Scaling

by Grigory Isaakovich Barenblatt

• ISBN13: 9780521826570
• ISBN10: 0521826578
• Format: Hardcover
• Copyright: 2003-12-08
• Publisher: Cambridge University Press

Summary

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