Nonlinear Flow Phenomena and Homotopy Analysis: Fluid Flow and Heat Transfer

197 Pages·2012·2.188 MB·English

by  Kuppalapalle Vajravelu, Robert A. van Gorder (auth.)

Kuppalapalle Vajravelu
Robert A. Van Gorder
Nonlinear Flow Phenomena
and Homotopy Analysis
Fluid Flow and Heat Transfer
123
Kuppalapalle Vajravelu
Robert A. Van Gorder
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(cid:66)(cid:79)(cid:69)(cid:1)(cid:41)(cid:80)(cid:78)(cid:80)(cid:85)(cid:80)(cid:81)(cid:90)(cid:1)(cid:34)(cid:79)(cid:66)(cid:77)(cid:90)(cid:84)(cid:74)(cid:84)
Fluid Flow and Heat Transfer
Kuppalapalle Vajravelu
Robert A. Van Gorder
Nonlinear Flow Phenomena
and Homotopy Analysis
Fluid Flow and Heat Transfer
With 71 figures
3
Authors
KuppalapalleVajravelu RobertA.vanGorder
DepartmentofMathematics DepartmentofMathematics
UniversityofCentralFlorida UniversityofCentralFlorida
OrlandoFlorida32816-1364,USA OrlandoFlorida32816-1364,USA
E-mail:kvajravelu@cfl.rr.com E-mail:[email protected]
ISBN978-7-04-035449-2
HigherEducationPress,Beijing
ISBN978-3-642-32101-6 ISBN978-3-642-32102-3 (eBook)
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012943603
(cid:2)c HigherEducationPress,BeijingandSpringer-VerlagBerlinHeidelberg2012
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Foreword
“Theessenceofmathematicsliesentirelyinitsfreedom”
byGeorgCantor(1845—1918).
Solvingnonlinearproblemsisinherentlydifficult.Perturbationtechniquesaremostly
usedtogainanalyticapproximationsofnonlinearequations.Unfortunately,pertur-
bationmethodsdependtooheavilyonsmallphysicalparameters,andperturbative
resultsarevalidonlyincasesofweaknonlinearity.Althoughsomenon-perturbation
techniques were developed to overcome the restrictions of perturbation methods,
neitherofthemcanguaranteetheconvergenceofapproximationseries.
Thehomotopyanalysismethod(HAM)isapromisinganalyticmethodforhighly
nonlinearequations,whichhasbeensuccessfullyappliedinscience,appliedmath-
ematics,finance,andengineering.Basedontheconceptofhomotopyintopology,
thehomotopyanalysismethodisbeingdevelopedtosolvenonlinearproblemsinde-
pendentofanysmallphysicalparameters.Especially,theHAMintroducesatotally
newconcept“convergence-control”bymeansofthe“convergence-controlparam-
eter” that provides a simple way to guarantee the convergence of approximation
series.Infact,itis the“convergence-controlparameter”thatmakestheHAMdif-
ferentfromallofotheranalyticmethods.Asaresult,unlikeotheranalyticmethods,
theHAMisvalidforhighlynonlinearproblems.
Thankstocontributionsofmanyresearchersindozensofcountries,theHAMhas
beendevelopedandmodifiedgreatlyintheory,andwidelyappliedinmanyfields.
Written by two outstanding scholars, the book “Nonlinear Flow Phenomena and
Homotopy Analysis” describes some theoretical developments and new attempts
oftheHAM, togetherwith typicalapplicationsinflowandheattransferoffluids.
Especially,theauthorsdiscussthechoiceofinitialguess,auxiliarylinearoperator,
auxiliaryfunction,convergence-controlparameterandso on,fromthe viewpoints
ofappliedmathematician.Theselectedexamplesarefundamentalinnatureandare
importantfornewuserstounderstandandusetheHAM.Obviously,thisbookisof
benefittotheadvancementandwideapplicationoftheHAM.
ShijunLiao
May26th2012
Preface
Overthelastdecade,thehomotopyanalysismethodhascomeintoprevalenceasit
allowsonetoconstructreasonablyaccurateapproximationstononlineardifferential
equations.Awidevarietyofmathematicalproblems,appearinginareasasdiverse
asfluidmechanics,chemicalengineering,biology,finance,theoreticalphysicsand
aerospaceengineering,havebeensolvedbymeansofhomotopyanalysis.Though,
thehomotopyanalysismethodisfrequentlyemployedintheliterature,therearestill
anumberofquestionswhichremainopenregardingthemethod,dueinparttothe
generalityofthemethod.
