Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes : Numerical Solutions

190 Pages·2016·4.738 MB·English

by  Mukhopadhyay, Swati, Vajravelu, Kuppalapalle

Fluid Flow, Heat and Mass Transfer
at Bodies of Different Shapes
Fluid Flow, Heat and
Mass Transfer at Bodies
of Different Shapes
Numerical Solutions
Kuppalapalle Vajravelu
The University of Central Florida, USA
and
Swati Mukhopadhyay
The University of Burdwan, India
AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON
NEW YORK (cid:129) OXFORD (cid:129) PARIS (cid:129) SAN DIEGO
SAN FRANCISCO (cid:129) SINGAPORE (cid:129) SYDNEY (cid:129) TOKYO
Academic Press is an imprint of Elsevier
AcademicPressisanimprintofElsevier
125LondonWall,London,EC2Y5AS,UK
525BStreet,Suite1800,SanDiego,CA92101–4495,USA
225WymanStreet,Waltham,MA02451,USA
TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK
©2016ElsevierLtd.Allrightsreserved.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,
electronicormechanical,includingphotocopying,recording,oranyinformationstorageand
retrievalsystem,withoutpermissioninwritingfromthepublisher.Detailsonhowtoseek
permission,furtherinformationaboutthePublisher’spermissionspoliciesandour
arrangementswithorganizationssuchastheCopyrightClearanceCenterandtheCopyright
LicensingAgency,canbefoundatourwebsite:www.elsevier.com/permissions.
Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe
Publisher(otherthanasmaybenotedherein).
Notices
Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchand
experiencebroadenourunderstanding,changesinresearchmethods,professionalpractices,or
medicaltreatmentmaybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgein
evaluatingandusinganyinformation,methods,compounds,orexperimentsdescribedherein.
Inusingsuchinformationormethodstheyshouldbemindfuloftheirownsafetyandthesafety
ofothers,includingpartiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,
assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproducts
liability,negligenceorotherwise,orfromanyuseoroperationofanymethods,products,
instructions,orideascontainedinthematerialherein.
ISBN:978-0-12-803733-1
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressControlNumber:2015939630
ForinformationonallAcademicPresspublications
visitourwebsiteathttp://store.elsevier.com/
Preface
Fluid mechanics is one of the oldest branches of applied mathematics. It is also the
foundationoftheunderstandingofvariousaspectsofscienceandengineering.Awide
varietyofmathematicalproblems,appearinginareas asdiverse asfluidmechanics,
mechanical engineering, chemical engineering, theoretical physics, and aerospace
engineering,havebeensolvedbymeansofanalyticalornumericalmethods.Though
analysisofdifferenttypesoffluidflowandheatormasstransferproblemsareavail-
ableintheopenliterature,therearestillanumberofgapsthataretobefilledup.There
areseveralexcellentbookscoveringdifferentaspectsoffluidflowandheatormass
transfer.Yetonestilllooksforasystematicandsequentialanalysisthathelpsinunder-
standing thisparticular area ofinterest.
Tohelpstudentsandresearchersacquireadeeperunderstandingofthecharacter-
isticsoffluidflowandheatandmasstransfer,thismonographaimstopresent,ingen-
eral,astudyoftransportphenomena.Itiswellknownthatforexternalflows,theshape
oftheobjectinfluencestheflowoveranobject(i.e.,abody)significantly.Asaresult,
itaffectstheheatandmasstransfercharacteristics.Inotherwords,thebookaimsto
helpreadersdeveloptheirunderstandinginthisparticularfieldwithoutspendinghuge
timeinsearchingtheendlessliteratureonthisarea.Tohelpdevelopaclearinsight,we
discussseveralflowfeatures.Bymaintainingtheapplicabilityoftheobtainedresults,
we alsodiscussseveral cases ofphysical problems.
Inselectingspecificproblemstoworkthrough,wehaverestrictedourattentionto
thephenomenaoffluidflowandheatormasstransferassuchproblemsintroducea
widevarietyofmathematicalproblemsofinterest.Hence,inordertoillustratevarious
propertiesandtoolsusefulinanalyzingtheproblems,wehaveselectedrecentresearch
inthe area offluid flow and heat ormass transfer.
