Stability of Fluid Motions II

ISBN-13: 9783642809965 / Angielski / Miękka / 2011 / 276 str.

D. D. Joseph

Opis

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.

Spis treści

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 Bénard 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 Bénard 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.