potential flows of viscous and viscoelastic fluids

CUFX112-Joseph et al October 31, 2007 10:28

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POTENTIAL FLOWS OF VISCOUS AND VISCOELASTIC FLUIDS

The goal of this book is to show how potential flows enter into the general theoryof motions of viscous and viscoelastic fluids. Traditionally, the theory of potentialflow is presented as a subject called potential flow of an inviscid fluid; when thefluid is incompressible, these fluids are, curiously, said to be perfect or ideal.This type of presentation is widespread; it can be found in every book and in alluniversity courses on fluid mechanics, but it is deeply flawed. It is never necessaryand typically not useful to put the viscosity of fluids in potential (irrotational) flowto zero. The dimensionless description of potential flows of fluids with a nonzeroviscosity depends on the Reynolds number, and the theory of potential flow ofan inviscid fluid can be said to rise as the Reynolds number tends to infinity. Thetheory given here can be described as the theory of potential flows at finite andeven small Reynolds numbers.

Daniel Joseph is a Professor of Aerospace Engineering and Mechanics at the Uni-versity of Minnesota since 1963, where he has served as the Russel J. PenroseProfessor and Regents Professor. He is presently a Distinguished Adjunct Profes-sor of Aerospace and Mechanical Engineering at the University of California,Irvine, and an Honorary Professor at Xian Jiaotong University in China. Hehas authored 10 patents, 400 journal articles, and six books. He is a GuggenheimFellow; amember of theNational Academy of Engineering, theNational Academyof Sciences; and theAmericanAcademy ofArts and Sciences,G. I. TaylorMedalist,Society of Engineering Science; a Fellow of the American Physical Society; winnerof the TimoshenkoMedal of the ASME, the Schlumberger Foundation Award, theBingham Medal of the Society of Rheology, and the Fluid Dynamics Prize of theAPS. He is listed in Thompson Scientific-ISIs Highly Cited ResearchersTM.

Toshio Funada is a Professor of Digital Engineering at the Numazu College ofTechnology in Japan and has served as both Dean and Department Head. Hereceived his doctorate at Osaka University under the guidance of Prof Kakatuni.He is an expert in the theory of stability, bifurcation, and dynamical systems. Hehas worked on the potential flows of viscous fluids with Professor Joseph for nearlyten years.

Jing Wang is a Post Doctoral Fellow at the University of Minnesota. He receivedthe Best Dissertation Award in Physical Sciences and Engineering for 2006 atthe University of Minnesota.

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CAMBRIDGE AEROSPACE SERIES

EditorsWei Shyy and Michael J. Rycroft

1. J. M. Rolfe and K. J. Staples (eds.): Flight Simulation

2. P. Berlin: The Geostationary Applications Satellite

3. M. J. T. Smith: Aircraft Noise

4. N. X. Vinh: Flight Mechanics of High-Performance Aircraft

5. W. A. Mair and D. L. Birdsall: Aircraft Performance

6. M. J. Abzug and E. E. Larrabee: Airplane Stability and Control

7. M. J. Sidi: Spacecraft Dynamics and Control

8. J. D. Anderson: A History of Aerodynamics

9. A. M. Cruise, J. A. Bowles, C. V. Goodall, and T. J. Patrick: Principlesof Space Instrument Design

10. G. A. Khoury and J. D. Gillett (eds.): Airship Technology

11. J. Fielding: Introduction to Aircraft Design

12. J. G. Leishman: Principles of Helicopter Aerodynamics, 2nd Edition

13. J. Katz and A. Plotkin: Low Speed Aerodynamics, 2nd Edition

14. M. J. Abzug and E. E. Larrabee: Airplane Stability and Control: A Historyof the Technologies that Made Aviation Possible, 2nd Edition

15. D. H. Hodges and G. A. Pierce: Introduction to Structural Dynamicsand Aeroelasticity

16. W. Fehse: Automatic Rendezvous and Docking of Spacecraft

17. R. D. Flack: Fundamentals of Jet Propulsion with Applications

18. E. A. Baskharone: Principles of Turbomachinery in Air-Breathing Engines

19. Doyle D. Knight: Elements of Numerical Methods for High-Speed Flows

20. C. Wagner, T. Huettl, and P. Sagaut: Large-Eddy Simulation for Acoustics

21. D. Joseph, T. Funada, and J. Wang: Potential Flows of Viscousand Viscoelastic Fluids

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POTENTIAL FLOWS OF VISCOUSAND VISCOELASTIC FLUIDS

Daniel Joseph

University of Minnesota

Toshio Funada

Numazu College of Technology

Jing Wang

University of Minnesota

v

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-87337-6

ISBN-13 978-0-511-50805-9

Daniel Joseph, Toshio Funada, Jing Wang 2008

2008

Information on this title: www.cambridge.org/9780521873376

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.

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.

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

eBook (NetLibrary)

hardback


Contents

Preface page xv

List of Abbreviations xvii

1 Introduction 1

1.1 Irrotational flow, Laplaces equation 21.2 Continuity equation, incompressible fluids, isochoric flow 31.3 Eulers equations 31.4 Generation of vorticity in fluids governed by Eulers equations 41.5 Perfect fluids, irrotational flow 41.6 Boundary conditions for irrotational flow 51.7 Streaming irrotational flow over a stationary sphere 6

2 Historical notes 8

2.1 NavierStokes equations 82.2 Stokes theory of potential flow of viscous fluid 92.3 The dissipation method 102.4 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 164.2 NavierStokes equations for the decomposition 174.3 Self-equilibration of the irrotational viscous stress 194.4 Dissipation function for the decomposed motion 204.5 Irrotational flow and boundary conditions 204.6 Examples from hydrodynamics 21

4.6.1 Poiseuille flow 214.6.2 Flow between rotating cylinders 214.6.3 Stokes flow around a sphere of radius a in a uniform stream U 224.6.4 Streaming motion past an ellipsoid 234.6.5 HadamardRybyshinsky solution for streaming flow past a liquid

sphere 23

vii


viii Contents

4.6.6 Axisymmetric steady flow around a spherical gas bubble at finiteReynolds numbers 24

4.6.7 Viscous decay of free-gravity waves 244.6.8 Oseen flow 254.6.9 Flows near internal stagnation points in viscous incompressible

fluids 264.6.10 Hiemenz boundary-layer solution for two-dimensional flow

toward a stagnation point at a rigid boundary 294.6.11 JeffreyHamel flow in diverging and converging channels 314.6.12 An irrotational Stokes flow 324.6.13 Lighthills 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 427.2 Experiments 467.3 Conclusions 50

8 Ellipsoidal model of the rise of a Taylor bubble in a round tube 51

8.1 Introduction 518.1.1 Unexplained and paradoxical features 528.1.2 Drainage 538.1.3 Browns analysis of drainage 548.1.4 Viscous potential flow 55

8.2 Ellipsoidal bubbles 568.2.1 Ovary ellipsoid 568.2.2 Planetary ellipsoid 608.2.3 Dimensionless rise velocity 61

8.3 Comparison of theory and experiment 638.4 Comparison of theory and correlations 668.5 Conclusion 68

9 RayleighTaylor instability of viscous fluids 70

9.1 Acceleration 719.2 Simple thought experiments 719.3 Analysis 71

9.3.1 Linear theory of Chandrasekhar 739.3.2 Viscous potential flow 74

9.4 Comparison of theory and experiments 769.5 Comparison of the stability theory with the experiments on drop breakup 769.6 Comparison of the measured wavelength of corrugations on the drop surface

with the prediction of the stability theory 81


Contents ix

9.7 Fragmentation of Newtonian and viscoelastic drops 849.8 Modeling RayleighTaylor instability of a sedimenting suspension of

several thousand circular particles in a direct numerical simulation 89

10 The force on a cylinder near a wall in viscous potential flows 90

10.1 The flow that is due to the circulation about the cylinder 9010.2 The streaming flow past the cylinder near a wall 9310.3 The streaming flow past a cylinder with circulation near a wall 95

11 KelvinHelmholtz instability 100

11.1 KH instability on an unbounded domain 10011.2 Maximum growth rate, Hadamard instability, neutral curves 102

11.2.1 Maximum growth rate 10211.2.2 Hadamard instability 10211.2.3 The regularization of Hadamard instability 10211.2.4 Neutral curves 103

11.3 KH instability in a channel 10311.3.1 Formulation of the problem 10411.3.2 Viscous potential flow analysis 10511.3.3 KH instability of inviscid fluid 10911.3.4 Dimensionless form of the dispersion equation 11011.3.5 The effect of liquid viscosity and surface tension on growth rates

and neutral curves 11211.3.6 Comparison of theory and experiments in rectangular ducts 11411.3.7 Critical viscosity and density ratios 11811.3.8 Further comparisons with previous results 11911.3.9 Nonlinear effects 12111.3.10 Combinations of RayleighTaylor and KelvinHelmholtz

instabilities 123

12 Energy equation for irrotational theories of gasliquid flow: viscous potentialflow, viscous potential flow with pressure correction, and dissipation method 126

12.1 Viscous potential flow 12612.2 Dissipation method according to Lamb 12612.3 Drag on a spherical gas bubble calculated from the viscous dissipation

of an irrotational flow 12712.4 The idea of a pressure correction 12712.5 Energy equation for irrotational flow of a viscous fluid 12812.6 Viscous correction of viscous potential flow 13012.7 Direct derivation of the viscous correction of the normal stress balance

for the viscous decay of capillary-gravity waves 132

13 Rising bubbles 134

13.1 The dissipation approximation and viscous potential flow 13413.1.1 Pressure correction formulas 134

13.2 Rising spherical gas bubble 135


x Contents

13.3 Rising oblate ellipsoidal bubble 13613.4 A liquid drop rising in another liquid 13713.5 Purely irrotational analysis of a toroidal bubble in a viscous fluid 139