WeappreciatethesupportandmotivationoftheeditorGlynJonesandtheeditorial
projectmanagerStevenMathews.WealsoacknowledgetheroleofElsevier(Oxford)
formakingthisbookareality.ThankstoMr.SudiptaGhosh(PhDstudentofDr.Swati
Mukhopadhyay) for his help in drawing some of the figures. We thank Prof. Mike
Taylor for reading the entire manuscript and suggesting some needed changes. The
authorsaregratefultoalltheauthorsofthearticleslistedinthebibliographyofthis
book. The authors are also very much thankful to their coauthors. Finally, we very
much like to acknowledge the encouragement, patience, and support provided by
the membersofourfamilies.
K.Vajravelu
Orlando, Florida
S. Mukhopadhyay
Burdwan,India
2015
Introduction
Air andwater are the mostimportant constituentsof the environment we livein, so
thatalmosteverythingwedoisconnectedtothescienceoffluidmechanics.Forexam-
ple,theflightofbirdsintheairandthemotionoffishinthewatercanbeexplained
from the perspective of fluid mechanics [1]. The designs of airplanes and ships are
basedonthetheoryoffluidmechanics.Fluidmechanicsisoneoftheoldestbranches
ofappliedmathematics,andthefoundationoftheunderstandingofdifferentaspects
of science and engineering [2–5]. From the nineteenth century, the scope of fluid
mechanics has steadily broadened, as the study of hydraulics was associated with
the growth of the fields of civil engineering and naval architecture. In recent times,
thedevelopmentofthedifferentbranchesofengineering,namely,aeronautical,chem-
ical,andmechanicalengineering,havegivenadditionalstimulitothestudyoffluid
mechanics.Itnowranksasoneofthemostimportantbasicsubjectsnotonlyinapplied
mathematicsbutalsoinengineering[6].Now,itisasubjectofwidespreadinterestin
almost all fields of engineering as well as in astrophysics, meteorology, physical
chemistry, plasma physics,geophysics, biology, and biomedicine[7–9].
Innature,fluidflowoverbodiesoccursfrequentlyandgivesrisetonumerousphys-
icalphenomena,forexample,dragforceactingontrees,underwaterpipelines,auto-
mobiles,theliftgeneratedbyairplanewings,upwarddraftofrains,dustparticlesin
highwinds,andtransportationofredbloodcellsinbloodflow(see,[6]).Sometimes,
fluidmovesoverastationarybody,forexample,windblowingoverabuilding,ora
bodymovingthroughaquiescentfluidorabusmovingthroughair.Suchmotionsare
referred to as flows over bodies or external flows [10]. The shape of the object has
profound influence on the flow over a body and thus affects significantly the heat
andmasstransfercharacteristics.Flowpastbodiescanbeclassifiedintoincompress-
ibleandcompressibleflows.Compressibilityeffectsareneglectedatvelocitiesbelow
360km/hour,andsuchtypesofflowsareknownasincompressibleflows.Inthisbook,
we are concerned withincompressible fluidflows [11–17].
Becauseoftherecenthighdemandintheneedforunderstandingandanalyzingthe
problemswecomeacrossinscienceandengineering,wefeelthatthereisaneedfora
bookofthiskind.Theunderlyingaimofthisbookistopresenttransportphenomena
that will help students and researchers in the field of fluid mechanics in acquiring
a deeper understanding of the characteristics of flow and heat and mass transfer
(see,e.g.,[18]).Obviously,partofthematerialinthebookcanbeconvenientlyused
asanintroductorycoursematerialforresearchersworkinginboundarylayertheory,
flow,andheatandmasstransfer[19–45].Also,thebookisintendedforgraduatestu-
dentsinmathematics,engineering,andinthemathematicalsciences.Inaddition,the
materialinthebookmaybeofinteresttoresearchersworkinginphysicalchemistry,
soil physics, meteorology, andnanotechnology.
x Introduction
Thebookisdesignedtoaccommodateseveraltopicsofvaryingemphasis,andthe
chapters comprise fairly self-contained material from which one can make various
coherentselections.Thereareseveralunderlyingthemesthatbecomeapparentwhen
oneexaminestheliteratureonthesubject.Wehopetobringoutclearinsightbydis-
cussingseveralflowfeatures.Also,wediscussseveralcasesofphysicalproblemsin
general.
