Singularities: Formation, Structure, and Propagation

Description

Many key phenomena in physics and engineering are described as singularities in the solutions to the differential equations describing them. Examples covered thoroughly in this book include the formation of drops and bubbles, the propagation of a crack and the formation of a shock in a gas. Aimed at a broad audience, this book provides the mathematical tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main concepts and techniques, using the most elementary mathematics possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more specialized methods of partial differential equations, complex analysis and asymptotic techniques. The book may be used for advanced fluid mechanics courses and as a complement to a general course on applied partial differential equations. Table of Contents Preface Part I. Setting the Scene: 1. What are singularities all about? 2. Blow-up 3. Similarity profile 4. Continuum equations 5. Local singular expansions 6. Asymptotic expansions of PDEs Part II. Formation of Singularities: 7. Drop break-up 8. A numerical example: drop pinch-off 9. Slow convergence 10. Continuation Part III. Persistent Singularities – Propagation: 11. Shock waves 12. The dynamical system 13. Vortices 14. Cusps and caustics 15. Contact lines and cracks Appendix A. Vector calculus Appendix B. Index notation and the summation convention Appendix C. Dimensional analysis References Index.