Scaling (Cambridge Texts in Applied Mathematics, 34)


Scaling (Cambridge Texts in Applied Mathematics, 34)

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  • 製本 Paperback:紙装版/ペーパーバック版/ページ数 186 p.
  • 言語 ENG
  • 商品コード 9780521533942
  • DDC分類 530.1595


Advanced textbook on how to discover and use scaling laws in natural sciences and engineering.

Full 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

Foreword by A.J. Chorin                            ix
Preface xi
Introduction 1 (11)
Chapter 1 Dimensional analysis and physical 12 (40)
1.1 Dimensions 12 (10)
1.2 Dimensional analysis 22 (15)
1.3 Physical similarity 37 (15)
Chapter 2 Self-similarity and intermediate 52 (17)
2.1 Gently sloping groundwater flow. A 52 (3)
mathematical model
2.2 Very intense concentrated flooding: the 55 (5)
self-similar solution
2.3 The intermediate asymptotics 60 (5)
2.4 Problem: very intense groundwater pulse 65 (4)
flow -the self-similar
intermediate-asymptotic solution
Chapter 3 Scaling laws and self-similar 69 (13)
solutions that cannot be obtained by
dimensional analysis
3.1 Formulation of the modified groundwater 69 (2)
flow problem
3.2 Direct application of dimensional 71 (1)
analysis to the modified problem
3.3 Numerical experiment. Sell-similar 72 (6)
intermediate asymptotics
3.4 Self-similar limiting solution. The 78 (4)
nonlinear eigenvalue problem
Chapter 4 Complete and incomplete similarity. 82 (12)
Self-similar solutions of the first and second
4.1 Complete and incomplete similarity 82 (5)
4.2 Self-similar solutions of the first and 87 (4)
second kind
4.3 A practical recipe for the application of 91 (3)
similarity analysis
Chapter 5 Scaling and transformation groups. 94 (15)
Renormalization group
5.1 Dimensional analysis and transformation 94 (2)
5.2 Problem: the boundary layer on a flat 96 (6)
plate in uniform flow
5.3 The renormalization group and incomplete 102(7)
Chapter 6 Self-similar phenomena and travelling 109(14)
6.1 Travelling waves 109(2)
6.2 Burgers' shock waves - steady travelling 111(2)
waves of the first kind
6.3 Flames - steady travelling waves of the 113(6)
second kind. Nonlinear eigenvalue problem
6.4 Self-similar interpretation of solitons 119(4)
Chapter 7 Scaling laws and fractals 123(14)
7.1 Mandelbrot fractals and incomplete 123(6)
7.2 Incomplete similarity of fractals 129(3)
7.3 Scaling relationship between the 132(5)
breathing rate of animals and their mass.
Fractality of respiratory organs
Chapter 8 Scaling laws for turbulent 137(26)
wall-bounded shear flows at very large Reynolds
8.1 Turbulence at very large Reynolds numbers 137(3)
8.2 Chorin's mathematical example 140(2)
8.3 Steady shear flows at very large Reynolds 142(8)
numbers. The intermediate region in pipe flow
8.4 Modification of Izakson-Millikan-von 150(4)
Mises derivation of the velocity distribution
in the intermediate region The
vanishing-viscosity asymptotics
8.5 Turbulent boundary layers 154(9)
References 163(7)
Index 170