13.5.1 Prior work, experiments 13913.5.2 The energy equation 14113.5.3 The impulse equation 14513.5.4 Comparison of irrotational solutions for inviscid and viscous

fluids 14513.5.5 Stability of the toroidal vortex 14813.5.6 Boundary-integral study of vortex ring bubbles in a viscous

liquid 15213.5.7 Irrotational motion of a massless cylinder under the combined

action of KuttaJoukowski lift, acceleration of added mass, andviscous drag 153

13.6 The motion of a spherical gas bubble in viscous potential flow 15513.7 Steady motion of a deforming gas bubble in a viscous potential flow 15713.8 Dynamic simulations of the rise of many bubbles in a viscous potential

flow 157

14 Purely irrotational theories of the effect of viscosity on the decay of waves 159

14.1 Decay of free-gravity waves 15914.1.1 Introduction 15914.1.2 Irrotational viscous corrections for the potential flow solution 16014.1.3 Relation between the pressure correction and Lambs exact

solution 16214.1.4 Comparison of the decay rate and the wave velocity given by the

exact solution, VPF, and VCVPF 16314.1.5 Why does the exact solution agree with VCVPF when k < kc and

with VPF when k > kc ? 16614.1.6 Conclusion and discussion 16814.1.7 Quasi-potential approximation vorticity layers 169

14.2 Viscous decay of capillary waves on drops and bubbles 17014.2.1 Introduction 17114.2.2 VPF analysis of a single spherical drop immersed in another fluid 17214.2.3 VCVPF analysis of a single spherical drop immersed in another

fluid 17614.2.4 Dissipation approximation (DM) 18014.2.5 Exact solution of the linearized free-surface problem 18114.2.6 VPF and VCVPF analyses for waves acting on a plane interface

considering surface tension comparison with Lambs solution 18314.2.7 Results and discussion 18514.2.8 Concluding remarks 192

14.3 Irrotational dissipation of capillary-gravity waves 19314.3.1 Correction of the wave frequency assumed by Lamb 19314.3.2 Irrotational dissipation of nonlinear capillary-gravity waves 195


Contents xi

15 Irrotational Faraday waves on a viscous fluid 197

15.1 Introduction 19815.2 Energy equation 19915.3 VPF and VCVPF 200

15.3.1 Potential flow 20015.3.2 Amplitude equations for the elevation of the free surface 201

15.4 Dissipation method 20415.5 Stability analysis 20415.6 RayleighTaylor instability and Faraday waves 20615.7 Comparison of purely irrotational solutions with exact solutions 21015.8 Bifurcation of Faraday waves in a nearly square container 21315.9 Conclusion 213

16 Stability of a liquid jet into incompressible gases and liquids 215

16.1 Capillary instability of a liquid cylinder in another fluid 21516.1.1 Introduction 21516.1.2 Linearized equations governing capillary instability 21716.1.3 Fully viscous flow analysis 21816.1.4 Viscous potential flow analysis 21816.1.5 Pressure correction for viscous potential flow 21916.1.6 Comparison of growth rates 22216.1.7 Dissipation calculation for capillary instability 23016.1.8 Discussion of the pressure corrections at the interface of two

viscous fluids 23216.1.9 Capillary instability when one fluid is a dynamically inactive gas 23416.1.10 Conclusions 237

16.2 Stability of a liquid jet into incompressible gases: Temporal, convective,and absolute instabilities 23816.2.1 Introduction 23916.2.2 Problem formulation 24016.2.3 Dispersion relation 24116.2.4 Temporal instability 24316.2.5 Numerical results of temporal instability 25016.2.6 Spatial, absolute, and convective instability 25116.2.7 Algebraic equations at a singular point 25516.2.8 Subcritical, critical, and supercritical singular points 25616.2.9 Inviscid jet in inviscid fluid (Re , m = 0) 26116.2.10 Exact solution; comparison with previous results 26216.2.11 Summary and discussion 266

16.3 Viscous potential flow of the KelvinHelmholtz instability of a cylindricaljet of one fluid into the same fluid 26716.3.1 Mathematical formulation 26716.3.2 Normal modes; dispersion relation 26816.3.3 Growth rates and frequencies 26916.3.4 Hadamard instabilities for piecewise discontinuous profiles 269


xii Contents

17 Stress-induced cavitation 272

17.1 Theory of stress-induced cavitation 27317.1.1 Mathematical formulation 27317.1.2 Cavitation threshold 275

17.2 Viscous potential flow analysis of stress-induced cavitation in anaperture flow 27817.2.1 Analysis of stress-induced cavitation 27917.2.2 Stream function, potential function, and velocity 28117.2.3 Cavitation threshold 28217.2.4 Conclusions 28617.2.5 NavierStokes simulation 287

17.3 Streaming motion past a sphere 28717.3.1 Irrotational flow of a viscous fluid 29017.3.2 An analysis for maximum K 293

17.4 Symmetric model of capillary collapse and rupture 29717.4.1 Introduction 29717.4.2 Analysis 29917.4.3 Conclusions and discussion 30417.4.4 Appendix 308

18 Viscous effects of the irrotational flow outside boundary layers on rigid solids 310

18.1 Extra drag due to viscous dissipation of the irrotational flow outsidethe boundary layer 31118.1.1 Pressure corrections for the drag on a circular gas bubble 31218.1.2 A rotating cylinder in a uniform stream 31518.1.3 The additional drag on an airfoil by the dissipation method 32418.1.4 Discussion and conclusion 327

18.2 Glauerts solution of the boundary layer on a rapidly rotating cylinder in auniform stream revisited 32918.2.1 Introduction 33018.2.2 Unapproximated governing equations 33418.2.3 Boundary-layer approximation and Glauerts equations 33418.2.4 Decomposition of the velocity and pressure field 33518.2.5 Solution of the boundary-layer flow 33618.2.6 Higher-order boundary-layer theory 34718.2.7 Discussion and conclusion 350

18.3 Numerical study of the steady-state uniform flow past a rotating cylinder 35218.3.1 Introduction 35318.3.2 Numerical features 35518.3.3 Results and discussion 35918.3.4 Concluding remarks 372

19 Irrotational flows that satisfy the compressible NavierStokes equations 374

19.1 Acoustics 37519.2 Spherically symmetric waves 377


Contents xiii

19.3 Liquid jet in a high-Mach-number airstream 37819.3.1 Introduction 37819.3.2 Basic partial differential equations 37919.3.3 Cylindrical liquid jet in a compressible gas 38019.3.4 Basic isentropic relations 38019.3.5 Linear stability of the cylindrical liquid jet in a compressible gas;

dispersion equation 38119.3.6 Stability problem in dimensionless form 38319.3.7 Inviscid potential flow 38619.3.8 Growth-rate parameters as functions of M for different

viscosities 38619.3.9 Azimuthal periodicity of the most dangerous disturbance 38719.3.10 Variation of the growth-rate parameters with the Weber number 38819.3.11 Convective/absolute instability 38919.3.12 Conclusions 393

20 Irrotational flows of viscoelastic fluids 395

20.1 Oldroyd B model 39520.2 Asymptotic form of the constitutive equations 396

20.2.1 Retarded motion expansion for the UCM model 39620.2.2 The expanded UCM model in potential flow 39720.2.3 Potential flow past a sphere calculated with the expanded

UCM model 39720.3 Second-order fluids 39820.4 Purely irrotational flows 40020.5 Purely irrotational flows of a second-order fluid 40020.6 Reversal of the sign of the normal stress at a point of stagnation 40120.7 Fluid forces near stagnation points on solid bodies 402

20.7.1 Turning couples on long bodies 40220.7.2 Particleparticle interactions 40220.7.3 Spherewall interactions 40320.7.4 Flow-induced microstructure 404

20.8 Potential flow over a sphere for a second-order fluid 40620.9 Potential flow over an ellipse 408

20.9.1 Normal stress at the surface of the ellipse 40920.9.2 The effects of the Reynolds number 41020.9.3 The effects of 1/( a 2 ) 41220.9.4 The effects of the aspect ratio 412

20.10 The moment on the ellipse 41320.11 The reversal of the sign of the normal stress at stagnation points 41420.12 Flow past a flat plate 41620.13 Flow past a circular cylinder with circulation 41620.14 Potential flow of a second-order fluid over a triaxial ellipsoid 41720.15 Motion of a sphere normal to a wall in a second-order fluid 418

20.15.1 Low Reynolds numbers 419


xiv Contents

20.15.2 Viscoelastic Potential Flow 42220.15.3 Conclusions 425

21 Purely irrotational theories of stability of viscoelastic fluids 426

21.1 RayleighTaylor instability of viscoelastic drops at high Weber numbers 42621.1.1 Introduction 42621.1.2 Experiments 42721.1.3 Theory 42821.1.4 Comparison of theory and experiment 437

21.2 Purely irrotational theories of the effects of viscosity and viscoelasticityon capillary instability of a liquid cylinder 44321.2.1 Introduction 44321.2.2 Linear stability equations and the exact solution 44421.2.3 Viscoelastic potential flow 44621.2.4 Dissipation and the formulation for the additional pressure

contribution 44721.2.5 The additional pressure contribution for capillary instability 44821.2.6 Comparison of the growth rate 44921.2.7 Comparison of the stream functions 45121.2.8 Discussion 456

21.3 Steady motion of a deforming gas bubble in a viscous potential flow 460

22 Numerical methods for irrotational flows of viscous fluid 461

22.1 Perturbation methods 46122.2 Boundary-integral methods for inviscid potential flow 46222.3 Boundary-integral methods for viscous potential flow 464

Appendix A. Equations of motion and strain rates for rotational and irrotationalflow in Cartesian, cylindrical, and spherical coordinates 465

Appendix B. List of frequently used symbols and concepts 471

References 473

Index 487


Preface

Potential flows of incompressible fluids with constant properties are irrotational solutionsof the NavierStokes equations that satisfy Laplaces equation. How do these solutionsenter into the general problem of viscous fluid mechanics? Under certain conditions,the Helmholtz decomposition says that solutions of the NavierStokes equations can bedecomposed into a rotational part and an irrotational part satisfying Laplaces equation.The irrotational part is required for satisfying the boundary conditions; in general, theboundary conditions cannot be satisfied by the rotational velocity, and they cannot besatisfied by the irrotational velocity; the rotational and irrotational velocities are bothrequired and they are tightly coupled at the boundary. For example, the no-slip conditionfor Stokes flow over a sphere cannot be satisfied by the rotational velocity; harmonicfunctions that satisfy Laplaces equation subject to a Robin boundary condition in whichthe irrotational normal and tangential velocities enter in equal proportions are required.

