Hydrodynamic Stability Theory MECHANICS : ANALYSIS Editors: V.J.Mizel and G.Le

Hydrodynamic stability theory MECHANICS : ANALYSIS Editors: V.J.Mizel and G.lE. Oravas

M.A. Krasnosel'skii, P.P. Zabreyko, E.I. Pustylnik and P.E. Sobolevski, Integral Operators in Spaces of Sum• mabie Functions. 1976. ISBN 90-286-0294-1. V.V. Ivanov, The Theory of Approximate Methods and Their Application to the Numerical Solution of Singular Integral Equations. 1976. ISBN 90-286-0036-1 . A. Kufner, J. Oldrich and F.Cl. Svatopluk (eds), Function Spaces. 1977. ISBN 90-286-0015-9. S.G. Mikhlin, Approximation on a Rectangular Grid. With Application to Finite Element Methods and Other Prob• lems. 1979. ISBN 90-286-0008-6. D.G.B. Edelen, Isovector Methods for Equations of Balance. With Programs for Computer Assistance in Operator Calculations and an Exposition of Practical Topics of the Exterior Calculus. 1980. ISBN 90-286-0420-0. R.S. Anderssen, F.R. de Hoog and M.A. Lukas (eds)~ The Application and Numerical Solution of Integral Equations. 1980. ISBN 90-286-0450-2. R.Z. Has'minski, Stochastic Stability of Differential Equa• tions. 1980. ISBN 90-286-0100-7. A.I. Vol'pert and S.I. Hudjaey, Analysis in Classes of Discontinuous Functions and :Equations of Mathematical Physics. 1985. ISBN 90-247-3109-7. A. Georgescu, Hydrodynamic Stability Theory. 1985. ISBN 90-247-3120-8. Hydrodynamic stability theory

By Adelina Georgescu Institute of Mathematics Bucharest, Romania

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. Library of Congress Cataloging-in-Publication Data Georgescu, Adelina. Hydrodynamic stability theory. (Mechanics-analysis; 9) Revised translation of: Teo ria sta bilitatii hidrodinamice. ' Includes bibliographies and index. 1. Hydrodynamics. 2. Navier-Stokes equations• Numerical solutions. 3. Stability. I. Title. II. Series. QA911.G4413 1985 532'.5 85-18903 ISBN 978-90-481-8289-3 ISBN 978-94-017-1814-1 (eBook) DOI 10.1007/978-94-017-1814-1

Book information Revised, updated translation of the Romanian edition "Teoria stabili• tatii hidrodinamice", first published by Editura ~tiintifica ~i Enciclopedica, Bucharest, 1976. Translated by dr. Adelina Georgescu. Translation edited by Professor David Sattinger.

Copyright © 1985 by Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1985 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publishers, Editura ~tiintifica ~i Enciclopedica, Piata Scinteii 1, Bucharest 33, Ro• mania and Martinus Nijhoff Publishers, P.O. Box 163, 3300 AD Dordrecht, The Netherlands. CONTENTS

Foreword ...... 9 Introduction ...... 11

Chapter 1. Classical hydrodynamic stability ...... 17 § 1.1. Setting of the problem ...... 17 1.1. 1. Stability in the small ...... 17 1.1.2. Stability in the mean ...... • ...... 21 1.1.3. Linear stability ...... 26 1.1.4. Rayleigh and Squire's theorems...... 37 1. 1. 5. Global stability ...... 40 1.1.6. Stability of the mean motion in transition regime 42 § 1.2. Orr-Sommerfeld equation 45 1.2.1. Non"Fiscous Orr-Sommerfeld equation...... 45 1.2.2. Tollmien's solution of the Blasius' boundary layer prohlem ...... 47 1.2.3 Relationship between Tollmien's and Heisenberg's solutions ...... 52 § 1.3. Criteria of hydrodynamic stability...... 54 1.3. 1 Serrin's universal criteria...... 55 1.3.2. Synge's criterion for the Couette flow between rotating cylinders ...... 58 1.3.3. Synge's criterion for plane parallel flows...... 61 1.3.4. Joseph's theorems ...... 64 1.3.5. The envelope method ...... 71

