Potential Flows of Viscous and Viscoelastic Liquids (Cambridge Aerospace Series, 21)

517 Pages·2007·10.98 MB·English

by  Daniel Joseph

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POTENTIALFLOWSOFVISCOUSANDVISCOELASTICFLUIDS
Thegoalofthisbookistoshowhowpotentialflowsenterintothegeneraltheory
ofmotionsofviscousandviscoelasticfluids.Traditionally,thetheoryofpotential
flowispresentedasasubjectcalled“potentialflowofaninviscidfluid”;whenthe
fluidisincompressible,thesefluidsare,curiously,saidtobe“perfect”or“ideal.”
This type of presentation is widespread; it can be foundin every bookand in all
universitycoursesonfluidmechanics,butitisdeeplyflawed.Itisnevernecessary
andtypicallynotusefultoputtheviscosityoffluidsinpotential(irrotational)flow
tozero.Thedimensionlessdescriptionofpotentialflowsoffluidswithanonzero
viscosity depends on the Reynolds number, and the theory of potential flow of
aninviscidfluidcanbesaidtoriseastheReynoldsnumbertendstoinfinity.The
theory given here can be described as the theory of potential flows at finite and
evensmallReynoldsnumbers.
DanielJosephisaProfessorofAerospaceEngineeringandMechanicsattheUni-
versity of Minnesota since 1963, where he has served as the Russel J. Penrose
ProfessorandRegent’sProfessor.HeispresentlyaDistinguishedAdjunctProfes-
sor of Aerospace and Mechanical Engineering at the University of California,
Irvine, and an Honorary Professor at Xi’an Jiaotong University in China. He
hasauthored10patents,400journalarticles,andsixbooks.HeisaGuggenheim
Fellow;amemberoftheNationalAcademyofEngineering,theNationalAcademy
ofSciences;andtheAmericanAcademyofArtsandSciences,G.I.TaylorMedalist,
SocietyofEngineeringScience;aFellowoftheAmericanPhysicalSociety;winner
oftheTimoshenkoMedaloftheASME,theSchlumbergerFoundationAward,the
BinghamMedaloftheSocietyofRheology,andtheFluidDynamicsPrizeofthe
APS.HeislistedinThompsonScientific-ISI’sHighlyCitedResearchersTM.
Toshio Funada is a Professor of Digital Engineering at the Numazu College of
Technology in Japan and has served as both Dean and Department Head. He
receivedhisdoctorateatOsakaUniversityundertheguidanceofProfKakatuni.
Heisanexpertinthetheoryofstability,bifurcation,anddynamicalsystems.He
hasworkedonthepotentialflowsofviscousfluidswithProfessorJosephfornearly
tenyears.
JingWangisaPostDoctoralFellowattheUniversityofMinnesota.Hereceived
the “Best Dissertation Award” in Physical Sciences and Engineering for 2006 at
theUniversityofMinnesota.
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CAMBRIDGEAEROSPACESERIES
Editors
WeiShyyandMichaelJ.Rycroft
1. J.M.RolfeandK.J.Staples(eds.):FlightSimulation
2. P.Berlin:TheGeostationaryApplicationsSatellite
3. M.J.T.Smith:AircraftNoise
4. N.X.Vinh:FlightMechanicsofHigh-PerformanceAircraft
5. W.A.MairandD.L.Birdsall:AircraftPerformance
6. M.J.AbzugandE.E.Larrabee:AirplaneStabilityandControl
7. M.J.Sidi:SpacecraftDynamicsandControl
8. J.D.Anderson:AHistoryofAerodynamics
9. A.M.Cruise,J.A.Bowles,C.V.Goodall,andT.J.Patrick:Principles
ofSpaceInstrumentDesign
10. G.A.KhouryandJ.D.Gillett(eds.):AirshipTechnology
11. J.Fielding:IntroductiontoAircraftDesign
12. J.G.Leishman:PrinciplesofHelicopterAerodynamics,2ndEdition
13. J.KatzandA.Plotkin:LowSpeedAerodynamics,2ndEdition
14. M.J.AbzugandE.E.Larrabee:AirplaneStabilityandControl:AHistory
oftheTechnologiesthatMadeAviationPossible,2ndEdition
15. D.H.HodgesandG.A.Pierce:IntroductiontoStructuralDynamics
andAeroelasticity
16. W.Fehse:AutomaticRendezvousandDockingofSpacecraft
17. R.D.Flack:FundamentalsofJetPropulsionwithApplications
18. E.A.Baskharone:PrinciplesofTurbomachineryinAir-BreathingEngines
19. DoyleD.Knight:ElementsofNumericalMethodsforHigh-SpeedFlows