The literature that focuses on the computation of layers of vorticity in flows that areelsewhere irrotational describes boundary-layer solutions in the Helmholtz decomposedforms. These kinds of solutions require small viscosity and, in the case of gasliquid flows,are said to give rise to weak viscous damping. It is true that viscous effects arising fromthese layers are weak, but the main effects of viscosity in so many of these flows are purelyirrotational, and they are not weak.

The theory of purely irrotational flows of a viscous fluid is an approximate theorythat works well especially in gasliquid flows of liquids of high viscosity at low Reynoldsnumbers. The theory of purely irrotational flows of a viscous fluid can be seen as a verysuccessful competitor to the theory of purely irrotational flows of an inviscid fluid. Wehave come to regard every solution of free-surface problems in an inviscid liquid asan opportunity for a new study. There are hundreds of such opportunities that are stillavailable.

The theory of irrotational flows of viscous and viscoelastic liquids that is developed hereis embedded in a variety of fluid mechanics problems ranging from cavitation, capillarybreakup and rupture, RayleighTaylor and KelvinHelmholtz instabilities, irrotationalFaraday waves on a viscous fluid, flow-induced structure of particles in viscous and vis-coelastic fluids, boundary-layer theory for flow over rigid solids, rising bubbles, and othertopics. The theory of stability of free-surface problems developed here is a great improve-ment of what was available previously and could be used as supplemental text in courseson hydrodynamic stability.

xv


xvi Preface

We have tried to assemble here all the literature bearing on the irrotational flow ofviscous liquids. For sure, it is not a large literature, but it is likely that despite an honesteffort we missed some good works.

Wearehappy to acknowledge the contributions of personswhohavehelpedus.TerrenceLiaomadevery important contributions toour earlyworkon this subject in the early 1990s.More recently, Juan Carlos Padrino joined our group and has made truly outstandingcontributions to problems described here. In a sense, Juan Carlos could be consideredto be an author of this book and we are lucky that he came along. We are indebted toG. I. Barenblatt and to K. R. Sreenivasan for their support and help in promoting viscouspotential flow as a topic at the foundation of fluid mechanics. The National ScienceFoundation has supported our work from the beginning.

We worked day and night on this research; Funada in his day and our night and Josephand Wang in their day and his night. The whole effort was a great pleasure.


List of Abbreviations

2D two-dimensional3D three-demensionalBEM boundary-element methodBU Benjamin and UnsellC/A convectiveabsolutec.c. complex conjugateDM dissipation methodES exact solutionFHS fully hydrodynamic systemFVF fully viscous flowIPF inviscid potential flowJBB Joseph, Belanger, and BeaversJBF Joseph, Beavers, and FunadaKH KelvinHelmholtzKT Kumar and TuckermanLHC Longuet-Higgins and CokeletMVK Miksis, Vanden-Broeck, and KellerODE ordinary differential equationPAA polyacrylamidePDE partial differential equationPISO pressure implicit with splitting of operatorsPNSCC principal normal stress cavitation criterionPO or PEO polyox or polyethylene oxideQUICK quadratic upwind interpolation for convective kinematics (scheme)RT RayleighTaylorTVF Taylor vortex flowVCVPF viscous correction of VPFVPF viscous potential flow

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CUFX112-Joseph et al November 15, 2007 20:25

1

Introduction

The theory of potential flow is a topic in both the study of fluid mechanics and in mathe-matics. The mathematical theory treats properties of vector fields generated by gradientsof a potential. The curl of a gradient vanishes. The local rotation of a vector field is pro-portional to its curl so that potential flows do not rotate as they deform. Potential flowsare irrotational.

The mathematical theory of potentials goes back to the 18th century (see Kellogg,1929). This elegant theory has given rise to jewels of mathematical analysis, such as thetheory of a complex variable. It is a well-formed or mature theory, meaning that the bestresearch results have already been obtained. We are not going to add to the mathematicaltheory; our contributions are to the fluid mechanics theory, focusing on effects of viscosityand viscoelasticity. Two centuries of research have focused exclusively on the motions ofinviscid fluids.Among the 131,000,000 hits that comeup under potential flows onGooglesearch are mathematical studies of potential functions and studies of inviscid fluids. Thesestudies can be extended to viscous fluids at small cost and great profit.

The fluid mechanics theory of potential flow goes back to Euler in 1761 (see Truesdell,1954, 36). The concept of viscosity was not known in Eulers time. The fluids he studiedwere driven by pressures, not by viscous stresses. The effects of viscous stresses wereintroduced by Navier (1822) and Stokes (1845). Stokes (1851) considered potential flowof a viscous fluid in an approximate sense, but most later authors restrict their attentionto potential flow of an inviscid fluid. All the books on fluid mechanics and all coursesin fluid mechanics have chapters on potential flow of inviscid fluids and none on thepotential flow of a viscous fluid.

An authoritative and readable exposition of irrotational flow theory and its applicationscan be found in chapter 6 of the book on fluid dynamics by Batchelor (1967). He speaksof the role of the theory of flow of an inviscid fluid:

In this and the following chapter, various aspects of the flow of a fluid regarded as entirelyinviscid (and incompressible) will be considered. The results presented are significant onlyinasmuch as they represent an approximation to the flow of a real fluid at large Reynoldsnumber, and the limitations of each result must be regarded as important as the result itself.

In this book we consider irrotational flows of a viscous fluid. We are of the opinion thatwhen one is considering irrotational solutions of the NavierStokes equations it is nevernecessary and typically not useful for one to put the viscosity to zero. This observationruns counter to the idea frequently expressed that potential flow is a topic that is useful

1


2 Introduction

only for inviscid fluids; many people think that the notion of a viscous potential flow isan oxymoron. Incorrect statements like . . . irrotational flow implies inviscid flow but notthe other way around can be found in popular textbooks.

Irrotational flows of a viscous fluid scale with the Reynolds number as do rotationalsolutions of the NavierStokes equations generally. The solutions of the NavierStokesequations, rotational and irrotational, are thought to become independent of theReynoldsnumber at large Reynolds numbers. Unlike the theory of irrotational flows of inviscidfluids, the theory of irrotational flow of a viscous fluid can be considered as a descriptionof flow at a finite Reynolds number.

Most of the classical theorems reviewed in chapter 6 of Batchelors 1967 book do notrequire that the fluid be inviscid. These theorems are as true for viscous potential flowas they are for inviscid potential flow. Kelvins minimum-energy theorem holds for theirrotational flow of a viscous fluid. The theory of the acceleration reaction leads to theconcept of addedmass; it follows from the analysis of unsteady irrotational flow.The theoryapplies to viscous and inviscid fluids alike.

It can be said that every theorem about potential flow of inviscid incompressible fluidsapplies equally to viscous fluids in regions of irrotational flow. Jeffreys (1928) derived anequation [his (20)] that replaces the circulation theorem of classical (inviscid) hydrody-namics. When the fluid is homogeneous, Jeffreys equation may be written as

dCdt

=

curl dl, (1.0.1)

where

= curl u, C (t) =

u dl,

is the circulation around a closed material curve drawn in the fluid. This equation showsthat

. . . the initial value of dC/dt around a contour in a fluid originally moving irrotationally iszero, whether or not there is a moving solid within the contour. This at once provides anexplanation of the equality of the circulation about an aeroplane and that about the vortexleft behind when it starts; for the circulation about a large contour that has never been cutby the moving solid or its wake remains zero, and therefore the circulations about contoursobtained by subdividing it must also add up to zero. It also indicates why the motion is ingeneral nearly irrotational except close to a solid of to fluid that has pass near one.

Saint-Venant (1869) interpreted the result of Lagrange (1781) about the invariance ofcirculation dC/dt = 0 to mean that

vorticity cannot be generated in the interior of a viscous incompressible fluid, subject toconservative extraneous force, but is necessarily diffused inward from the boundaries.

Circulation formula (1.0.1) is an important result in the theory of irrotational flows ofa viscous fluid. A particle that is initially irrotational will remain irrotational in motionsthat do not enter into the vortical layers at the boundary.

1.1 Irrotational flow, Laplaces equation

A potential flow is a velocity field u () given by the gradient of a potential :

u () = . (1.1.1)


1.3 Eulers equations 3

Potential flows have a zero curl:

curl u = curl = 0. (1.1.2)Fields that are curl free, satisfying (1.1.2), are called irrotational.

Vector fields satisfying the equation

divu = 0 (1.1.3)are said to be solenoidal. Solenoidal flows that are irrotational are harmonic; the potentialsatisfies Laplaces equation:

divu () = div = 2 = 0. (1.1.4)The theory of irrotational flow needed in this book is given in many books; for example,

Lamb (1932), Milne-Thomson (1968), Batchelor (1967), and Landau and Lifshitz (1987).No-slip cannot be enforced in irrotational flow. However, eliminating all the irrotationaleffects of viscosity by putting = 0 to reconcile our desire to satisfy no-slip at the cost ofreal physics is like throwing out the baby with the bathwater. This said, we can rely on thebook by Batchelor and others for the results we need in our study of irrotational flow ofviscous fluids.