References 73

Chapter 2. Generalized solutions in hydrodynamic stability ...... 77 § 2. 1. Function spaces ...... 77 2.1.1. Spaces of continuous functions ...... 78 2.1.2. The LP spaces ...... 79 2.1.3. Generalized deri·rati·res ...... 82 2.1.4. Sobolev spaces...... 84 2.1.5. Embedding theorems .... , ...... 85 2.1.6. Compactness in the LP spaces...... 88 2.1.7. Spaces of vector functions...... 89 2. 1.8. Solenoidal vectors ...... 90 2.1.9. Functions of time ...... 92 § 2.2. Types of solutions in hydrodynamic stability theory. . . . 93 2.2.1. Classical solutions ...... 93 2.2.2. Generalized solutions of the linear problem ...... 94 2.2.3. Generalized solutions of the nonlinear problem.... 99 2.2.4. Existence of generalized solutions...... 104

5 § 2.3. Completeness of normal modes ...... 112 2.3.1. Motions in bounded domains ...... 112 2.3.2. Motions in unbounded domains ...... 115 § 2. 4. Linearization principle ...... 118 2.4.1. The finite-dimensional case ...... 118 2.4.2. Linearization principle in hydrodynamic stability 120 2.4.3. Stability of plane Couette flows ...... 128 § 2.5. The principle of exchange of stabilities (P.E.S.) ...... 135 2.5.1. The neutral state and P.E.S...... 135 2.5.2. Proof of P.E.S. for particular motions...... 136 2.5.3. Branching (bifurcation) of solutions of hydrodynamic equations ...... 137 § 2.6. Universal criteria of hydrodynamic stability...... 1-!0 2.6.1. Stationary basic flows ...... 140 2.6.2. Nonstationary basic flows...... 143 References 147

Chapter 3. Branching znd stability of the solutions of the Navier-Stokes equations ...... , ...... 152 3.1. Topological degree method for nonlinear equations in Banach spaces (Leray-Schauder methcd) ...... 152 3. 1. 1. The finite-dimensional case ...... 152 3.1.2. The infinite-dimensional case (Leray-Schauder method) ...... 156 § 3.2. Branching of solutions of the Navier-Stokes equations by the Leray-Schauder method ...... 161 3.2.1. Convective motions ...... 161 3.2.2. Couette flow in the case of a fixed exterior cylinder 167 3.2.3. Flows between two cylinders rotating in the same direction ...... 170 3.2.4. Flows in bounded domains...... 174 3.2.5. Kolmogorov's flows ...... 174 § 3.3. Liapunov-Schmidt method 176 3.3.1. The case of integral equations ...... 176 3.3.2. The case of nonlinear equations in Banach spaces. . 180 § 3.4. Branching of solutions of the Navier-Stokes equations by the Liapunov-Schmidt method ...... 186 3.4. 1. Convective motions ...... 186 3.4.2. Couette motion ...... 191 3.4.3. Motions in bounded domains ...... l-94 3.4.4. The stability of branching solutions...... 196 § 3. 5. Hopf bifurcation by the Joseph-Sattinger method...... 200 3.5.1. Deduction of secondary solutions ...... 200 3.5.2. Stability of secondary solutions ...... 205 § 3.6. Generation of turbulence by instability and local branching 207 References 209

6 Chapter 4. Nature of turbulence 212 § 4.1. Leray model ...... 212 § 4.2. The Landau-Hopf conjecture ...... 213 § 4.3. The Ruelle-Takens theory ...... 216 4.3. 1. The case of the Navier-Stokes equations...... 216 4.3.2. The Lorenz model ...... 219 4.4. Generic finiteness of the set of the solutions of the Navier- Stokes equations ...... 221 § 4.5. Pattern formation; symmetry breaking instability. . . . 225 § 4.6. Concluding remarks; open problems ...... 225 References 228

Chapter 5. The influence of the presence of a porous medium on hydrodynamic stability ...... 231 § 5.1. The mathematical problem ...... 231 § 5.2. Rayleigh-Taylor instability ...... 234 § 5.3. The Kelvin-Helmholtz instability ...... 236 § 5.4. The case of a vertical cylinder...... 243 References 246

Appendices 1. Operators in Hilbert spaces ...... 248 2. Semigroups of operators in Banach spaces...... 255 3. Spectral theory of linear operators ...... 256 4. Calculus of variations ...... 261 5. Geometric methods in branching theory...... 265 6. New methods for solving the Orr-Sommerfeld equation...... 269 7. Analytical methods to solve some eigenvalue problems in hydrodynamic and hydromagnetic stability theory...... 274 8. Stability of nonstationary fluid flows...... 297 Afterword 299 Index ...... 305 FOREWORD

