Stability of Fluid Motions I

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.