1.2 Continuity equation, incompressible fluids, isochoric flow

The equation governing the evolution of the density ,

ddt

+ divu = 0,(1.2.1)

ddt

= t

+ (u ) ,

is called the continuity equation. It guarantees that themass of a fluid element is conserved.If the fluid is incompressible, then is constant and div u = 0. Flows of compressible flu-

ids for which div u = 0 are called isochoric. Low-Mach-number flows are nearly isochoric.Incompressible and isochoric flows are solenoidal.

1.3 Eulers equations

Eulers equations of motion are given by

dudt

= p+ g,(1.3.1)

dudt

= ut

+ (u ) u,

where g is a body force per unit mass. If is constant and g = G has a force potential,then

p+ g = p, (1.3.2)where p = p G can be called the pressure head. If g is gravity, then

G = g x.


4 Introduction

To simplify the writing of equations we put the body force to zero except in casesfor which it is important. If the fluid is compressible, then an additional relation, fromthermodynamics, relating p to is required. Such a relation can be found for isentropicflow or isothermal flow of a perfect gas. In such a system there are five unknowns, p, ,and u, and five equations.

The effects of viscosity are absent in Eulers equations of motion. The effects of viscousstresses are absent in Eulers theory; the flows are driven by pressure. The NavierStokesequations reduce to Eulers equations when the fluid is inviscid.

1.4 Generation of vorticity in fluids governed by Eulers equations

Eulers equations (1.3.1) may be written as

ut

u+ 12 |u|2 = g p

, (1.4.1)

where we have used the vector identity

(u ) u = 12 |u|2 u (1.4.2)

and

= [u] = curl u.We can obtain the vorticity equation by forming the curl of (1.4.1):

t+ u = u curl p

+ curl g. (1.4.3)

If curl ( p/) = [ 1p() ] = 0 and curl g = 0, then [u] = 0; curl g = 0 if g is

given by a potential g = G. Flows for which curl ( p/) = 0 are said to be barotropic.Barotropic flows governed by Eulers equations with conservative body forces g = Gcannot generate vorticity. If the fluid is incompressible, then the flow is barotropic.

1.5 Perfect fluids, irrotational flow

Inviscid fluids that are also incompressible are called perfect or ideal. Perfect fluids sat-isfy Eulers equations. Perfect fluids with conservative body forces give rise to Bernoullisequation:

[

(

t+ 1

2||2

)+ p G

]= 0,

where

(

t+ 1

2||2

)+ p G = f (t) . (1.5.1)

We may absorb the function f (t) into the potential = + t f (t) dt without changingthe velocity:

= [+

tf (t) dt

]= = u. (1.5.2)


1.6 Boundary conditions for irrotational flow 5

Bernoullis equation relates p and before any problem is solved; actually curlu = 0is a constraint on solutions. The velocity u is determined by satisfying 2 = 0 and theboundary conditions.

1.6 Boundary conditions for irrotational flow

Irrotational flows have a potential, and if the flow is solenoidal the potential is harmonic,2 = 0. This book reveals the essential role of harmonic functions in the flow of incom-pressible viscous and viscoelastic fluids. All the books on partial differential equations(PDEs) have sections devoted to the mathematical analysis of the Laplaces equation.Laplaces equation may be solved for prescribed data on the boundary of the flow regionincluding infinity for flows on unbounded domains. It can be solved for Dirichlet data inwhich values ofare prescribed on the boundary or forNeumann data inwhich the normalcomponent of is prescribed on the boundary. Almost any combination of Dirichlet andNeumann data all over the boundary will lead to unique solutions of Laplaces equation.

It is important that unique solutions can be obtained when only one condition isprescribed for at each point on the boundary. The problem is overdetermined whentwo conditions are prescribed. We can solve the problem when Dirichlet conditions areprescribed or when Neumann conditions are prescribed but not when both are prescribed.In the case in which Neumann conditions are prescribed over the whole the boundarythe solution is unique up to the addition of any constant. If the boundary condition is notspecified at each point on the boundary, then the problem may be not overdeterminedwhen two conditions are prescribed.

This point can be forcefully made within the framework of fluid dynamics. Consider,for example, an irrotational streaming flow over a body. The velocity U at infinity in thedirection x is given by the potential = U x. The tangential component of velocity onthe boundary S of the body is given by

uS = eS u = eS , (1.6.1)

where eS is a unit lying entirely in S. The normal component of velocity on S is given by

un = n u = n , (1.6.2)

where n is the unit normal pointing from body to fluid.Laplaces equation for streaming flow can be solved if

is prescribed on S, (1.6.3)

or if

n is prescribed on S. (1.6.4)

It cannot be solved if and n are simultaneously prescribed on S.The prescription of the tangential velocity on S is aDirichlet condition. If is prescribed

on S the tangential derivatives can be computed. It is possible to solve Laplaces equationfor a linear combination of and n ; say n + is prescribed on S. This bind ofcondition is called a Robin boundary condition.


6 Introduction

U x

r

Figure 1.1. Axisymmetric flow over a sphere of radius a. The flow depends on the radius r and the polar angle .

The no-slip condition of viscous fluid mechanics requires that

uS = 0, un = 0 (1.6.5)

simultaneously on S. These conditions cannot be satisfied by potential flow. In fact, theseconditions cannot be satisfied by solutions of Eulers equations even when they are notirrotational.

The point of view that has been universally adopted by researchers and students of fluidmechanics for several centuries is that the normal component should be enforced so thatthe fluid does not penetrate the solid. The fluid must then slip at the boundary becausethere is no other choice. To reconcile this with no-slip, researchers put the viscosity to zero.The resolution of these difficulties lies in the fact that real flows have a nonzero rotationalvelocity at boundaries generated by the no-slip condition. The no-slip condition usuallycannot be satisfied by the rotational velocity alone; the irrotational velocity is also needed(see chapter 4).

1.7 Streaming irrotational flow over a stationary sphere

A flow of speed U in the direction x = r cos streams past a sphere of radius a (seefigure 1.1). A solution of Laplaces equation 2 = 0 in spherical polar coordinates is

= Ur cos + cr2

cos . (1.7.1)

The normal and tangential components of the velocity at r = a are, respectively,

r= U cos 2c

a3cos , (1.7.2)

1a

= U sin c

a3sin . (1.7.3)

If the normal velocity is prescribed to be zero on the sphere, then c = Ua3/2 and

1 = U(r + a

3

2r2

)cos . (1.7.4)


1.7 Streaming irrotational flow over a stationary sphere 7

The corresponding tangential velocity is 3U sin /2. If the tangential velocity is pre-scribed to be zero on the sphere, then c = Ua3 and

2 = U(r a

3

r2

)cos . (1.7.5)

The corresponding normal velocity is 3U cos .The preceding analysis and uniqueness of solutions of Laplaces equation show that

wemay obtain the solution1 with a zero normal velocity by prescribing the nonzero func-tion 32U sin for the tangential velocity. We may obtain the solution 2 with a zerotangential velocity by prescribing the nonzero function 3U cos for the normal velocity.

More complicated conditions for harmonic solutions on rigid bodies are encountered inexact solutions of the NavierStokes equations in which irrotational and rotational flowsare tightly coupled at the boundary. The boundary conditions cannot be satisfied withoutirrotational flow and they cannot be satisfied by irrotational flow only [cf. equation (4.6.6)].


2

Historical notes

Potential flows of viscous fluids are an unconventional topic with a niche history assembledin a recent review article (Joseph, 2006a, 2006b).

2.1 NavierStokes equations

The history of NavierStokes equations begins with the 1822 memoir of Navier, whoderived equations for homogeneous incompressible fluids from a molecular argument.Using similar arguments, Poisson (1829) derived the equations for a compressible fluid.The continuum derivation of the NavierStokes equation is due to Saint-Venant (1843)and Stokes (1845). In his 1851 paper, Stokes wrote as follows:

Let P1, P2, P3 be the three normal, and T1, T2, T3 the three tangential pressures in the directionof three rectangular planes parallel to the co-ordinate planes, and let D be the symbol ofdifferentiation with respect to t when the particle and not the point of space remains thesame. Then the general equations applicable to a heterogeneous fluid (the equations (10) ofmy former (1845) paper), are

(DuDt

X)

+ dP1dx

+ dT3dy

+ dT2dz

= 0, (132)

with the two other equations which may be written down from symmetry. The pressures P1,T1, etc. are given by the equations

P1 = p 2(dudx

)

, T1 = (dvdz

+ dwdy

), (133)

and four other similar equations. In these equations

3 = dudx

+ dvdy

+ dwdz

. (134)

The equations written by Stokes in his 1845 paper are the same ones we use today:

(dudt

X)

= divT, (2.1.1)

8


2.2 Stokes theory of potential flow of viscous fluid 9

where X is presumably a body force, which was not specified by Stokes, and

T =(p 2

3 div u

)1+ 2D[u], (2.1.2)

dudt

= ut

+ (u ) u, (2.1.3)

D[u] = 12

[u+ uT] , (2.1.4)ddt

+ divu = 0. (2.1.5)

Stokes assumed that the bulk viscosity 23 is selected so that the deviatoric part of T

vanishes and traceT = 3p.Inviscid fluids are fluids with zero viscosity. Viscous effects on the motion of fluids were

not understood before the notion of viscosity was introduced by Navier in 1822. Perfectfluids, following the usage of Stokes and other 19th-century English mathematicians, areinviscid fluids that are also incompressible. Statements like Truesdells (1954),

In 1781 Lagrange presented his celebrated velocity-potential theorem: if a velocity potentialexists at one time in a motion of an inviscid incompressible fluid, subject to conservativeextraneous force, it exists at all past and future times.

though perfectly correct, could not have been asserted by Lagrange, because the conceptof an inviscid fluid was not available in 1781.