The outline of the book will be as follows. Chapter 1 deals with the numerical
method(s)adoptedintheseworks.InChapters2–4,whichcomprisePartIofthebook,
wepresenttheflowpastsurfacesofdifferenttypes,namely,stretching,shrinking,and
flat surfaces. This first set of chapters provides explanations intended for general
readers and can be directly employed for problems in engineering, applied physics,
and other applied sciences. We keep the discussion broad based so as to provide a
frameworkforresearchers.Inordertomotivatethereaderandprovideagoodunder-
standingofthesubjectmatter,attheendofChapters2–4,weprovidemultipleexam-
ples of problemsthat have been solved numerically.
InPartIIofthebook,Chapters5–6,weshiftthefocustoconcreteexamplesand
problemsrelatedtobluffbodiesinfluidmechanicsandheatandmasstransfer.Here
the governingequationsofthe problems are highlynonlineardifferentialequations.
TheproblemsconsideredinthisPartwillhelpthereaderunderstandproblemsofphys-
icalrelevanceandapplyittotheotherphysicalfieldsofinterest.Wegroupsuchprob-
lems into three chapters: general fluid flow past a cylinder in Chapter 5, fluid flow
over a sphere in Chapter 6, and finally problems related to flow past a wedge in
Chapter7.
References
[1] YuanSW.Foundationsoffluidmechanics.NewDelhi:Prentice-HallofIndia;1969.
[2] Abbas Z, Wang Y, Hayat T, Oberlack M. Slip effects and heat transfer analysis in a
viscous fluid over an oscillatory stretching surface. Int J Numer Meth Fluid 2009;59:
443–58.
[3] AnderssonHI,BechKH,DandapatBS.Magnetohydrodynamicflowofapowerlawfluid
overastretchingsheet.IntJNon-LinearMech1992;27:929–36.
[4] ArielPD,HayatT,AsgharS.Theflowofanelastico-viscousfluidpastastretchingsheet
withpartialslip.ActaMechanica2006;187:29–35.
[5] BachokN,IshakA,PopI.Mixedconvectionboundarylayerflowoverapermeablever-
tical flat plate embedded in an anisotropic porous medium. Math Problems Eng
2010;659023:http://dx.doi.org/10.1155/2010/659023.
[6] BatchelorGK.Anintroductiontofluiddynamics.London:CambridgeUniversityPress;
2006.
[7] Bhattacharyya K, Mukhopadhyay S, Layek GC. Slip effects on an unsteady boundary
layerstagnation-pointflowandheattransfertowardsastretchingsheet.ChinPhysLett
2011;28(9):094702.
[8] BhattacharyyaK,MukhopadhyayS,LayekGC.MHDboundarylayerslipflowandheat
transferoveraflatplate.ChinPhysLett2011;28(2):024701.
[9] BhattacharyyaK,VajraveluK.Stagnation-pointflowandheattransferoveranexponen-
tiallyshrinkingsheet.CommNonlinearSciNumerSimulat2012;17(7):2728–34.
Introduction xi
[10] Cebeci T, Bradshow P. Momentum transfer in boundary layer. Washington, DC:
McGraw-HillHemisphere;1977.
[11] ChamkhaAJ,AlyAM,MansourMA.Similaritysolutionforunsteadyheatandmasstrans-
fer from a stretching surface embedded in a porous medium with suction/injection and
chemicalreactioneffects.ChemEngComm2010;197:846–58.
[12] CortellR.Flowandheattransferofafluidthroughaporousmediumoverastretching
surface with internal heat generation/absorption and suction/blowing. Fluid Dyn Res
2005;37:231–45.
[13] Cortell R. Effects of viscous dissipation and work done by deformation on the MHD
flow and heat transfer of a viscoelastic fluid over a stretching sheet. Phys Lett A
2006;357:298–305.
[14] CraneLJ.Flowpastastretchingplate.ZAngewMathPhys1970;21:645–7.
