Stability of fluid motions
Author(s)
Bibliographic Information
Stability of fluid motions
（Springer tracts in natural philosophy, v. 27-28）
Springer-Verlag, 1976
- v. 1 : gw
- v. 1 : us
- v. 2 :gw
- v. 2 :us
Available at 74 libraries
Note
Bibliography: v. 1, p. [267]-277; v. 2, p. [262]-269
Includes index
Description and Table of Contents
- Volume
-
v. 1 : gw ISBN 9783540075141
Description
The study of stability aims at understanding the abrupt changes which are observed in fluid motions as the external parameters are varied. It is a demanding study, far from full grown"whose most interesting conclusions are recent. I have written a detailed account of those parts of the recent theory which I regard as established. Acknowledgements I started writing this book in 1967 at the invitation of Clifford Truesdell. It was to be a short work on the energy theory of stability and if I had stuck to that I would have finished the writing many years ago. The theory of stability has developed so rapidly since 1967 that the book I might then have written would now have a much too limited scope. I am grateful to Truesdell, not so much for the invitation to spend endless hours of writing and erasing, but for the generous way he has supported my efforts and encouraged me to higher standards of good work. I have tried to follow Truesdell's advice to write this work in a clear and uncomplicated style. This is not easy advice for a former sociologist to follow; if I have failed it is not due to a lack of urging by him or trying by me.
My research during the years 1969-1970 was supported in part by a grant from the Guggenheim foundation to study in London.
Table of Contents
- 1. Global Stability and Uniqueness.- 1. The Initial Value Problem and Stability.- 2. Stability Criteria-the Basic Flow.- 3. The Evolution Equation for the Energy of a Disturbance.- 4. Energy Stability Theorems.- 5. Uniqueness.- Notes for Chapter I.- (a) The Reynolds Number.- (b) Bibliographical Notes.- II. Instability and Bifurcation.- 6. The Global Stability Limit.- 7. The Spectral Problem of Linear Theory.- 8. The Spectral Problem and Nonlinear Stability.- 9. Bifurcating Solutions.- 10. Series Solutions of the Bifurcation Problem.- 11. The Adjoint Problem of the Spectral Theory.- 12. Solvability Conditions.- 13. Subcritical and Supercritical Bifurcation.- 14. Stability of the Bifurcating Periodic Solution.- 15. Bifurcating Steady Solutions
- Instability and Recovery of Stability of Subcritical Solutions.- 16. Transition to Turbulence by Repeated Supercritical Bifurcation.- Notes for Chapter II.- III. Poiseuille Flow: The Form of the Disturbance whose Energy Increases Initially at the Largest Value of v.- 17. Laminar Poiseuille Flow.- 18. The Disturbance Flow.- 19. Evolution of the Disturbance Energy.- 20. The Form of the Most Energetic Initial Field in the Annulus.- 21. The Energy Eigenvalue Problem for Hagen-Poiseuille Flow.- 22. The Energy Eigenvalue Problem for Poiseuille Flow between Concentric Cylinders.- (a) Parabolic Poiseuille Flow.- (b) Poiseuille Flow in an Annular Pipe.- 23. Energy Eigenfunctions-an Application of the Theory of Oscillation kernels.- 24. On the Absolute and Global Stability of Poiseuille Flow to Disturbances which are Independent of the Axial Coordinate.- 25. On the Growth, at Early Times, of the Energy of the Axial Component of Velocity.- 26. How Fast Does a Stable Disturbance Decay.- IV. Friction Factor Response Curves for Flow through Annular Ducts.- 27. Responce Functions and Response Functionals.- 28. The Fluctuation Motion and the Mean Motion.- 29. Steady Causes and Steady Effects.- 30. Laminar and Turbulent Comparison Theorems.- 31. A Variational Problem for the Least Pressure Gradient in Statistically Stationary Turbulent Poiseuille Flow with a Given Mass Flux Discrepancy.- 32. Turbulent Plane Poiseuille Flow-a Lower Bound for the Response Curve.- 33. The Response Function Near the Point of Bifurcation.- 34. Construction of the Bifurcating Solution.- (a) The Spectral Problem.- (b) The Perturbation Series.- (c) Some Properties of the Bifurcating Solution.- 35. Comparison of Theory and Experiment.- (a) Instability of Laminar Poiseuille Flow.- (b) Description of the Diagrams.- (c) Inferences and Conjectures.- Notes for Chapter IV.- V. Global Stability of Couette Flow between Rotating Cylinders.- 36. Couette Flow, Taylor Vortices, Wavy Vortices and Other Motions which Exist between the Cylinders.- 37. Global Stability of Nearly Rigid Couette Flows.- 38. Topography of the Response Function, Rayleigh's Discriminant...- 39. Remarks about Bifurcation and Stability.- 40. Energy Analysis of Couette Flow