2.2 Stokes theory of potential flow of viscous fluid

The theory of potential flow of a viscous fluid was introduced by Stokes in 1851. All ofhis work on this topic is framed in terms of the effects of viscosity on the attenuationof small-amplitude waves on a liquidgas surface. Everything he said about this prob-lem is subsequently cited. The problem treated by Stokes was solved exactly by Lamb(1932), who used the linearized NavierStokes equations, without assuming potentialflow.

Stokes discussion is divided into three parts, discussed in 51, 52, and 53 of his (1851)paper:

(1) The dissipation method in which the decay of the energy of the wave is computedfrom the viscous dissipation integral in which the dissipation is evaluated on potentialflow (51).

(2) The observation that potential flows satisfy the NavierStokes equations togetherwith the notion that certain viscous stresses must be applied at the gasliquid surfaceto maintain the wave in permanent form (52).

(3) The observation that, if the viscous stresses required formaintaining the irrotationalmotion are relaxed, the work of those stresses is supplied at the expense of the energyof the irrotational flow (53).


10 Historical notes

Lighthill (1978) discussed Stokes ideas but he did not contribute more to the theory ofirrotational motions of a viscous fluid. On page 234 he notes that

Stokes ingenious idea was to recognise that the average value of this rate of working

2[(/x) 2/xz+ (/z) 2/z2]z=0

required to maintain the unattenuated irrotational motions of sinusoidal waves must exactlybalance the rate at which the same waves when propagating freely would lose energy byinternal dissipation!.

Lamb (1932) gave an exact solution of the problem considered by Stokes in whichvorticity and boundary layers are not neglected. Wang and Joseph (2006a) did purelyirrotational theories of Stokes problem that are in good agreement with Lambs exactsolution.

In fact, Stokes idea called ingenious by Lighthill has serious defects in implemen-tation. The unattenuated irrotational waves move with a speed independent of viscosityas would be true for waves on an inviscid fluid. Lambs 1932 application of Stokes ideagives rise to a good approximation of the decay rates due to viscosity for long waves butdoes not correct the wave speeds for the effects of viscosity. Moreover, the cutoff betweenlong and short waves, which is defined by a condition (say large viscosity) for which thewave speed vanishes and progressive waves become standing waves, cannot be obtainedfrom the dissipation calculation proposed by Stokes and implemented by Lamb and allother authors. A correct irrotational approximation, called VCVPF, which gives an ap-proximation to Lambs exact solution for all wavenumbers, including the dependence ofthe wave speed on viscosity and the all important cutoff value, was implemented byWangand Joseph (2006b). Padrino and Joseph (2007) have demonstrated that all these resultsmay be obtained by a revised and rigorous application of the dissipation method to theproblem of viscous decay of capillary-gravity waves. They show that the effects of viscosityon the speed of progressive waves are sensible for waves near the cutoff value which arenot too long and they obtain the irrotational approximation of this cutoff value (see figure14.14).

2.3 The dissipation method

In his 1851 paper, Stokes writes:

51. By means of the expression given in Art. 49, for the loss of vis viva due to internal friction,we may readily obtain a very approximate solution of the problem: To determine the rateat which the motion subsides, in consequence of internal friction, in the case of a series ofoscillatory waves propagated along the surface of a liquid. Let the vertical plane of xy beparallel to the plane of motion, and let y be measured vertically downwards from the meansurface; and for simplicitys sake suppose the depth of the fluid very great compared with thelength of a wave, and the motion so small that the square of the velocity may be neglected.In the case of motion which we are considering, udx + vdy is an exact differential d whenfriction is neglected, and

= cmy sin (mx nt) , (140)where c,m, n are three constants, of which the last two are connected by a relation which it isnot necessary towrite down.Wemay continue to employ this equation as a near approximation


2.4 The distance a wave will travel before it decays by a certain amount 11

when friction is taken into account, provided we suppose c, instead of being constant, to beparameter which varies slowly with the time. Let V be the vis viva of a given portion of thefluid at the end of the time t , then

V = c2m2

e2mydxdydz. (141)

But by means of the expression given in Art.49, we get for the loss of vis viva during the timedt , observing that in the present case is constant, w = 0, = 0, and udx + vdy = d, where is independent of z,

4dt {(

d2dx2

)2+(d2dy2

)2+ 2

(d2dxdy

)2}dxdydz

which becomes, on substituting for its value,

8c2m4dt

e2mydxdydz.

But we get from (141) for the decrement of vis viva of the samemass arising from the variationof the parameter c

2m2cdcdt

dt

e2mydxdydz.

Equating the two expressions for the decrement of vis viva, putting for m its value 21,where is the length of a wave, replacing by , integrating, and supposing c0 to be theinitial value of c, we get1

c = c0e162 t

2 .

It will presently appear that the value of for water is about 0.0564, an inch and a second

being the units of space and time. Suppose first that is two inches, and t ten seconds. Then162t2 = 1.256, and c: c0::1:0.2848, so that the height of the waves, which varies as c, isonly about a quarter of what it was. Accordingly, the ripples excited on a small pool by a puffof wind rapidly subside when the exciting cause ceases to act.

Now suppose that is to fathoms or 2880 inches, and that t is 86400 seconds or a whole day.In this case 162t2 is equal to only 0.005232, so that by the end of an entire day, in whichtime waves of this length would travel 574 English miles, the height would be diminished bylittle more than the one two hundredth part in consequence of friction. Accordingly, the longswells of the ocean are but little allayed by friction, and at last break on some shore situatedat the distance of perhaps hundreds of miles from the region where they were first excited.

2.4 The distance a wave will travel before it decays by a certain amount

The observationsmade by Stokes about the distance awavewill travel before its amplitudedecays by a given amount point the way to a useful frame for the analysis of the effects ofviscosity on wave propagation. Many studies of nonlinear irrotational waves can be foundin the literature, but the only study known to us of the effects of viscosity on the decayof these waves is due to Longuet-Higgins (1997), who used the dissipation method todetermine the decay that is due to viscosity of irrotational steep capillary-gravity waves indeep water. He finds that the limiting rates of decay for small-amplitude solitary waves are

1 In a footnote on page 624, Lamb notes that Through an oversight in the original calculation the value2/162 was too small by one half. The value 16 should be 8.


12 Historical notes

twice those for linear periodic waves computed by the dissipation method. The dissipationof very steep waves can be more than 10 times that of linear waves because of the sharplyincreased curvature in wave troughs. He assumes that the nonlinear wave maintains itssteady form while decaying under the action of viscosity. The wave shape could changeradically from its steady shape in very steep waves.

Stokes (1880) studied the motion of nonlinear irrotational gravity waves in two di-mensions that are propagated with a constant velocity, and without change of form. Thisanalysis led Stokes (1880) to the celebrated maximum wave, whose asymptotic form givesrise to a pointed crest of angle 120. The effects of viscosity on such extreme waves hasnot been studied, but they may be studied by the dissipation method or the same potentialflow theory used by Stokes (1851) for inviscid fluids with the caveat that the normal stresscondition that p vanish on the free surface be replaced with the condition that

p+ 2un/n = 0on the free surface with normal n, where the velocity component un = /n is given bythe potential.


3

Boundary conditions for viscous fluids

Boundary conditions for incompressible viscous fluids cannot usually be satisfied withoutcontributions from potential flows [Joseph and Renardy, 1991, 1.2(d)].

(1) No-slip conditions are required at the boundary S of a rigid solid:

u = U for x S, (3.0.1)whereU is the velocity of the solid.

(2) At the interface S between two fluids, the fluid velocities are continuous, the shearstress is continuous, and the stress is balanced by surface tension.

Now we express the condition just mentioned with equations. Let the position of thesurface S as it moves through the surface be F [x(t), t] = 0, where x (t) S for all t andx (t) uS is the velocity of points of S. Because F = 0 is an identity in t , we have

dFdt

= Ft

+ uS F = 0; (3.0.2)

in fact dmF/dtm = 0 for all m. Now, the normal n to S (figure 3.1) is n = F/ |F |;hence

uS F = (n uS) |F | . (3.0.3)The tangential component of uS is irrelevant, and

n uS = n u, (3.0.4)where u is the velocity of a material point on x S.

We now set a convention for jumps in the value of variables as we cross S at x. Define

[[]] = ()1 ()2 ,where ()1 means to evaluate () at x in the fluid 1 and likewise for 2.

Combining (3.0.2), (3.0.3), and (3.0.4), we get the kinematic equation for the evolu-tion of F :

dFdt

= Ft

+ u F = 0. (3.0.5)

Because [[F/t]] = 0, we have[[u n]] = [[u]] n = 0. (3.0.6)

13


14 Boundary conditions for viscous fluids

n = F/ F

x(t)

F[x(t), t] = 0

Fluid 2

Fluid 1

Figure 3.1. Interface S between two fluids.

The normal component of velocity is continuous. If the tangential component of velocityis also continuous across S, we have

n [[u]] = 0, x S. (3.0.7)Turning now to the stress,

T = p1+ 2D[u],(3.0.8)

Ti j = pi j + (

uixj

+ ujxi

),

we express the continuity of the shear stress as

eS [[2D[u]]] n = 0, (3.0.9)where eS is a unit vector tangent to S; eS F = 0.

