Applied Mathematical Sciences Volume 91
Editors S.S. Antman J.E. Marsden L. Sirovich
Advisors J.K. Hale P. Holmes J. Keener J. Keller B.J. Matkowsky A. Mielke C.S. Peskin K.R. Sreenivasan
Springer Science+ Business Media, LLC
Applied Mathematical Sciences
1. John: Partial Differential Equations, 4th ed. 33. Grenander: Regular Structures: Lectures in 2. Sirovich: Techniques of Asymptotic Analysis. Pattern Theory, Vol. III. 3. Hale: Theory of Functional Differential Equations, 34. Kevorkian!Cole: Perturbation Methods in Applied
2nd ed. Mathematics. 4. Percus: Combinatorial Methods. 35. Carr: Applications of Centre Manifold Theory. 5. von Mises/Friedrichs: Fluid Dynamics. 36. Bengtsson/Ghil/Kiillen: Dynamic Meteorology: 6. Freiberger!Grenander: A Short Course in Data Assimilation Methods.
Computational Probability and Statistics. 37. Saperstone: Semidynamical Systems in Infinite 7. Pipkin: Lectures on Viscoelasticity Theory. Dimensional Spaces. 8. Giacaglia: Perturbation Methods in Non-linear 38. Lichtenberg/Lieberman: Regular and Chaotic
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Equations. Dynamical Systems, and Bifurcations of Vector 14. Yoshizawa: Stability Theory and the Existence of Fields.
Periodic Solution and Almost Periodic Solutions. 43. Ockendon!Taylor: Inviscid Fluid Flows. 15. Braun: Differential Equations and Their 44. Pazy: Semigroups of Linear Operators and
Applications, 3rd ed. Applications to Partial Differential Equations. 16. Lefschetz: Applications of Algebraic Topology. 45. Glashoff!Gustafson: Linear Operations and 17. Col/atz/Wetterling: Optimization Problems. Approximation: An Introduction to the Theoretical 18. Grenander: Pattern Synthesis: Lectures in Pattern Analysis and Numerical Treatment of Semi-
Theory, Vol. I. Infmite Programs. 19. Marsden!McCracken: Hopf Bifurcation and Its 46. Wilcox: Scattering Theory for Diffraction
Applications. Gratings. 20. Driver: Ordinary and Delay Differential 47. Hale/Magalhiies/0/iva: Dynamics in Infinite
Equations. Dimensions, 2nd ed. 21. Courant!Friedrichs: Supersonic Flow and Shock 48. Murray: Asymptotic Analysis.
Waves. 49. Ladyzhenskaya: The Boundary-Value Problems of 22. Rouche/Habets!Laloy: Stability Theory by Mathematical Physics.
Liapunov's Direct Method. 50. Wilcox: Sound Propagation in Stratified Fluids. 23. Lamperti: Stochastic Processes: A Survey of the 51. Golubitsky/Schaeffer: Bifurcation and Groups in
Mathematical Theory. Bifurcation Theory, Vol. I. 24. Grenander: Pattern Analysis: Lectures in Pattern 52. Chipot: Variational Inequalities and Flow in
Theory, Vol. II. Porous Media. 25. Davies: Integral Transforms and Their 53. Majda: Compressible Fluid Flow and Systems of
Applications, 2nd ed. Conservation Laws in Several Space Variables. 26. Kushner/Clark: Stochastic Approximation 54. Wasow: Linear Turning Point Theory.
Methods for Constrained and Unconstrained 55. Yosida: Operational Calculus: A Theory of Systems. Hyperfunctions.
27. de Boor: A Practical Guide to Splines: Revised 56. Chang/Howes: Nonlinear Singular Perturbation Edition. Phenomena: Theory and Applications.
28. Kei/son: Markov Chain Models-Rarity and 57. Reinhardt: Analysis of Approximation Methods Exponentiality. for Differential and Integral Equations.
29. de Veubeke: A Course in Elasticity. 58. Dwoyer!Hussaini!Voigt (eds): Theoretical 30. Sniatycki: Geometric Quantization and Quantum Approaches to Turbulence.
Mechanics. 59. Sanders/Verhulst: Averaging Methods in 31. Reid: Sturmian Theory for Ordinary Differential Nonlinear Dynamical Systems.
Equations.