[15] DandapatBS,SantraB,VajraveluK.Theeffectsofvariablefluidpropertiesandthermo-
capillarityontheflowofathinfilmonanunsteadystretchingsheet.IntJHeatMassTrans-
fer2007;50:991–6.
[16] GuptaPS,GuptaAS.Heatandmasstransferonastretchingsheetwithsuctionorblowing.
CanJChemEng1977;55:744–6.
[17] HayatT,AbbasZ,SajidM.Seriessolutionfortheupper-convectedMaxwellfluidovera
porousstretchingplate.PhysLettA2006;358:396–403.
[18] InghamDB,PopI,editors.Transportphenomenainporousmedia.Oxford:Elsevier;2005,
vol.III.
[19] IshakA.Mixedconvectionboundarylayerflowoveraverticalcylinderwithprescribed
surfaceheatflux.JPhysA:MathTheor2009;42:195501.
[20] KumaranV,BanerjeeAK,KumarAV,VajraveluK.MHDflowpastastretchingperme-
ablesheet.ApplMathComput2008;210:26–32.
[21] LiuIC,AnderssonHI.Heattransferinaliquidfilmonanunsteadystretchingsheet.IntJ
ThermalSci2008;47:766–72.
[22] MukhopadhyayS.Effectofthermalradiationonunsteadymixedconvectionflowandheat
transfer over a porous stretching surface in porous medium. Int J Heat Mass Transfer
2009;52:3261–5.
[23] MukhopadhyayS.Effectsofsliponunsteadymixedconvectiveflowandheattransferpast
astretchingsurface.ChinPhysLett2010;27(12):124401.
[24] MukhopadhyayS.HeattransferanalysisforunsteadyMHDflowpastanon-isothermal
stretchingsurface.NuclEngDes2011;241:4835–9.
[25] MukhopadhyayS.Chemicallyreactivesolutetransferinboundarylayerslipflowalonga
stretchingcylinder.FrontChemSciEng2011;5:385–91.
[26] Mukhopadhyay S, Vajravelu K. Effects of transpiration and internal heat generation/
absorptionontheunsteadyflowofaMaxwellfluidatastretchingsurface.ASMEJAppl
Mech2012;79:http://dx.doi.org/10.1115/1.4006260,044508-1-6.
[27] Mukhopadhyay S, Vajravelu K, Van Gorder RA. Flow and heat transfer in a moving
fluid over a moving non-isothermal surface. Int J Heat Mass Transfer 2012;55(23-24):
6632–7.
[28] Prasad KV, Vajravelu K, Datti PS. Mixed convection heat transfer over a non-linear
stretchingsurfacewithvariablefluidproperties.IntJNon-LinearMech2010;45:320–30.
[29] PrasadKV,VajraveluK,DattiPS.Theeffectsofvariablefluidpropertiesonthehydro-
magnetic flow and heat transfer over a non-linearly stretching sheet. Int J Thermal Sci
2010;49:603–10.
[30] PrasadKV,SujathaA,VajraveluK,PopI.MHDflowandheattransferofaUCMfluid
overastretchingsurfacewithvariablethermophysicalproperties.Meccanica2012;47:
1425–39.
xii Introduction
[31] SajidM,AhmadI,HayatT,AyubM.Unsteadyflowandheattransferofasecondgrade
fluidoverastretchingsheet.CommNonlinearSciNumerSimulat2009;14:96–108.
[32] SchlichtingH.Boundary-layertheory.NewYork:McGraw-Hill;2008.
[33] SharidanS,MahmoodT,PopI.Similaritysolutionsfortheunsteadyboundarylayerflow
andheattransferduetoastretchingsheet.IntJApplMechEng2006;11:647–54.
[34] SiddiquiAM,ZebA,GhoriQK,BenharbitAM.Homotopyperturbationmethodforheat
transfer flow of a third grade fluid between parallel plates. Chaos Solitons Fractals
2008;36:182–92.
[35] SoundalgekarVM,TakharHS,VighnesamNV.Thecombinedfreeandforcedconvection
flowpastasemi-infiniteplatewithvariablesurfacetemperature.NuclEngDes1988;110:
95–8.