20. C.Wagner,T.Huettl,andP.Sagaut:Large-EddySimulationforAcoustics
21. D.Joseph,T.Funada,andJ.Wang:PotentialFlowsofViscous
andViscoelasticFluids
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POTENTIAL FLOWS OF VISCOUS
AND VISCOELASTIC FLUIDS
Daniel Joseph
UniversityofMinnesota
Toshio Funada
NumazuCollegeofTechnology
Jing Wang
UniversityofMinnesota
v
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521873376
© Daniel Joseph, Toshio Funada, Jing Wang 2008
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2008
ISBN-13 978-0-511-50805-9 eBook (NetLibrary)
ISBN-13 978-0-521-87337-6 hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
CUFX112-Josephetal October31,2007 10:28
Contents
Preface page xv
List of Abbreviations xvii
1 Introduction 1
1.1 Irrotational flow, Laplace’s equation 2
1.2 Continuity equation, incompressible fluids, isochoric flow 3
1.3 Euler’s equations 3
1.4 Generation of vorticity in fluids governed by Euler’s equations 4
1.5 Perfect fluids, irrotational flow 4
1.6 Boundary conditions for irrotational flow 5
1.7 Streaming irrotational flow over a stationary sphere 6
2 Historical notes 8
2.1 Navier–Stokes equations 8
2.2 Stokes theory of potential flow of viscous fluid 9
2.3 The dissipation method 10
2.4 The distance a wave will travel before it decays by a certain amount 11
3 Boundary conditions for viscous fluids 13
4 Helmholtz decomposition coupling rotational to irrotational flow 16
4.1 Helmholtz decomposition 16
4.2 Navier–Stokes equations for the decomposition 17
4.3 Self-equilibration of the irrotational viscous stress 19
4.4 Dissipation function for the decomposed motion 20
4.5 Irrotational flow and boundary conditions 20
4.6 Examples from hydrodynamics 21
4.6.1 Poiseuille flow 21
4.6.2 Flow between rotating cylinders 21
4.6.3 Stokes flow around a sphere of radius a in a uniform stream U 22
4.6.4 Streaming motion past an ellipsoid 23
4.6.5 Hadamard–Rybyshinsky solution for streaming flow past a liquid
sphere 23
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viii Contents
4.6.6 Axisymmetric steady flow around a spherical gas bubble at finite
Reynolds numbers 24
4.6.7 Viscous decay of free-gravity waves 24
4.6.8 Oseen flow 25
4.6.9 Flows near internal stagnation points in viscous incompressible
fluids 26
4.6.10 Hiemenz boundary-layer solution for two-dimensional flow
toward a “stagnation point” at a rigid boundary 29
4.6.11 Jeffrey–Hamel flow in diverging and converging channels 31
4.6.12 An irrotational Stokes flow 32
4.6.13 Lighthill’s approach 32
4.7 Conclusion 33
5 Harmonic functions that give rise to vorticity 35
6 Radial motions of a spherical gas bubble in a viscous liquid 39
7 Rise velocity of a spherical cap bubble 42
7.1 Analysis 42
7.2 Experiments 46
7.3 Conclusions 50
8 Ellipsoidal model of the rise of a Taylor bubble in a round tube 51
8.1 Introduction 51
8.1.1 Unexplained and paradoxical features 52
8.1.2 Drainage 53
8.1.3 Brown’s analysis of drainage 54
8.1.4 Viscous potential flow 55
8.2 Ellipsoidal bubbles 56
8.2.1 Ovary ellipsoid 56
8.2.2 Planetary ellipsoid 60
8.2.3 Dimensionless rise velocity 61
8.3 Comparison of theory and experiment 63
8.4 Comparison of theory and correlations 66
8.5 Conclusion 68
9 Rayleigh–Taylor instability of viscous fluids 70
9.1 Acceleration 71
9.2 Simple thought experiments 71
9.3 Analysis 71
9.3.1 Linear theory of Chandrasekhar 73
9.3.2 Viscous potential flow 74
9.4 Comparison of theory and experiments 76
9.5 Comparison of the stability theory with the experiments on drop breakup 76
9.6 Comparison of the measured wavelength of corrugations on the drop surface
with the prediction of the stability theory 81