32. Meis/Markowitz: Numerical Solution of Partial Differential Equations. (continued following index)
Brian Straughan
The Energy Method, Stability, and Nonlinear Convection Second Edition
With 30 Illustrations
i Springer
Brian Straughan Department of Mathematical Sciences Science Laboratories University of Durham Durham DHI 3LE UK brian.straughan@ durham.ac.uk
Editors: S.S. Antman Department of Mathematics and Institute for Physical Science
and Technology University of Maryland College Park, MD 20742-4015 USA ssa @math.umd.edu
J.E. Marsden Control and Dynamical
Systems, 107-81 California Institute of
Technology Pasadena, CA 9 I I 25 USA marsden @cds.caltech.edu
L. Sirovich Division of Applied
Mathematics Brown University Providence, RI 02912 USA [email protected]
Mathematics Subject Classification (2000): 76Exx, 76Rxx, 35K55, 35Q35
Library of Congress Cataloging-in-Publication Data Straughan, B. (Brian)
The energy method, stability, and nonlinear convection I Brian Straughan. p. em.- (Applied mathematical sciences ; 91)
Includes bibliographical references and index.
I. Fluid dynamics. 2. Differential equations, Nonlinear-Numerical solutions. 3. Heat-Convection-Mathematical models. I. Title. II. Applied mathematical sciences (Springer-Verlag New York Inc.); v. 91. QAI.A647 vol. 91 [QA9llj 510 s-dc21 [532'.05'01515355] 2003053004
ISBN 978-I-4419-1824-6 ISBN 978-0-387-21740-6 (eBook) DOI 10.1007/978-0-387-21740-6
© 2004, 1991 Springer Science+ Business Media New York Originally published by Springer-Verlag New York, Inc. in 2004 Softcover reprint of the hardcover 2nd edition 2004 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag Berlin Heidelberg GmbH), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
9 8 7 6 5 4 3 2 I SPIN 10911914
Typesetting: Pages Created by the Author in LaTeX 2e using Springer's svsing6.sty macro.
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Preface
This book is a revised edition of my earlier book of the same title. The current edition adopts the structure of the earlier version but is much changed. The introduction now contains definitions of stability. Chapters 2 to 4 explain stability and the energy method in more depth and new sections dealing with porous media are provided. Chapters 5 to 13 are revisions of those in the earlier edition. However, chapters 6 to 12 are substantially revised, brought completely up to date, and have much new material in.
Throughout the book new results are provided which are not available elsewhere.
Six new chapters, 14 - 19, are provided dealing with topics of current interest. These cover the topics of multi-component convection diffusion, convection in a compressible fluid, convection with temperature dependent viscosity and thermal conductivity, the subject of penetrative convection whereby part of the fluid layer can penetrate into another, nonlinear stability in the oceans, and finally in chapter 19 practical methods for solving numerically the eigenvalue problems which arise are presented.
The book presents convection studies in a variety of fluid and porous media contexts. It should be accessible to a wide audience and begins at an elementary level. Many new references are provided.
It is a pleasure to thank Magda Carr of Durham University for finding several misprints in an early version of this book. It is also a pleasure to thank Achi Dosanjh of Springer for her advice with editorial matters, and also to thank Frank Ganz of Springer for his help in sorting out latex problems for me.
v1 Preface
Some of the research work reported in this book was supported in part by the Leverhulme Research Grant number RF & G/9/2000/226. This support is very gratefully acknowledged.
Durham, UK June 2003
Brian Straughan
Contents
Preface
1 Introduction
2 Illustration of the energy method 2.1 The diffusion equation ..... 2.2 The diffusion equation with a linear source term
2.2.1 Spatial Region x E JR. .......... . 2.2.2 Finite Spatial Region. . ........ . 2.2.3 Energy Stability for a Solution to (2.9) ..
2.3 Conditional energy stability ....... . 2.4 Weighted energy and boundedness .... . 2.5 Weighted energy and unconditional decay 2.6 A stronger force in the diffusion equation . 2. 7 A polynomial heat source in three dimensions 2.8 Sharp conditional stability ...... . 2.9 Interaction diffusion systems ..... .
2.9.1 A general Segel-Jackson system 2.9.2 Exchange of Stabilities ..... . 2.9.3 Stability in a Two-Constituent System 2.9.4 Specific Segel-Jackson systems 2.9.5 Numerical results ........... .
v
1
7 7 9
10 10 11 14 16 19 20 22 24
27 27 29 30 33 37
vm Contents
3 The Navier-Stokes equations and the Benard problem 40 3.1 Energy stability for the Navier-Stokes equations 40
3.1.1 Stability of the zero solution. 10 3.1.2 Steady Solutions. . . . . . . . . . . . . . 42 3.1.3 Nonlinear (Energy) Stability of V. 43
3.2 Balance of Energy and the Boussinesq Approximation. 47 3.3 Energy Stability and the Benard Problem 49 3.4 The equations for convection in a porous medium
3.4.1 The Darcy equations .... 3.4.2 The Brinkman equations . . . . . . . . .. 3.4.3 The Forchheimer equations . . . . . . . ..