[36] Takhar HS, Gorla RSR, Soundalgekar VM. Radiation effects on MHDfree convection
flow of a gas past a semi-infinite vertical plate. Int J Numer Meth Heat Fluid Flow
1996;6:77–83.
[37] TsaiR,HuangKH,HuangJS.Flowandheattransferoveranunsteadystretchingsurface
withanon-uniformheatsource.IntCommHeatMassTran2008;35:1340–3.
[38] VafaiK,TienCL.Boundaryandinertiaeffectsonflowandheattransferinporousmedia.
IntJHeatMassTransfer1981;24:195–204.
[39] VajraveluK,PrasadKV,Chiu-OnNg.Unsteadyflowandheattransferinathinfilmof
Ostwald–deWaeleliquidoverastretchingsurface.CommNonlinearSciNumerSimulat
2012;17:4163–73.
[40] Van Gorder RA, Vajravelu K. Hydromagnetic stagnation point flow of a second grade
fluidoverastretchingsheet.MechResComm2010;37:113–8.
[41] VanGorderRA,SweetE,VajraveluK.Nanoboundarylayersoverstretchingsurfaces.
CommNonlinearSciNumerSimulat2010;15:1494–500.
[42] VanGorderRA,VajraveluK,AkyildizFT.Existenceanduniquenessresultsforanon-
linear differential equation arising in viscous flow over a nonlinearly stretching sheet.
ApplMathLett2011;24:238–42.
[43] Van Gorder RA, Vajravelu K. Convective heat transfer in a conducting fluid over a
permeablestretchingsurfacewithsuctionandinternalheatgeneration/absorption.Appl
MathComput2011;217:5810–21.
[44] VleggaarJ.Laminarboundary-layerbehavioroncontinuousacceleratingsurfaces.Chem
EngSci1977;32:1517–25.
[45] WangCY.Flowduetoastretchingboundarywithpartialslip—anexactsolutionofthe
Navier-Stokesequations.ChemEngSci2002;57:3745–7.
1
Numerical methods
The governing equations of fluid flow problems are generally of a nonlinear and
boundary value type. Usually, the exact solutions of the boundary value problems
(BVPs) are very difficult to obtain, and so we have to use numerical methods.
For some special classes of flow problems, a set of partial differential equations
aretransformedintoasetofordinarydifferentialequationswiththehelpofsimilarity
variables.Thetransformed(final)equationsthencanbesolvedanalyticallyornumer-
ically. The procedure for finding a numerical solution for a BVP is generally more
difficultthanthatofaninitialvalueproblem(IVP).Anumberofmethodscanbeused
to solve linear BVPs. The method of differences is useful in such cases. But this
methodcannotbeusedfornonlinearequations.Othermethodscanbeusedtoobtain
linearlyindependentsolutions,whichcanthenbecombinedinsuchawaythatthey
satisfytheboundaryconditions(Mukhopadhyay[1]).Forsuchproblems,thediffer-
ence method can be adapted. The most popular numerical method is the shooting
method. The shooting method can be used for both linear and nonlinear problems.
Because the convergence of the method depends on a good initial guess, there is
noguaranteethatthemethodwillconverge.Butthemethodiseasytoapply,andwhen
itdoesconverge,itisusuallymoreefficientthanothermethods(Mukhopadhyay[1],
MukhopadhyayandLayek[2]).Moreover,theshootingmethodgivesmoreaccurate
resultsif guess values (slopes) are chosencorrectly.
ThebasicideaofashootingmethodistoreplacetheBVPbysomeIVPwherethe
slopeattheinitialpointisobviouslyunknown.Wecanguessthisunknownquantity,
and then, usingiteration, the guess valuecan beimproved.
The shooting method consists ofthe following steps:
1. transformationofthegivenBVPtoanIVP,
2. findingasolutionoftheIVP,and
3. findingasolutionofthegivenBVP.
(cid:1) (cid:3)
Letusconsideranonlinearsecond-orderdifferentialequationy==¼f x,y,y= with
the boundary conditions yðaÞ¼y and yðbÞ¼y .