In the present monograph,we highlight some of the key points which need to
be understoodby those workingin appliedmathematics, physics and applied sci-
enceandengineeringinordertoapplythehomotopyanalysismethod.Thatis,the
book helps the reader to developthe toolset needed to apply the method,without
siftingthroughtheendlessliteratureonthearea.Issuesoftheoptimalselectionof
theauxiliarylinearoperator,auxiliaryfunction,convergencecontrolparameter,and
initialapproximations,arediscussedheuristically.Furthermore,advancedandless
frequentlyseenmethods,suchasmultiplehomotopiesandnonlinearauxiliaryoper-
ations,arediscussed.Asmentionedabove,thereareverymanyapplicationsofthe
homotopy analysis method in the literature. In selecting applications and specific
problems to work through, we have restricted our attention to the fluid phenom-
ena of fluid flow and heat transfer as such problems introduce a wide variety of
mathematicalproblemsyetallowforasufficientlynarrowfocus.Hence,inorderto
illustratevariouspropertiesandtoolsusefulwhenapplyingthehomotopyanalysis
method,wehaveselectedrecentresearchintheareaoffluidflowandheattransfer.
Weappreciatethesupportandmotivationofprof.A.C.-J.Luo.Wealsoacknowl-
edgetheroleofHigherEducationPress(China)andSpringerformakingthisbook
areality.K.V.Prasadhelpedintypingsomeoftheexamples.
Orlando,Florida KuppalapalleVajravelu
2012 RobertA.VanGorder
Contents
1 Introduction................................................... 1
References..................................................... 3
2 PrinciplesofHomotopyAnalysis................................. 7
2.1 Principlesofhomotopyandthehomotopyanalysismethod........ 7
2.2 Constructionofthedeformationequations...................... 9
2.3 Constructionoftheseriessolution ............................ 11
2.4 Conditionsfortheconvergenceoftheseriessolutions ............ 12
2.5 Existenceanduniquenessofsolutionsobtainedbyhomotopy
analysis................................................... 14
2.6 Relationsbetweenthe homotopyanalysismethodandother
analyticalmethods.......................................... 14
2.7 HomotopyanalysismethodfortheSwift-Hohenbergequation ..... 14
2.7.1 Applicationofthehomotopyanalysismethod ............ 16
2.7.2 Convergenceoftheseriessolutionanddiscussionof
results.............................................. 17
2.8 Incompressibleviscousconductingfluidapproachinga
permeablestretchingsurface ................................. 21
2.8.1 Exactsolutionsforsomespecialcases................... 23
2.8.2 ThecaseofG(cid:2)=0.................................... 25
2.8.3 ThecaseofG=0.................................... 29
2.8.4 Numericalsolutionsanddiscussionoftheresults.......... 32
2.9 Hydromagneticstagnationpointflowofasecondgradefluidover
astretchingsheet........................................... 34
2.9.1 Formulationofthemathematicalproblem................ 38
2.9.2 Exactsolutions ...................................... 38
2.9.3 Constructinganalyticalsolutionsviahomotopyanalysis.... 40
References..................................................... 46
x Contents
3 MethodsfortheControlofConvergenceinObtainedSolutions...... 53
3.1 Selectionofthe auxiliarylinear operatorandbase function
representation.............................................. 53
3.1.1 Methodoflinearpartitionmatching..................... 56
3.1.2 Methodofhighestorderdifferentialmatching ............ 57
3.1.3 Methodofcompletedifferentialmatching................ 58
3.1.4 Initialversusboundaryvalueproblems .................. 59
3.1.5 Additionaloptionsfortheselectionofanauxiliary
linearoperator....................................... 60
3.1.6 Remarksonthesolutionexpression..................... 60
3.2 Theroleoftheauxiliaryfunction ............................. 61
3.3 Selectionoftheconvergencecontrolparameter.................. 63
3.4 OptimalconvergencecontrolparametervalueandtheLane-Emden
equationofthefirstkind..................................... 65
3.4.1 Physicalbackground ................................. 65
3.4.2 AnalyticsolutionsviaTaylorseries ..................... 66
3.4.3 Analyticsolutionsviahomotopyanalysis ................ 69
References..................................................... 75
4 AdditionalTechniques .......................................... 77
4.1 Constructionofmultiplehomotopiesforcoupledequations ....... 77
4.2 Selectionofanauxiliarynonlinearoperator..................... 79
4.3 Validationoftheconvergencecontrolparameter................. 80
