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.- 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 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 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 Flow-a 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.