56 56 57 57
3.4.4 Darcy equations with anisotropic permeability 57
4 Symmetry, competing effects, and coupling parameters 59 4.1 Coupling parameters and the classical Benard problem 59 4.2 Energy stability in porous convection . . . . . . . . ..
4.2.1 The Benard problem for the Darcy equations 4.2.2 Benard problem for the Forchheimer equations .
64 64 69
4.2.3 Darcy equations with anisotropic permeability 70
4.3 4.4 4.5
4.6 4.7
4.2.4 Benard problem for the Brinkman equations Symmetry and competing effects . . ... Convection with internal heat generation Convection in a variable gravity field . 4.5.1 Nonlinear energy stability . . . 4.5.2 Discussion of numerical results. Multiparameter eigenvalue problems Finite geometries and numerical basic solutions
74 77 87 92 95 96 !)9
103
5 Convection problems in a half space 105 5.1 Existence in the energy maximum problem . . . . . . . 105 5.2 The salinity gradient heated vertical sidewall problem . 109
6 Generalized energies and the Lyapunov method 6.1 The stabilizing effect of rotation . . . . ..... . 6.2
6.3 6.4 6.5
Construction of a generalised energy . . . . . . . 6.2.1 Open problems for unconditional stability 6.2.2 Problems solved for unconditional stability Nonlinear stability in rotating porous convection . Bio-Convection . . . . . . . . . . . Convection in a porous vertical slab 6.5.1 Brinkman's equation .. 6.5.2 A nonlinear density law 6.5.3 Variable fluid properties
112 112 116 118 119 120 122 126 129 1:n 133
Contents IX
7 Geophysical problems 135 7.1 Patterned (or polygonal) ground formation . . . . . . 135 7.2 Mathematical models for patterned ground formation 140
7.2.1 Linear instability . . . . . 141 7.2.2 Nonlinear energy stability. . . . . . . . . . . 142 7.2.3 A cubic density law . . . . . . . . . . . . . . 143 7.2.4 A model incorporating phase change effects 144 7.2.5 Patterned ground formation under water 145
7.3 Conclusions for patterned ground formation 146 7.4 Convection in thawing subsea permafrost 148 7.5 The model of Harrison and Swift 149 7.6 The energy stability maximum problem . 152 7. 7 Decay of the energy . . . . . . . . . . . . 155 7.8 Other models for thawing subsea permafrost 156
7.8.1 Unconditional stability with a general density 157 7.9 Other models for geophysical phenomena . . . . . . . 158
8 Surface tension driven convection 161 8.1 Energy stability and surface tension driven convection 161 8.2 Surface film driven convection . . . . . . . . . . 165 8.3 Conclusions from surface film driven convection 171 8.4 Energy stability and nonlinear surface tension 171 8.5 A fluid overlying a porous material . . . . . . 174
9 Convection in generalized fluids 180 9.1 The Benard problem for a micropolar fluid . . . . . . 180 9.2 Oscillatory instability . . . . . . . . . . . . . . . . . . 184 9.3 Nonlinear energy stability for micropolar convection . 185 9.4 Micropolar convection and other effects 187 9.5 Generalized fluids . . . . . . . . . . 190
9.5.1 Generalized heat conduction 191
10 Time dependent basic states 193 10.1 Convection problems with time dependent basic states 193 10.2 Time-varying gravity and surface temperatures 194 10.3 Patterned ground formation . . . . . . . . . . . . . 198
11 Electrohydrodynamic and magnetohydrodynamic convection 201 11.1 The MHD Benard problem and symmetry . . . . . 201 11.2 Thermo-convective electrohydrodynamic instability 203
11.2.1 The investigations of Roberts and of Turnbull 204 11.3 Stability in thermo-convective EHD . . . . . . . 208
11.3.1 Charge injection induced instability . . . 209 11.3.2 Energy stability for non-isothermal EHD 211
x Contents
11.4 Unconditional stability in thermo-convective EHD 212 11.4.1 Exchange of stabilities and nonlinear stability 214
12 Ferrohydrodynamic convection 12.1 The basic equations of ferrohydrodynamics 12.2 Thermo-convective instability in FHD .. 12.3 Unconditional nonlinear stability in FHD 12.4 Other models and results in FHD
13 Reacting viscous fluids 13.1 Chemical convective instability ......... . 13.2 Quasi-equilibrium thermodynamics . . . . . . .. 13.3 Basic equations for a chemically reacting mixture 13.4 A model for reactions far from equilibrium .