The great number of varied approaches to hy drodynamic stability theory appear as a bulk of results whose classification and discussion are well-known in the literature. Several books deal with one aspect of this theory alone (e.g. the linear case, the influence of temperature and magnetic field, large classes of globally stable fluid motions etc.). The aim of this book is to provide a complete mathe• matical treatment of hydrodynamic stability theory by combining the early results of engineers and applied mathematicians with the recent achievements of pure mathematicians. In order to ensure a more operational frame to this theory I have briefly outlined the main results concerning the stability of the simplest ty pes of flow. I have attempted several definitions of the stability of fluid flows with due consideration of the connections between them. On the other hand, as the large number of initial and boundary value problems in hydrodynamic stability theory requires appropriate treat• ments, most of this book is devoted to the main concepts and methods used in hydrodynamic stability theory. Open problems are expressed in both mathematical and physical terms. In order to understand hydrodynamic stability theory and the reality underlying the Navier-Stokes model the reader is of necessity assumed to be highly conversant with mathematics (the calculus of variations, differential equations, spectral theory of linear operators; nonlinear functional analysis, differential topology etc.). Additio• nally, the mathematical formulation of the problems of hy drody• namic stability theory makes use of highly abstract concepts from the above theories. Practice has taught us that whereas undergraduates or graduates in mathematics can deal with these concepts at a high level of abstraction, they are unable to take good account of these abstract formulations in tackling particular cases from mechanics and eventually to grasp the phenomena through equations. Besides, most of the talks I had with experts in boundary layer theory, turbu• lence, chemistry, only reinforced my belief that most of these special• ists cannot abstract the physical meanning from the beautiful re• sults of pure mathematicians. I have attempted to overcome these difficulties by an appropriate presentation and explanation of ab• stract problems and by associating physical facts to the mathema• tical concepts discussed.

9 The main topics discussed in this book refer to the mathematical linear and nonlinear stability of fluid flows. In order to manage a unified treatment of this theory, I have discussed the linear case only after elucidating its connection with the nonlinear one. Besides I have made use of both classical and modern methods of treatment. The first three chapters provide a general account of hydrody• namic stability theory. Thus, after reducing the nonlinear stability problem to a nonlinear spectral one (i.e. after the completeness of normal modes has been proven) results of nonlinear stability are derived from linear theory in virtue of the linearization principle. In that respect Chapter 1 is a brief exposition of the classical linear theory. Chapter 2 analyses the principle of exchange of stabilities, which relates stability with bifurcation for the solutions of the Navier-Stokes equations. Stress is laid on the generalized theories developed by 0. A. Ladyzhenskaya, G. Iooss, C. Foias, G. Prodi and to the energetic methods of ]. Serrin, ]. L. Synge and D. D. Jo• seph. Tliis generalized frame is a suitable background for proving the principle of exchange of stabilities and the linearization principle. Chapter 3 is a treatment of the bifurcation of solutions of the Na• vier-Stokes equations. The stabillity of secondary flows (including basic results of D. D. Joseph, D. H. Sattinger, W. Velte, V. Yudo• vich) is also discussed. Both stability and bifurcation are used to explain the origin of turbulence. Three phenomenological theories on the nature of turbulence (Leray, Landau-Hop£, Ruelle-Takens) and Sattinger's theory of symmetry breaking and pattern formation are the toP.ic of Chapter 4. The contribution of stability and bifurca• tion in selecting from the solutions of the Navier-Stokes equations, that solution which corresponds to the real motion is duly emphasized. The last chapter discusses Gheorghitza's linear theory of stability in the presence of a porous medium. Compared with the first edition (1976) this book adds a chapter on the nature of turbulence which presents the results of stability and bifurcation theory obtained over the last decade. Appendices 3-S give several recent methods for solving various problems in hydrodynamic stability theory. I am grateful to Professor St. I. Gheorghitza, unfortunately no longer with us, for his authoritative guidance of my studies in fluid mechanics and for his encouragement in writing this book. Particular thanks and acknowledgements are due to Dr. Mihnea Moroianu who made valuable comments and remarks and gave much helpful cri• ticism to most of the mathematics involved in this book. Last but not least, I would like to express my gratitude to Professor David Sattinger who has substantially contributed to the publication of this edition.