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Book Description

Many phenomena in nature, engineering or society when seen at an intermediate distance, in space or time, exhibit the remarkable property of self-similarity: they reproduce themselves as scales change, subject to so-called scaling laws. It's crucial to know the details of these laws, so that mathematical models can be properly formulated and analysed, and the phenomena in question can be more deeply understood. In this 2003 book, the author describes and teaches the art of discovering scaling laws, starting from dimensional analysis and physical similarity, which are here given a modern treatment. He demonstrates the concepts of intermediate asymptotics and the renormalisation group as natural attributes of self-similarity and shows how and when these notions and tools can be used to tackle the task at hand, and when they cannot. Based on courses taught to undergraduate and graduate students, the book can also be used for self-study by biologists, chemists, astronomers, engineers and geoscientists.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Dedication
  5. Contents
  6. Foreword by A. J. Chorin
  7. Preface
  8. Introduction
  9. Chapter 1 Dimensional Analysis and Physical Similarity
    1. 1.1 Dimensions
    2. 1.2 Dimensional Analysis
    3. 1.3 Physical Similarity
  10. Chapter 2 Self-Similarity and Intermediate Asymptotics
    1. 2.1 Gently Sloping Groundwater Flow. A Mathematical Model
    2. 2.2 Very Intense Concentrated Flooding: The Self-Similar Solution
    3. 2.3 The Intermediate Asymptotics
    4. 2.4 Problem: Very Intense Groundwater Pulse Flow – The Self-Similar Intermediate-Asymptotic Solution
  11. Chapter 3 Scaling Laws and Self-Similar Solutions That Cannot Be Obtained By Dimensional Analysis
    1. 3.1 Formulation of The Modified Groundwater Flow Problem
    2. 3.2 Direct Application of Dimensional Analysis to The Modified Problem
    3. 3.3 Numerical Experiment. Self-Similar Intermediate Asymptotics
    4. 3.4 Self-Similar Limiting Solution. The Nonlinear Eigenvalue Problem
  12. Chapter 4 Complete and Incomplete Similarity. Self-Similar Solutions of The First and Second Kind
    1. 4.1 Complete and Incomplete Similarity
    2. 4.2 Self-Similar Solutions of The First and Second Kind
    3. 4.3 A Practical Recipe for The Application of Similarity Analysis
  13. Chapter 5 Scaling and Transformation Groups. Renormalization Group
    1. 5.1 Dimensional Analysis and Transformation Groups
    2. 5.2 Problem: The Boundary Layer on A Flat Plate in Uniform Flow
    3. 5.3 The Renormalization Group and Incomplete Similarity
  14. Chapter 6 Self-Similar Phenomena and Travelling Waves
    1. 6.1 Travelling Waves
    2. 6.2 Burgers’ Shock Waves – Steady Travelling Waves of The First Kind
    3. 6.3 Flames: Steady Travelling Waves of The Second Kind. Nonlinear Eigenvalue Problem
    4. 6.4 Self-Similar Interpretation of Solitons
  15. Chapter 7 Scaling Laws and Fractals
    1. 7.1 Mandelbrot Fractals and Incomplete Similarity
    2. 7.2 Incomplete Similarity of Fractals
    3. 7.3 Scaling Relationship Between The Breathing Rate of Animals and their Mass. Fractality of Respiratory Organs
  16. Chapter 8 Scaling Laws for Turbulent Wall-Bounded Shear Flows at Very Large Reynolds Numbers
    1. 8.1 Turbulence at Very Large Reynolds Numbers
    2. 8.2 Chorin’s Mathematical Example
    3. 8.3 Steady Shear Flows at Very Large Reynolds Numbers. The Intermediate Region in Pipe Flow
    4. 8.4 Modification of Izakson–Millikan–von Mises Derivation of The Velocity Distribution in The Intermediate Region. The Vanishing-Viscosity Asymptotics
    5. 8.5 Turbulent Boundary Layers
  17. References
  18. Index