Stability of Fluid Motions I

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.

Inhalt

1. Global Stability and Uniqueness.- § 1. The Initial Value Problem and Stability.- §2. Stability Criteriathe Basic Flow.- § 3. The Evolution Equation for the Energy of a Disturbance.- § 4. Energy Stability Theorems.- § 5. Uniqueness.- Notes for Chapter I.- 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.- § 23. Energy Eigenfunctionsan 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 andResponse 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 Flowa Lower Bound for the Response Curve.- § 33. The Response Function Near the Point of Bifurcation.- § 34. Construction of the Bifurcating Solution.- § 35. Comparison of Theory and Experiment.- 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 RotatingPlane 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.- 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.