- Nonlinear Extension of Synge's Theorem.- 41. The Optimum Energy Stability Boundary for Axisymmetric Disturbances of Couette Flow.- 42. Comparison of Linear and Energy Limits.- VI. Global Stability of Spiral Couette-Poiseuille Flows.- 43. The Basic Spiral Flow. Spiral Flow Angles.- 44. Eigenvalue Problems of Energy and Linear Theory.- 45. Conditions for the Nonexistence of Subcritical Instability.- 46. Global Stability of Poiseuille Flow between Cylinders which Rotate with the Same Angular Velocity.- 47. Disturbance Equations for Rotating Plane Couette Flow.- 48. The Form of the Disturbance Whose Energy Increases at the Smallest R.- 49. Necessary and Sufficient Conditions for the Global Stability of Rotating Plane Couette Flow.- 50. Rayleigh's Criterion for the Instability of Rotating Plane Couette Flow, Wave Speeds.- 51. The Energy Problem for Rotating Plane Couette Flow when Spiral Disturbances are Assumed from the Start.- 52. Numerical and Experimental Results.- VII. Global Stability of the Flow between Concentric Rotating Spheres.- 53. Flow and Stability of Flow between Spheres.- (a) Basic Flow.- (b) Stability Analysis.- (c) Experimental and Numerical Results.- Appendix A. Elementary Properties of Almost Periodic Functions.- Appendix B. Variational Problems for the Decay Constants and the Stability Limit.- B 1. Decay Constants and Minimum Problems.- B 2. Fundamental Lemmas of the Calculus of Variations.- B 6. Representation Theorem for Solenoidal Fields.- B 8. The Energy Eigenvalue Problem.- B 9. The Eigenvalue Problem and the Maximum Problem.- Notes for Appendix B.- Appendix C. Some Inequalities.- Appendix D. Oscillation Kernels.- Appendix E. Some Aspects of the Theory of Stability of Nearly Parallel Flow.- E 1. Orr-Sommerfeld Theory in a Cylindrical Annulus.- E 2. Stability and Bifurcation of Nearly Parallel Flows.- References.
- Volume
-
v. 2 :gw ISBN 9783540075165
Description
The study of stability aims at understanding the abrupt changes which are observed in fluid motions as the external parameters are varied. It is a demanding study, far from full grown, whose most interesting conclusions are recent. I have written a detailed account of those parts of the recent theory which I regard as established. Acknowledgements I started writing this book in 1967 at the invitation of Clifford Truesdell. It was to be a short work on the energy theory of stability and if I had stuck to that I would have finished the writing many years ago. The theory of stability has developed so rapidly since 1967 that the book I might then have written would now have a much too limited scope. I am grateful to Truesdell, not so much for the invitation to spend endless hours of writing and erasing, but for the generous way he has supported my efforts and encouraged me to higher standards of good work. I have tried to follow Truesdell's advice to write this work in a clear and uncomplicated style. This is not easy advice for a former sociologist to follow; if I have failed it is not due to a lack of urging by him or trying by me.
My research during the years 1969-1970 was supported in part by a grant from the Guggenheim foundation to study in London.
Table of Contents
- VIII. The Oberbeck-Boussinesq Equations. The Stability of Constant Gradient Solutions of the Oberbeck-Boussinesq Equations.- 54. The Oberbeck-Boussinesq Equations for the Basic Flow.- 55. Boundary Conditions.- (a) Temperature Conditions.- (b) Concentration Boundary Conditions.- (c) Velocity Boundary Conditions.- 56. Equations Governing Disturbances of Solutions of the OB Equations.- 57. The ? Family of Energy Equations.- 58. Kinematic Admissibility, Sufficient Conditions for Stability.- 59. Motionless Solutions of the Oberbeck-Boussinesq Equations.- 60. Physical Mechanisms of Instability of the Motionless State.- 61. Necessary and Sufficient Conditions for Stability.- 62. The Benard Problem.- 63. Plane Couette Flow Heated from below.- 64. The Buoyancy Boundary Layer.- IX. Global Stability of Constant Temperature-Gradient and Concentration-Gradient States of a Motionless Heterogeneous Fluid.- 65. Mechanics of the Instability of the Conduction-Diffusion Solutions in a Motionless Heterogeneous Fluid.- 66. Energy Stability of Heated below and Salted above.- 67. Heated and Salted from below: Linear Theory.