The condition that the jump in the normal component of the stress be balanced bya surface-tension force may be expressed as

n [[T]] n = [[p]]+ 2n [[D[u]]] n = 2H II, (3.0.10)where H is the mean curvature,

2H = 1R1

+ 1R2

, (3.0.11)

and R1 and R2 are principal radii of curvature. The surface gradient is

II = n (n ) . (3.0.12)The curvature 2H is determined by the surface divergence of the normal (see Joseph

and Renardy, 1991):

2H = II n. (3.0.13)(3) Free surfaces. Free surfaces are two fluid interfaces for which one fluid is a dy-namically inactive gas. The density of the gas is usually much less than the density ofthe liquid, and the viscosity of the gas is usually much less than the viscosity of theliquid. In some cases we can put the viscosity and density of the gas to zero; then wehave a liquidvacuum interface. This procedure works well for problems of capillaryinstability and RayleighTaylor (RT) instability. However, for KelvinHelmholtz (KH)instabilities the kinematic viscosity = / is important, and for the liquid and gasare of comparable magnitude. The dynamic participation of the ambient is necessaryfor KH instability.


Boundary conditions for viscous fluids 15

For free-surface problems with a dynamically inactive ambient, no condition on thetangential velocity of the interface is required; the shear stress vanishes,

2eS D[u] n = 0, x S, (3.0.14)and the normal stress balance is

p+ 2n D[u] n = 2H II. (3.0.15)Sometimes the sign of themean curvature is ambiguous (we could have assigned n from2 to 1). A quick way to determine the sign is to look at the terms,

p = 2(

1R1

+ 1R2

)

,

making sure the pressure is higher inside the equivalent sphere.

Summarizing, the main equations on S are (3.0.5), (3.0.6), (3.0.7), (3.0.9), (3.0.10),(3.0.14), and (3.0.15). These equations are written for Cartesian, cylindrical, and polarspherical coordinates in Appendix A.


4

Helmholtz decomposition coupling rotationalto irrotational flow

In this chapter we present the form of the NavierStokes equations implied by theHelmholtz decomposition in which the relation of the irrotational and rotational ve-locity fields is made explicit. The idea of self-equilibration of irrotational viscous stressesis introduced. The decomposition is constructed first by selection of the irrotational flowcompatible with the flow boundaries and other prescribed conditions. The rotationalcomponent of velocity is then the difference between the solution of the NavierStokesequations and the selected irrotational flow. To satisfy the boundary conditions, the irrota-tional field is required, and it depends on the viscosity. Five unknown fields are determinedby the decomposed form of the NavierStokes equations for an incompressible fluid: thethree rotational components of velocity, the pressure, and the harmonic potential. Thesefive fields may be readily identified in analytic solutions available in the literature. It isclear from these exact solutions that potential flow of a viscous fluid is required for sat-isfying prescribed conditions, such as the no-slip condition at the boundary of a solid orcontinuity conditions across a two-fluid boundary. The decomposed form of the NavierStokes equations may be suitable for boundary layers because the target irrotational flowthat is expected to appear in the limit, say at large Reynolds numbers, is an explicit to-be-determined field. It can be said that equations governing the Helmholtz decompositiondescribe the modification of irrotational flow due to vorticity, but the analysis shows thetwo fields are coupled and cannot be completely determined independently.

4.1 Helmholtz decomposition

The Helmholtz decomposition theorem says that every smooth vector field u, definedeverywhere in space and vanishing at infinity, together with its first derivatives, can bedecomposed into a rotational part v and an irrotational part :

u = v + , (4.1.1)where

divu = div v + 2, (4.1.2)curl u = curl v. (4.1.3)

This decomposition leads to the theory of the vector potential, which is not followed here.The decomposition is unique on unbounded domains, and explicit formulas for the scalar

16


4.2 NavierStokes equations for the decomposition 17

and vector potentials are well known. A framework for embedding the study of potentialflows of viscous fluids, in which no special flow assumptions are made, is suggested by thisdecomposition (Joseph, 2006a, 2006c). If the fields are solenoidal, then

divu = div v = 0, 2 = 0. (4.1.4)Because is harmonic, we have from (4.1.1) and (4.1.4) that

2u = 2v. (4.1.5)The irrotational part of u is on the null space of the Laplacian, but in special cases, suchas plane shear flow, 2v = 0 but curl v = 0.

Unique decompositions are generated by solutions of NavierStokes equation (4.2.1)in the decomposed form of (4.2.2) for which the irrotational flows satisfy (4.1.1), (4.1.3),(4.1.4), and (4.1.5) and certain boundary conditions. The boundary conditions for theirrotational flows bear a heavyweight in all this. Simple examples of unique decomposition,taken from hydrodynamics, are presented later.

The decomposition of the velocity into rotational and irrotational parts holds at eachand every point and varies from point to point in the flow domain. Various possibilitiesfor the balance of these parts at a fixed point and the distribution of these balances frompoint to point can be considered:

(1) The flow is purely irrotational or purely rotational. These two possibilities do occuras approximations but are not typical.

(2) Typically the flow is mixed with rotational and irrotational components at eachpoint.

4.2 NavierStokes equations for the decomposition

To study solutions of the NavierStokes equations, it is convenient to express the NavierStokes equations for an incompressible fluid,

dudt

= p+ 2u, (4.2.1)in terms of the rotational and irrotational fields implied by the Helmholtz decomposition,

vt

+ (

t+

2||2 + p

)+ div (v + v + v v) = 2v (4.2.2)

or

vi

t+

xi

(

t+

2||2 + p

)+

xj

(v j

xi+ vi

xj+ viv j

)= 2vi ,

satisfying (4.1.4).To solve this problem in a domain , say, when the velocity u = V is prescribed on ,

we would need to compute a solenoidal field v satisfying (4.2.2) and a harmonic function satisfying 2 = 0 such that

v + = V on .Because this system of five equations in five unknowns is just the decomposed form of thefour equations in four unknowns that define the NavierStokes system for u, it ought to


18 Helmholtz decomposition coupling rotational to irrotational flow

be possible to study this problem with exactly the same mathematical tools that are usedto study the NavierStokes equations.

In the NavierStokes theory for incompressible fluid, the solutions are decomposedinto a space of gradients and its complement, which is a space of solenoidal vectors. Thegradient space is not, in general, solenoidal because the pressure is not solenoidal. If itwere solenoidal, then 2p = 0, but 2p = div u u satisfies Poissons equation.

It is not true that only the pressure is found on the gradient space. Indeed, equation(4.2.2) gives rise to Poissons equation for the Bernoulli function, not just the pressure:

2(

t+

2||2 + p

)=

2

xixj

(v j

xi+ vi

xj+ viv j

).

In fact, there may be hidden irrotational terms on the right-hand side of this equation.The boundary condition for solutions of (4.2.1) is

u a = 0 on ,where a is solenoidal field, a = V on . Hence,

v a + = 0. (4.2.3)The decomposition depends on the selection of the harmonic function ; the traditionalboundary condition,

n a = n on , (4.2.4)together with a Dirichlet condition at infinity when the region of flow is unbounded, anda prescription of the value of the circulation in doubly connected regions give rise to aunique . Then the rotational flow must satisfy

v n = 0, es (v a)+ es = 0 (4.2.5)on . The v determined in this way is rotational and satisfies (4.1.3). However, v maycontain other harmonic components.

Purely rotational velocities can be identified in the exact solutions exhibited in theexamples in which the parts of the solution that are harmonic and the parts that arenot are identified by inspection. Equation (4.6.6), in which the purely irrotational flow isidentified by selection of a parameter , is a good example. We have a certain freedom inselecting the harmonic functions used for the decomposition.

We can form a more general formulation of the boundary condition generating thepotential in the Helmholtz decomposition by replacing Neumann condition (4.2.4) with aRobin condition,

+ n = a n, (4.2.6)depending on two free parameters. The boundary conditions satisfied by the rotationalvelocity v are

v n+ n = a n,v es + es = a es,

(4.2.7)


4.3 Self-equilibration of the irrotational viscous stress 19

where is a harmonic function satisfying (4.2.6). The free parameters and of equation(4.2.6) can be determined to obtain an optimal result, for which optimal is a conceptthat is in need of further elaboration. Ideally, we would like to determine and so thatv is purely rotational. The examples show that purely rotational flows exist in specialcases. It remains to see if this concept makes sense in a general theory.

What is the value we add to solutions of NavierStokes equations (4.2.1) by solvingthem in the Helmholtz decomposed form1 of (4.2.2)? Certainly it is not easier to solvefor five rather than for four fields; if you cannot solve (4.2.1) then you certainly cannotsolve (4.2.2). However, if the decomposed solution could be extracted from solutions of(4.2.1) or computed directly, then the form of the irrotational solution that is determinedthrough coupling with the rotational solution and the changes in its distribution as theReynolds number changes would be revealed. There is nothing approximate about this; itis the correct description of the role of irrotational solutions in the theory of the NavierStokes equations, and it looks different and is different than the topic potential flow ofan inviscid fluid that we all learned in school.

The form (4.2.2) of the NavierStokes equations may be well suited to the study ofboundary layers of vorticity with irrotational flow of viscous fluid outside. It may beconjectured that in such layers v = 0, whereas v is relatively small in the irrotationalviscous flow outside. Rotational and irrotational flows are coupled in the mixed inertialterm on the left-hand side of (4.2.2). The irrotational flow does not vanish in the boundarylayer, and the rotational flow, though small, probably will not be zero in the irrotationalviscous flow outside. This feature is also in Prandtls theory of boundary layers, but thattheory is not rigorous, and the irrotational part is, so to say, inserted by hand and is notcoupled to the rotational flow at the boundary. The coupling terms are of considerableinterest, and they should play a strong role in the region of small vorticity at the edge ofthe boundary layer. The actions of irrotational flow in the exact boundary-layer solution ofHiemenz (1911) for a steady two-dimensional (2D) flow toward a stagnation point at arigid body (Batchelor, 1967, 2868), and of Hamel (1917) flow in diverging and convergingchannels in the Helmholtz decomposed form are discussed at the end of this chapter.