13.4.1 Chemical affinities and mass supplies 13.4.2 Thermodynamic deductions ... 13.4.3 Reduction of the energy equation 13.4.4 Selection of ma and 1/J
13.5 Convection in a layer ........ .
217 217 220 221 223
225 225 226 229 231 232 232 233 233 235
14 Multi-component convection diffusion 238 14.1 Convection with heating and salting below 238 14.2 Convection with three components 248 14.3 Overturning and pollution instability 258 14.4 Chemical convection . . . . . . . 265
15 Convection in a compressible fluid 269 15.1 Boussinesq approximation, convection in a deep layer 269
15.1.1 The Zeytounian model . 273 15.2 Slightly compressible convection . 277
15.2.1 Stability analysis . . . . . 280 15.2.2 The Berezin-Hutter model 284
15.3 The low Mach number approximation . 15.4 Stability in compressible convection
15.4.1 Nonlinear energy stability ...
285 286 289
16 Temperature dependent fluid properties 291 16.1 Depth dependent viscosity and symmetry . . . . . . . 291 16.2 Stability when the viscosity depends on temperature 293 16.3 Unconditional nonlinear stability . . . . . 294
16.3.1 Unconditional stability for 11 = 1 296 16.3.2 Unconditional stability for 11 = 1/2. 298
16.4 Quadratic viscosity dependence . . . . . 300 16.4.1 Unconditional nonlinear stability 302
16.5 Temperature dependent diffusivity . . . . 304
16.6 Unconditional stability in porous convection .. . 16.6.1 The quadratic Forchheimer model .... . 16.6.2 Unconditional stability for other problems
Contents xi
307 309 311
17 Penetrative convection 313 17.1 Unconditional stability for a quadratic buoyancy . 313 17.2 Unconditional stability with an internal heat source 316 17.3 More general equations of state . . 318 17.4 Convection with radiation heating . 321
17.4.1 Boundary heating . . . . . . 324 17.4.2 Linear instability equations 327 17.4.3 Energy stability for the Q model 328 17.4.4 Energy stability for the p(T2 ) model 329 17.4.5 Radiant heating with Q and p(T2 ) . 331
17.5 Effect of a heat source and nonlinear density 337 17.6 Penetrative convection in porous media . . . 338
17.6.1 Model for anisotropic penetrative convection 339 17.6.2 Linearized instability . . . . . . . . . . . . . 341 17.6.3 Unconditional stability and weighted energy 342 17.6.4 An internal heat source . 343
17.7 Radiation effects . . . . . . . . . 17.7.1 An optically thin model
17.8 Krishnamurti's model ..... .
344 346 347
18 Nonlinear stability in ocean circulation models 354 18.1 Langmuir circulations . . . . . . . . . . . . 354 18.2 Stommel-Veronis ( quasigeostrophic) model 358 18.3 Quasigeostrophic model . . . . . . . . . . . 360
19 Numerical solution of eigenvalue problems 363 19.1 The shooting method . . . . . . . . . . . . 363
19.1.1 A system: the Viola eigenvalue problem. 364 19.2 The compound matrix method . 365 19.3 The Chebyshev tau method . 368 19.4 Compound matrix calculation . 372 19.5 Chebyshev tau calculation . . . 375 19.6 Convection in thawing subsea permafrost 379 19.7 Penetrative convection in a porous medium. 382 19.8 Porous medium with inclined temperature gradient 383
19.8.1 Compound matrix method for Hadley flow 385 19.8.2 Chebyshev tau method for Hadley flow 385
A Useful inequalities 387 A.1 The Poincare inequality. 387 A.2 The Wirtinger inequality 387
xii Contents
A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.lO
The Sobolev inequality ............ . An inequality for the supremum of a function A Sobolev inequality for u4 . . . . . . . . . . .
A Sobolev inequality for u4 over a more general cell A two-dimensional surface inequality . . . . . Inequality (A.20) is false in three-dimensions . A boundary estimate for u2
A surface inequality for u4
References
Index
388 388 389 389 392 393 393 394
397
443
