎ

120597119901

120597119909

119889119909

(19)



1) occlusion










(a) 120581 = 3 (b) 120581 = 5

(c) 120581 = 12 (d) 120581 rarr infin







119879= 15 and 120601 =







1



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1205821 = 1 2 3 4

1210

86420


00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

120572 = 05

ΔP

(a)

00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

ΔP

10

5

0

minus5

minus10

120572 = 025 05 075 10

1205821 = 10

(b)

00 05 10 15 20QT

120572 = 05

120581 = 20

t = 10

1205821 = 10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

ΔP

120601 = 03 04 05 06

(c)


00 05 10 15 20QT

ΔP

120601 = 04

120572 = 05

t = 10

1205821 = 10

2

0

minus2

minus4

minus6

(d)



1205812

+ (1198672

minus 1205812

)

2

times (log ((120581 minus 119867) (120581 + 119867)))2

)

minus1

]

1198622=

(2119867 + 119902) (1198672

minus 1205812

)

2

log ((120581 minus 119867) (120581 + 119867))

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

1198623= minus ( (2119867 + 119902)


minus(119867 + 120581)2 log (119867 + 120581) ])


1205812

+ (1198672

minus 1205812

)

2

times(log ((120581 minus 119867) (120581 + 119867)))2

] )

minus1

1198624= minus

2 (2119867 + 119902)119867120581

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

(16)


120597119901

120597119883

(119883 119905)

= (1 + 120582120572

1

120597120572

120597119905120572)

times

8 (2119867 + 119902)119867

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2


120597119901

120597119883

(119883 119905)

=

8 (2119867 + 119902)119867

minus411986721205812+ (1198672minus 1205812)2

(log ((120581 minus 119867) (120581 + 119867)))2

times (1 + 120582120572

1

119905minus120572

Γ (1 minus 120572)

)

(18)


00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

8

4

6

2

0

minus2

minus4

120572 = 025 05 075 10

1205821 = 10

F

(a)

120572 = 05

00 05 10 15 20QT

8

4

6

2

0

minus2

minus4

120601 = 06

120581 = 20

t = 10

1205821 = 04 06 08 10

F

(b)

F

00 05 10 15 20QT

120572 = 05

120581 = 20

t = 10

1205821 = 10

6

5

4

3

2

1

0

minus2

minus1

minus4

minus3

120601 = 03 04 05 06

(c)

Curved channel

00 05 10 15 20QT

F

120601 = 04

120572 = 05

t = 10

1205821 = 10

2

0

1

4

5

3

minus1

Straight channel

(d)



Δ119901 = int

1

0

120597119901

120597119909

119889119909

119865 = int

1

0

minusℎ

120597119901

120597119909

119889119909

(19)



1) occlusion










(a) 120581 = 3 (b) 120581 = 5

(c) 120581 = 12 (d) 120581 rarr infin







119879= 15 and 120601 =







1



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00 05 10 15 20QT

120601 = 06

120581 = 20

t = 10

8

4

6

2

0

minus2

minus4

120572 = 025 05 075 10

1205821 = 10

F

(a)

120572 = 05

00 05 10 15 20QT

8

4

6

2

0

minus2

minus4

120601 = 06

120581 = 20

t = 10

1205821 = 04 06 08 10

F

(b)

F

00 05 10 15 20QT

120572 = 05

120581 = 20

t = 10

1205821 = 10

6

5

4

3

2

1

0

minus2

minus1

minus4

minus3

120601 = 03 04 05 06

(c)

Curved channel

00 05 10 15 20QT

F

120601 = 04

120572 = 05

t = 10

1205821 = 10

2

0

1

4

5

3

minus1

Straight channel

(d)



Δ119901 = int

1

0

120597119901

120597119909

119889119909

119865 = int

1

0

minusℎ

120597119901

120597119909

119889119909

(19)



1) occlusion










(a) 120581 = 3 (b) 120581 = 5

(c) 120581 = 12 (d) 120581 rarr infin







119879= 15 and 120601 =







1



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) 120581 = 3 (b) 120581 = 5

(c) 120581 = 12 (d) 120581 rarr infin







119879= 15 and 120601 =







1



References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article peristaltic motion of viscoelastic...

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