Effects on boundary layers on rigid solids arising from the viscosity of the fluid in theirrotational flow outside have been considered without the decomposition by Wang andJoseph (2006b, 2006c) and by Padrino and Joseph (2007).

4.3 Self-equilibration of the irrotational viscous stress

The stress in a Newtonian incompressible fluid is given by

T = p1+ (u+ uT) = p1+ (v + vT)+ 2 .Most flows have an irrotational viscous stress. The term 2v in (4.2.2) arises from therotational part of the viscous stress.

The irrotational viscous stress I = 2 does not give rise to a force-density termin (4.2.2). The divergence of I vanishes on each and every point in the domain V of flow.

1 A cultured lady asked a famous conductor of Baroque music if J. S. Bach was still composing: No madame,he is decomposing.



Even though an irrotational viscous stress exists, it does not produce a net force to drivemotions. Moreover,

div IdV =

n IdS = 0. (4.3.1)

The traction vectors n I have no net resultant on each and every closed surface in thedomain V of flow.We say that the irrotational viscous stresses, which do not drive motions,are self-equilibrated. Irrotational viscous stresses are not equilibrated at boundaries, andthey may produce forces there.

4.4 Dissipation function for the decomposed motion

Thedissipation function evaluatedon thedecomposedfield (4.1.1) sorts out into rotational,mixed, and irrotational terms given by

=V2Di j Di jdV

=

2Di j [v]Di j [v] dV + 4

Di j [v]2

xixjdV + 2

2

xixj

2

xixjdV. (4.4.1)

Most flows have an irrotational viscous dissipation. In regions V where v is small, we haveapproximately that

T = p1+ 2 , (4.4.2)

= 2V

2

xixj

2

xixjdV. (4.4.3)

Equation (4.4.3) has been widely used to study viscous effects in irrotational flows sinceStokes (1851).

4.5 Irrotational flow and boundary conditions

How is the irrotational flow determined? It frequently happens that the rotational mo-tion cannot satisfy the boundary conditions; this well-known problem is associated withdifficulties in forming boundary conditions for the vorticity. The potential is a harmonicfunction that can be selected so that the values of the sum of the rotational and irrotationalfields can be chosen to balance prescribed conditions at the boundary. The allowed irrota-tional fields can be selected from harmonic functions that enter into the purely irrotationalsolution of the same problem on the same domain. A very important property of potentialflow arises from the fact that irrotational viscous stresses do not give rise to irrotationalviscous forces in equations of motion (4.2.2). The interior values of the rotational velocityare coupled to the irrotational motion through Bernoulli terms evaluated on the potentialand inertial terms that couple the irrotational and rotational fields. The dependence of theirrotational field on viscosity can be generated by the boundary conditions.


4.6 Examples from hydrodynamics 21

4.6 Examples from hydrodynamics

4.6.1 Poiseuille flow

A simple example that serves as a paradigm for the relation of the irrotational androtational components of velocity in all the solutions of NavierStokes equations is planePoiseuille flow:

u =[ P

2

(b2 y2) , 0, 0] ,

curl u =(0, 0, P

y

),

v =(P

2y2, 0, 0

),

=( P

2b2, 0, 0

).

(4.6.1)

The rotational flow is a constrained field and cannot satisfy the no-slip boundary condition.To satisfy the no-slip condition we add the irrotational flow:

x= P

2b2.

The irrotational component is for uniform and unidirectional flow, chosen so that u = 0 atthe boundary.

4.6.2 Flow between rotating cylinders

u = eu (r) ,u = eaa at r = a,u = ebb at r = b,

u (r) = Ar + Br

,

A= a2a b2bb2 a2 ,

B = (a b) a2b2

b2 a2 ,v = eAr,

1r

= B

r.

(4.6.2)

The irrotational flow with v = 0 is an exact solution of the NavierStokes equations withno-slip at boundaries.



4.6.3 Stokes flow around a sphere of radius a in a uniform stream U

This axisymmetric flow, described with spherical polar coordinates (r, ,), is independentof . The velocity components can be obtained from a stream function (r, ) such that

u = (ur , u) = 1r sin (1r

,

r

). (4.6.3)

The stream function can be divided into rotational and irrotational parts:

= v + p, E2 p = 0 with E2 = 2

r2+ sin

r2

(1

sin

), (4.6.4)

where p is the conjugate harmonic function. The potential corresponding to p is givenby

=( A2r2

+ Cr)2 cos ,

where the parameters Aand C are selected to satisfy the boundary conditions on u.The Helmholtz decomposition of the solution of the problem of slow streaming motion

over a stationary sphere is given by

u = (ur , u) =(vr +

r, v + 1r

),

ur = U[1 3a

2r+ 1

2

(ar

)3]cos ,

u = U[1 3a

4r 1

4

(ar

)3]sin ,

curl u = U(0, 0, 3a

2r2sin

),

vr = 32arU cos ,

v = 34arU sin ,

= U(r 1

4a3

r2

)cos ,

p p = 32ar2U cos .

(4.6.5)

The potential for flow over a sphere is

= U(r

4a3

r2

)cos . (4.6.6)

The normal component of velocity vanishes when = 2 and the tangential componentvanishes when = 4. In the present case, to satisfy the no-slip condition, we take = 1.Both the rotational and irrotational components of velocity are required for satisfying theno-slip condition.



4.6.4 Streaming motion past an ellipsoid

The problem of the steady translation of an ellipsoid in a viscous liquid was solved byLamb (1932, 6045) in terms of the gravitational potential of the solid and anotherharmonic function corresponding to the case in which 2 = 0, finite at infinity with = 1 for the internal space, as stated by Lamb:

If the fluid be streaming past the ellipsoid, regarded as fixed, with the general velocity U inthe direction of x, we assume

u = A2

x2+ B

(x

x

)+U, v = A

2

xy+ Bx

y, w = A

2

xz+ Bx

z.

These satisfy the equation of continuity, in virtue of the relations

2 = 0, 2 = 0;and they evidently make u = U, v = 0, w = 0 at infinity. Again, they make

2u = 2B2

x2, 2v = 2B

2

xy, 2w = 2B

2

xz.

We note next that 2v = 2u; the irrotational part of u is on the null space of theLaplacian. It follows that the rotational velocity is associated with and is given by theterms proportional to B. The irrotational velocity is given by

= (A

x

).

The vorticity is given by

= ey(2B

z

)+ ez

(2B

y

).

4.6.5 HadamardRybyshinsky solution for streaming flow past a liquid sphere

As in the flow around a solid sphere, this problem is posed in spherical coordinates with astream function and potential function related by (4.6.3), (4.6.4), and (4.6.5). The streamfunction is given by

= f (r) sin2 , (4.6.7)f (r) = A

r+ Br + Cr2 + Dr4. (4.6.8)

The irrotational part of (4.6.7) is the part that satisfies E2 p = 0. From this it follows that

p =(Ar

+ Cr2)sin2 , (4.6.9)

v =(Br + Dr4) sin2 . (4.6.10)

The potential corresponding to p is given by (4.6.5).The solution of this problem is determined by continuity conditions at r = a. The inner

solution for r < a is designated by u, v, , , p, , .



The normal component of velocity,

vr + r

= vr + r

, (4.6.11)

is continuous at r = a. The normal stress balance is approximated by a static balancein which the jump of pressure is balanced by surface tension so large that the drop isapproximately spherical. The shear stress

e (v + vT + 2 ) er = e (v + vT + 2 ) er . (4.6.12)

The coefficients A, B, C, and D are determined by the condition that u = Uex as r and continuity conditions (4.6.11) and (4.6.12). We find that, when r a,

f (r) = +

14Ua2(r4 a2r2) , (4.6.13)

where the r2 term is associated with irrotational flow and, when r a,

f (r) = 12U(r2 ar)+

+ 14U(a3

r ar

), (4.6.14)

where the r2 and a2/r terms are irrotational.

4.6.6 Axisymmetric steady flow around a spherical gas bubble at finiteReynolds numbers

This problem is like the HadamardRybyshinski problem, with the internal motion of thegas neglected, but inertia cannot be neglected. The coupling conditions reduce to

vr + r

= 0, (4.6.15)e

(v + vT + 2 ) er = 0. (4.6.16)There is no flow across the interface at r = a, and the shear stress vanishes there.

The equations ofmotion are the r and components of (4.1.1) with time derivative zero.Since Levich (1949), it has been assumed that, at moderately large Reynolds numbers, theflow in the liquid is almost purely irrotational with a small vorticity layer where v = 0 inthe liquid near r = a. The details of the flow in the vorticity layer, the thickness of thelayer, and the presence and variation of viscous pressure contribution all are unknown.

It may be assumed that the irrotational flow in the liquid outside the sphere can beexpressed as a series of spherical harmonics. The problem then is to determine the partici-pation coefficients of the different harmonics, the pressure distribution and the rotationalvelocity v satisfying continuity conditions (4.6.15) and (4.6.16). The determination ofthe participation coefficients may be less efficient than a purely numerical simulation ofLaplaces equation outside a sphere subject to boundary conditions on the sphere that arecoupled to the rotational flow. This important problem has not yet been solved.

4.6.7 Viscous decay of free-gravity waves

Flows that depend on only two space variables, such as plane flows or axisymmetric flows,admit a stream function. Such flows may be decomposed into a stream function and a



potential function. Lamb (1932, 349) calculated an exact solution of the problem of theviscous decay of free-gravity waves as a free-surface problem of this type. He decomposesthe solution into a stream function and a potential function , and the solution is givenby

u = x

+ y

, v = y

x

,p

= t

gy, (4.6.17)

where

2 = 0, t

= 2 . (4.6.18)

This decomposition is a Helmholtz decomposition; it can be said that Lamb solved thisproblem in the Helmholtz formulation.

