Equilibrium Capillary Surfaces

ISBN-13: 9781461385868 / Angielski / Miękka / 2011 / 245 str.

Robert Finn

Opis

Capillarity phenomena are all about us; anyone who has seen a drop of dew on a plant leaf or the spray from a waterfall has observed them. Apart from their frequently remarked poetic qualities, phenomena of this sort are so familiar as to escape special notice. In this sense the rise of liquid in a narrow tube is a more dramatic event that demands and at first defied explanation; recorded observations of this and similar occur rences can be traced back to times of antiquity, and for lack of expla nation came to be described by words deriving from the Latin word "capillus," meaning hair. It was not until the eighteenth century that an awareness developed that these and many other phenomena are all manifestations of some thing that happens whenever two different materials are situated adjacent to each other and do not mix. If one (at least) of the materials is a fluid, which forms with another fluid (or gas) a free surface interface, then the interface will be referred to as a capillary surface."

Spis treści

1 Introduction.- 1.1. Mean Curvature.- 1.2. Laplace’s Equation.- 1.3. Angle of Contact.- 1.4. The Method of Gauss; Characterization of the Energies.- 1.5. Variational Considerations.- 1.6. The Equation and the Boundary Condition.- 1.7. Divergence Structure.- 1.8. The Problem as a Geometrical One.- 1.9. The Capillary Tube.- 1.10. Dimensional Considerations.- Notes to Chapter 1.- 2 The Symmetric Capillary Tube.- 2.1. Historical and General.- 2.2. The Narrow Tube; Center Height.- 2.3. The Narrow Tube; Outer Height.- 2.4. The Narrow Tube; Estimates Throughout the Trajectory.- 2.5. Height Estimates for Tubes of General Size.- 2.6. Meniscus Height; Narrow Tubes.- 2.7. Meniscus Height; General Case.- 2.8. Comparisons with Earlier Theories.- Notes to Chapter 2.- 3 The Symmetric Sessile Drop.- 3.1. The Correspondence Principle.- 3.2. Continuation Properties.- 3.3. Uniqueness and Existence.- 3.4. The Envelope.- 3.5. Comparison Theorems.- 3.6. Geometry of the Sessile Drop; Small Drops.- 3.7. Geometry of the Sessile Drop; Larger Drops.- Notes to Chapter 3.- 4 The Pendent Liquid Drop.- 4.1. Mise en Scène.- 4.2. Local Existence.- 4.3. Uniqueness.- 4.4. Global Behavior; General Remarks.- 4.5. Small |u0|.- 4.6. Appearance of Vertical Points.- 4.7. Behavior for Large |u0|.- 4.8. Global Behavior.- 4.9. Maximum Vertical Diameter.- 4.10. Maximum Diameter.- 4.11. Maximum Volume.- 4.12. Asymptotic Properties.- 4.13. The Singular Solution.- 4.14. Isolated Character of Global Solutions.- 4.15. Stability.- Notes to Chapter 4.- 5 Asymmetric Case; Comparison Principles and Applications.- 5.1. The General Comparison Principle.- 5.2. Applications.- 5.3. Domain Dependence.- 5.4. A Counterexample.- 5.5. Convexity.- Notes to Chapter 5.- 6 Capillary Surfaces Without Gravity.- 6.1. General Remarks.- 6.2. A Necessary Condition.- 6.3. Sufficiency Conditions.- 6.4. Sufficiency Conditions II.- 6.5. A Subsidiary Extremal Problem.- 6.6. Minimizing Sequences.- 6.7. The Limit Configuration.- 6.8. The First Variation.- 6.9. The Second Variation.- 6.10. Solution of the Jacobi Equation.- 6.11. Convex Domains.- 6.12. Continuous and Discontinuous Disappearance.- 6.13. An Example.- 6.14. Another Example.- 6.15. Remarks on the Extremals.- 6.16. Example 1.- 6.17. Example 2.- 6.18. Example 3.- 6.19. The Trapezoid.- 6.20. Tail Domains; A Counterexample.- 6.21. Convexity.- 6.22. A Counterexample.- 6.23. Transition to Zero Gravity.- Notes to Chapter 6.- 7 Existence Theorems.- 7.1. Choice of Venue.- 7.2. Variational Solutions.- 7.3. Generalized Solutions.- 7.4. Construction of a Generalized Solution.- 7.5. Proof of Boundedness.- 7.6. Uniqueness.- 7.7. The Variational Condition; Limiting Case.- 7.8. A Necessary and Sufficient Condition.- 7.9. A Limiting Configuration.- 7.10. The Case µ>µ0>1.- 7.11. Application: A General Gradient Bound.- Notes to Chapter 7.- 8 The Capillary Contact Angle.- 8.1. Everyday Experience.- 8.2. The Hypothesis.- 8.3. The Horizontal Plane; Preliminary Remarks.- 8.4. Necessity for ?.- 8.5. Proof that ? is Monotone.- 8.6. Geometrically Imposed Stability Bounds.- 8.7. A Further Kind of Instability.- 8.8. The Inclined Plane; Preliminary Remarks.- 8.9. Integral Relations, and Impossibility of Constant Contact Angle.- 8.10. The Zero-Gravity Solution.- 8.11. Postulated Form for ?.- 8.12. Formal Analytical Solution.- 8.13. The Expansion; Leading Terms.- 8.14. Computer Calculations.- 8.15. Discussion.- 8.16. Further Discussion.- Notes to Chapter 8.- 9 Identities and Isoperimetric Relations.