4.3.1 Convergencecontrolparameterplots(“h¯-plots”) .......... 80
4.3.2 Minimizedresidualerrors ............................. 80
4.3.3 Minimizedapproximateresidualerrors.................. 82
4.4 Multiplehomotopiesandtheconstructionofsolutionstothe
Fo¨ppl-vonKa´rma´nequationsgoverningdeflectionsofathin
flatplate .................................................. 82
4.4.1 Physicalbackground ................................. 82
4.4.2 Linearizationandconstructionofperturbationsolutions.... 84
4.4.3 Recursivesolutionsfortheclampededgeboundarydata.... 85
4.4.4 Specialcase:Thethinplatelimith→0,ν2→1........... 87
4.4.5 Controloferrorandselectionoftheconvergencecontrol
parameters.......................................... 88
4.4.6 Results............................................. 90
4.5 Nonlinearauxiliaryoperatorsandlocalsolutionstothe
Drinfel’d-Sokolovequations ................................. 91
4.6 RecentworkonadvancedtechniquesinHAM .................. 97
4.6.1 Mathematicalpropertiesof h¯-curveintheframeworkof
thehomotopyanalysismethod ......................... 97
4.6.2 Predictorhomotopyanalysismethodanditsapplicationto
somenonlinearproblems.............................. 98
4.6.3 Anoptimalhomotopy-analysisapproachforstrongly
nonlineardifferentialequations......................... 98
Contents xi
4.6.4 On the homotopymultiple-variablemethodandits
applicationsintheinteractionsofnonlineargravity
waves.............................................. 98
References..................................................... 99
5 ApplicationoftheHomotopyAnalysisMethodtoFluidFlow
Problems......................................................101
5.1 ThinfilmflowofaSiskofluidonamovingbelt .................102
5.1.1 Mathematicalanalysisoftheproblem ...................103
5.1.2 Applicationofthehomotopyanalysismethod ............106
5.1.3 Numericalresultsanddiscussion .......................108
5.2 Nanoboundarylayersoverstretchingsurfaces ..................112
5.2.1 Formulationoftheproblem............................113
5.2.2 Applicationofthehomotopyanalysismethod ............115
5.2.3 Analyticalsolutionsviathehomotopyanalysismethod ....116
5.2.4 Numericalsolutions..................................118
5.2.5 Discussionoftheresults ..............................121
5.3 Rotatingflowofathirdgradefluidbyhomotopyanalysis
method ...................................................123
5.3.1 Mathematicalformulation.............................124
5.3.2 Solutionoftheproblem...............................126
5.3.3 Resultsanddiscussions ...............................129
5.4 Homotopyanalysisforboundarylayerflowofamicropolarfluid
throughaporouschannel ....................................133
5.4.1 Descriptionoftheproblem ............................134
5.4.2 HAMsolutionsforvelocityandmicro-rotationfields ......136
5.4.3 Convergenceofthesolutions ..........................139
5.4.4 Resultsanddiscussion................................140
5.5 Solution of unsteady boundary-layerflow caused by an
impulsivelystretchingplate ..................................142
5.5.1 Mathematicaldescription .............................143
5.5.2 Homotopyanalyticsolution ...........................145
5.5.3 Resultsanddiscussion................................147
References.....................................................151
6 FurtherApplicationsoftheHomotopyAnalysisMethod............157
6.1 Seriessolutionsofanonlinearmodelofcombinedconvectiveand
radiativecoolingofaspherical ...............................157
6.1.1 Basicequations......................................157
6.1.2 SeriessolutionsgivenbytheHAM .....................159
6.1.3 Resultanalysis ......................................164
6.1.4 Conclusionsanddiscussions...........................169
6.2 Anill-posedproblemrelatedtotheflowofathinfluidfilmover
asheet....................................................171
6.2.1 Introductionandphysicalmotivation....................171
xii Contents
6.2.2 Formulationofthethree-parameterproblem..............173
6.2.3 Arelatedfour-parameterill-posedproblem...............175
6.2.4 Analyticalsolutionfor f(η)viathehomotopyanalysis
method.............................................178
6.2.5 Resultsanddiscussion................................181
References.....................................................184
SubjectIndex ......................................................185
AuthorIndex ......................................................187