- 68. Heated and Salted below: Energy Stability Analysis.- 69. Heated and Salted below: Generalized Energy Analysis.- Addendum for Chapter IX: Generalized Energy Theory of Stability for Hydromagnetic Flows.- X. Two-Sided Bifurcation into Convection.- 70. The DOB Equations for Convention of a Fluid in a Container of Porous Material.- 71. The Spectral Problem, the Adjoint Spectral Problem and the Energy Theory of Stability.- 72. Two-Sided Bifurcation.- (a) Simple Eigenvalue.- (b) Multiple Eigenvalues.- (c) Stability of Bifurcating Solutions at Eigenvalues of Higher Multiplicity.- 73. Conditions for the Existence of Two-Sided Bifurcation.- (a) Axisymmetric Convection in Round Containers.- (b) Nonaxisymmetric Convection in Round Containers.- (c) Convection in a Hexagonal Container.- (d) Stability of Solutions Bifurcating at an Eigenvalue of Multiplicity N.- (e) Stability of Bifurcating Hexagonal Convection.- (f) One-Sided Convection in Containers of Rectangular Cross-Section.- (g) The Benard Problem for a DOB Fluid in a Container.- 74. Two-Sided Bifurcation between Spherical Shells.- 75. Stability of the Conduction Solution in a Container Heated below and Internally.- 76. Taylor Series in Two Parameters.- 77. Two-Sided Bifurcation in a Laterally Unbounded Layer of Fluid.- (a) Spectral Crowding.- (b) Cellular Convection.- (c) Stability and the Sign of the Motion in Cellular Convection.- Addendum to Chapter X: Bifurcation Theory for Multiple Eigenvalues.- (a) Membrane Eigenvalues Perturbed by a Nonlinear Term.- (b) Bifurcation from a Simple Eigenvalue.- (c) Bifurcation from a Multiple Eigenvalue.- (d) The Orthogonal Decomposition.- (e) Solvability Conditions.- (f) Perturbation of a Linear Problem at a Double Eigenvalue.- (g) Bifurcation from a Double Eigenvalue: An Example where the Initiating Solvability Condition Occurs at Order l =1 (?1 ? 0).- (h) Bifurcation from a Double Eigenvalue: An Example where the Initiating Solvability Condition Occurs at Order l = 2(?1 = 0, ?2 ? 0).- XI. Stability of Supercritical Convection-Wave Number Selection Through Stability.- 78. Statistically Stationary Convection and Steady Convection.- 79. Stability of Rolls to Noninteracting Three-Dimensional Disturbances.- 80. Nonlinear Neutral Curves for Three-Dimensional Disturbances of Roll Convection.- (a) Oblique-Roll and Cross-Roll Instabilities.- (b) Varicose Instabilities.- (c) Sinuous Instabilities.- 81. Computation of Stability Boundaries by Numerical Methods.- 82. The Amplitude Equation of Newell and Whitehead.- XII. The Variational Theory of Turbulence Applied to Convection in Porous Materials Heated from below.- 83. Bounds on the Heat Transported by Convection.- 84. The Form of the Admissible Solenoidal Fluctuation Field Which Minimizes ? [u, ?
- ?].- 85. Mathematical Properties of the Multi-? Solutions.- 86. The single-? Solution and the Situation for Small ?.- 87. Boundary Layers of the Single-? Solution.- 88. The Two-? Solution.- 89. Boundary-Layers of the Multi-? Solutions.- 90. An Improved Variational Theory Which Makes Use of the Fact that B is Small.- 91. Numerical Computation of the Single-? and Two-? Solution. Remarks about the Asymptotic Limit ? ? ?.- 92. The Heat Transport Curve: Comparison of Theory and Experiment.- XIII. Stability Problems for Viscoelastic Fluids.- 93. Incompressible Simple Fluids. Functional Expansions and Stability.- (a) Functional Expansions of ?, Stability and Bifurcation.- (b) Generation of the History of a Motion.- (c) Stability and Bifurcation of Steady Flow.- 94. Stability and Bifurcation of the Rest State.- (a) Slow Motion.- (b) Time-Dependent Perturbations of the Rest State.- (c) Stability of the Rest State.- (d) Bifurcation of the Rest State of a Simple Fluid Heated from below.- 95. Stability of Motions of a Viscoelastic Fluid.- (a) The Climbing Fluid Instability.- (b) Symmetry Breaking Instabilities of the Time-Periodic Motion Induced by Torsional Oscillations of a Rod.- (c) The Striping Instability.- (d) Tall Taylor Cells in Polyacrylamide.- XIV. Interfacial Stability.- 96. The Mechanical Energy Equation for the Two Fluid System.- 97. Stability of the Interface between Motionless Fluids When the Contact Line is Fixed.- 98. Stability of a Column of Liquid Resting on a Column of Air in a Vertical Tube-Static Analysis.- 99. Stability of a Column of Liquid Resting on a Column of Air in a Vertical Tube-Dynamic Analysis.- Notes for Chapter XIV.- References.
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