Wang and Joseph (2006a) constructed a purely irrotational solution of this problemthat is in very good agreement with the exact solution. The potential in the Lamb solutionis not the same as the potential in the purely irrotational solution because they satisfydifferent boundary conditions. It is worth noting that viscous potential flow rather thaninviscid potential flow is required for satisfying boundary conditions. The common ideathat the viscosity should be put to zero to satisfy boundary conditions is deeply flawed. Itis also worth noting that viscous component of the pressure does not arise in the boundarylayer for vorticity in the exact solution; the pressure is given by (4.6.17)

4.6.8 Oseen flow

Steady streaming flow of velocity U (Milne-Thomson, 1968, 6968, Lamb, 1932, 60816)of an incompressible fluid over a solid sphere of radius a that is symmetric about the xaxis satisfies

Uux

= 1 p+ 2u, (4.6.19)

where divu = 0 and U = 2k (for convenience). The inertial terms in this approximationare linearized but not zero. The equations of motion in decomposed form are

(U

x+ p

)+U v

x= 2v. (4.6.20)

Because div v = 0 and is harmonic, 2p = 0 and U[( curl v)/x] = 2 curl v. Therotational velocity is determined by a function :

v = 12k

ex ,

= a0r

exp [kr (1 cos )].(4.6.21)

The potential is governed by the Laplace equation 2 = 0, for which the solution isgiven by

= Ux + b0r

+ b1 x

(1r

)+ + = Ur cos + b0

r+ b11r2 cos + +,



where Ux is the uniform velocity term. With these, the velocity u = (ur , u) is expressedas

ur = r

+ 12k

r cos , u = 1r

+ 1

2k1r

+ sin .

These should be zero at the sphere surface (r = a) to give

b0 = a02k = 3Ua4k

= 3a2

, a0 = 3Ua2 , b1 =Ua3

4.

The higher-order terms in the potential vanish.To summarize, the decomposition of the velocity into rotational and irrotational com-

ponents is

u = (ur , u) =(vr +

r, v + 1r

),

curl u = curl v,vr = 12k

r cos ,

v = 12kr

+ sin ,

= Ux 3Ua4kr

Ua3

4r2cos .

(4.6.22)

4.6.9 Flows near internal stagnation points in viscous incompressible fluids

The fluid velocity relative to uniformmotion or rest vanishes at points of stagnation. Thesepoints occur frequently even in turbulent flow. When the flow is purely irrotational, thevelocity potential can be expanded near the origin as a Taylor series:

= 0 + ai xi + 12ai j xi xj + O(xi x3i

), (4.6.23)

where the tensor ai j is symmetric. At the stagnation point = 0, hence ai = 0, andbecause 2 = 0 everywhere, we have aii = 0. The velocity is

xi= ai j x j (4.6.24)

at leading order. The velocity components along the principal axes (x, y, z) of the tensorai j are

u1 = ax, u2 = by, u3 = (a + b)z, (4.6.25)where a and b are unknown constants relating to the flow field. Irrotational stagnationpoints in the plane are saddle points, centers are stagnation points around which the fluidrotates. Saddle points and centers are embedded in vortex arrays.

Taylor vortex flow (TVF) is a 2D (x, y) array of counterrotating vortices (figure 4.1)whose vorticity decays in time are due to viscous diffusion (t = 2). It is useful tointroduce the stream function in the analysis of this type of fluid motion. It is well known



Figure 4.1. (After Taylor, 1923). Streamlinesfor a system of eddies dying down underthe action of viscosity. All cell boundaries onwhich = 0 are irrotational. Near each andevery saddle point, say at (x, y) = 0, the veloc-ity in and out of the saddle point is linear, say(u, v) = (x, y). The vorticity in every quad-rant of this generic saddle point is constant onhyperbolae (xy = constant).

that a 2D incompressible flow can be written in terms of a stream function (x, y, t) as

u1 = y , u2 = x . (4.6.26)When this representation is used, the vorticity equation becomes

(/t 2)(2 )+ y x(2 ) x y(2 ) = 0. (4.6.27)Taylor (1923) showed that if (x, y, t) is in the form

(x, y, t) = exp(2t)(x, y) (4.6.28)and is a solution of the 2D equation for eigenmodes of the vibrating membrane

2 + 2 = 0, (4.6.29)then,

u1 = exp(2t)y, u2 = exp(2t)x (4.6.30)is a solution of the time-dependent, nonlinear, incompressible NavierStokes equations.The corresponding pressure is

p = 2exp(22t)(||2 + 22), (4.6.31)

and the vorticity is given by

= yu1 xu2 = 2 = k2 . (4.6.32)Consider the solution

= 0k2

cos(kxx) cos(kyy), (4.6.33)

which satisfies (4.6.29) with 2 = k2 = k2x + k2y ; kx and ky are the wavenumbers in the xand y directions and 0 is the initial maximum vorticity. This function has a maximum atthe origin. From (4.6.30), by use of (4.6.33), the instantaneous velocity components are

u1 = 0 kyk2 exp(k2t) cos(kxx) sin(kyy),

u2 = 0 kxk2 exp(k2t) sin(kxx) cos(kyy),

(4.6.34)

which gives the damped motion of a square array of counterrotating vortices (figure 4.1).



The vorticity vanishes at saddle points (X,Y). After expanding the solution in powerseries centered on a generic stagnation point we find that

u1 (X,Y, t) = 0kykxk2 X+ 1X3 + 1XY2 + +,

u2 (X,Y, t) = 0kykxk2 Y+ 2Y3 + 2YX2 + +.

(4.6.35)

The vorticity vanishes at (X,Y) = (0, 0) and appears first at second order:

(X,Y, t) = 0kxkyXYexp(k2t)+ +. (4.6.36)

The Helmholtz decomposition of the local solution near (X, Y) = (0, 0) is

u1 = X

+ v1, u2 = Y

+ v2, (4.6.37)

where (

X,

Y

)= 0kykx

k2(X, Y) , (4.6.38)

and

(v1, v2) =(1X3 + 1XY2 + +, 2Y3 + 2YX2 + +

). (4.6.39)

This solution is generally valid in eddy systems that are segregated into quadrants nearthe stagnation point as in figure 4.1.

The dissipation of vorticity at highReynolds numbers is an important topic in the theoryof turbulence. The dissipation integral is proportional to the viscosity but paradoxicallythis integral does not vanish as the Reynolds number tends to infinity. The role of potentialflows in an eddy structure is also of interest. The TVF is not a turbulent flow but it does giverise to an interesting result on the role of stagnation points in the decay of the dissipationin the potential flow limit.

Consider an array of square cells of side l in a fixed square array of side L= nl. Inthe square array kx = ky = k = 2/ = / l. The dissipation of this array is computed asfollows.

With the velocity field (4.6.34), the components of the strain-rate tensor are readilycomputed:

D11 = u1x

= 0kxkyk2

exp(k2t) sin(kxx) sin(kyy),

D22 = u2y

= D11,

D12 = D21 = 12(

u1y

+ u2x

)= 0

(k2x k2y2k2

)exp(k2t) cos(kxx) cos(kyy).



The dissipation is thus computed as

= 2 LL LL [D211 + D222 + 2D212] dxdy= 4

20

k4exp(2k2t)

LL

LL

[k2x k

2y sin

2(kyy) sin2(kxx)

+ (k2x k2y )2

4cos2(kxx) cos2(kyy)

]dxdy

= 420

k4exp(2k2t)L2

[k2x k

2y +

(k2x k2y )24

]= 20L2 exp

(2

k2t)

,

(4.6.40)

with k2 = k2x + k2y = 2k2. is maximum for a fixed t for a value of such thatdd

[ exp

(2

k2t)]

= 0.

Hence

max = 2k2t =2

162t. (4.6.41)

For this maximizing max we have

= 20L2

2k2texp(1) = 20L4

4n22texp(1) =

20L

4

16

(N + 1)22t exp(1), (4.6.42)

where n = 2(N + 1) is the number of cells and N is the number of stagnation points in thearray of side L= nl. The maximum dissipation of a Taylor vortex array of side Lat a fixedtime tends to zero as the number of stagnation points tends to infinity. The potential flowsat the stagnation points dominate the dissipation as the number of cells increases and thesize of the cells decreases in the limit (inviscid) in which max goes to zero as N goes toinfinity.

4.6.10 Hiemenz boundary-layer solution for two-dimensional flow towarda stagnation point at a rigid boundary

Stagnation points on solid bodies are very important because the pressures at such pointscan be very high. However, stagnation-point flow cannot persist all the way to the boun-dary because of the no-slip condition. Hiemenz (1911) looked for a boundary-layer solu-tion of the NavierStokes equations vanishing at y = 0 that tends to stagnation-point flowfor large y, expressed as

u = (u, v) , (u) ez = curl u.

(4.6.43)

The motion in the outer region is irrotational flow near a stagnation point at a planeboundary. The flow in the irrotational region is described by the stream function,

= kxy,



Figure 4.2. Steady 2D flow toward a stagnation point at a rigid boundary.

where x and y are rectilinear coordinates parallel and normal to the boundary (see fig-ure 4.2), with the corresponding velocity distribution

u = kx, v = ky, (4.6.44)where k is a positive constant that, in the case of a stagnation point on a body fixed in astream, must be proportional to the speed of the stream.

The next step is to determine the distribution of vorticity in the thin layer near theboundary from

u

x+ v

y=

(2

x2+

2

y2

), (4.6.45)

together with boundary conditions that u = 0 and v = 0 at y = 0 and that the flow tend toform (4.6.44) at the outer edge of the layer. Hiemen

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