0 1
Atfirst,wesety=¼pandp=¼fðx,y,pÞwithy¼y atx¼a.Actually,theequation
0
isrewrittenintermsofafirst-ordersystemoftwounknownfunctions.Becausethese
equationsarenonlinear,wecannotgetthesolutionbysuperpositionprinciple.Inorder
tointegratetheabovesystemasanIVP,werequireavalueforpatx¼athatisy/(a).
Generally, we take a guess for p(a) and use it to obtain a numerical solution.
Thencomparingthecalculatedvalueforyatx¼bwiththegivenboundarycondition
y¼y at x¼bandadjustingtheguessvalue,p(a),wegetabetterapproximationfor
1
thesolution.Thederivativeatx¼agivesthetrajectoryofcomputedsolution.Thatis
why, it is called shooting method. Basically, we seek a solution that also satisfies
the relation yðbÞ¼y . A suitable solution can be found by successively refining the
1
interval.Withthehelpoflinearinterpolationorotherroot-findingmethods,itisalso
FluidFlow,HeatandMassTransferatBodiesofDifferentShapes.http://dx.doi.org/10.1016/B978-0-12-803733-1.00001-6
©2016ElsevierLtd.Allrightsreserved.
4 FluidFlow,HeatandMassTransferatBodiesofDifferentShapes
possible to improve the obtained solution. To start the integration, the initial slope,
thatis,y/(a),isrequired,andtheshootingmethoddependsonthechoiceofthevalue
of y/(a).
To explain the method, we consider the equation
1
f===ðηÞ+ fðηÞf==ðηÞ¼0; (1.1)
2
along with the boundary conditions
fð0Þ¼0, f=ð0Þ¼0; (1.2)
f=ð1Þ¼1; (1.3)
which isa third-ordernonlinear BVP.
Now, the key factor is to choose an appropriate and suitable finite value of η as
η!1,sayη .Hereη¼η correspondstotheedgeoftheboundarylayer.Forcom-
1 1
putationalpurposes,η istobechosenarbitrarilylargerthantheboundarylayerthick-
1
ness.Themostimportantfactoroftheshootingmethodistochooseanappropriatefinite
valueofη .Inordertodetermineη fortheBVPstatedbyequations(1.1)–(1.3),we
1 1
startwithsomeinitialguessvalueαthatmustbedeterminedsothattheresultingsolu-
tions yield the prescribed value f=¼1atη ¼η for some particular set of physical
1
parameters (Mukhopadhyay and Layek [2]). We therefore guess at the initial slope,
andaniterativeprocedureissetupforconvergencetothecorrectslope.Anormally
betterapproximationtoαcannowbeobtainedbythefollowinglinearinterpolation
formula:
f=ðη Þ(cid:2)f=ðα ,η Þ
α ¼α +ðα (cid:2)α Þ 1 0 1 ;
2 0 1 0 f=ðα ,η Þ(cid:2)f=ðα ,η Þ
1 1 0 1
whereα ,α aretwoguessesattheinitialslopef//(0)andf=ðα ,η Þ,f=ðα ,η Þarethe
0 1 0 1 1 1
values of f/ at η¼η . We now integrate the differential equation using the initial
1
valuesfð0Þ¼0,f=ð0Þ¼0,andf==ð0Þ¼α toobtainf=ðα ,η Þ.Usinglinearinterpo-
3 2 1
lationbasedonα ,α ,wecanobtainanextapproximationα .Thisprocessisrepeated
1 2 3
until convergence is obtained. The convergence depends on a good initial guess
(Conte andBoor[3]).
Thesolution procedure is repeated with another large value of η until two suc-
1
cessive values of f//(0) differ only by the specified significant digit. The last value
of η is finally chosen to be the most appropriate value of the limit η . The value
1 1
ofη maychangeforanothersetofphysicalparameters,ifinvolvedintheproblem.
1
Oncethefinitevalueofη isdetermined,thentheintegrationiscarriedout.Wecom-
1
pare the calculated value for f/ at η¼15 (say) with the given boundary condition
f=ð15Þ¼1,andtheestimatedvalueisadjustedusingthesecantmethodtogetabetter
approximationfor the solution(Mukhopadhyay [4]).