Scaling

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Cambridge University Press, Nov 13, 2003 - Mathematics - 171 pages
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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 consequences 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.
  

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Contents

Dimensional analysis and physical similarity
12
12 Dimensional analysis
22
13 Physical similarity
37
Selfsimilarity and intermediate asymptotics
52
the selfsimilar solution
55
23 The intermediate asymptotics
60
very intense groundwater pulse flow the selfsimilar intermediateasymptotic solution
65
Scaling laws and selfsimilar solutions that cannot be obtained by dimensional analysis
69
53 The renormalization group and incomplete similarity
102
Selfsimilar phenomena and travelling waves
109
62 Burgers shock waves steady travelling waves of the first kind
111
steady travelling waves of the second kind Nonlinear eigenvalue problem
113
64 Selfsimilar interpretation of solitons
119
Scaling laws and fractals
123
72 Incomplete similarity of fractals
129
73 Scaling relationship between the breathing rate of animals and their mass Fractality of respiratory organs
132

32 Direct application of dimensional analysis to the modified problem
71
33 Numerical experiment Selfsimilar intermediate asymptotics
72
34 Selfsimilar limiting solution The nonlinear eigenvalue problem
78
Complete and incomplete similarity Selfsimilar solutions of the first and second kind
82
42 Selfsimilar solutions of the first and second kind
87
43 A practical recipe for the application of similarity analysis
91
Scaling and transformation groups Renormalization group
94
the boundary layer on a flat plate in a uniform flow
96
Scaling laws for turbulent wallbounded shear flows at very large Reynolds numbers
137
82 Chorins mathematical example
140
83 Steady shear flows at very large Reynolds numbers The intermediate region in pipe flow
142
84 Modification of IzaksonMillikanvon Mises derivation of the velocity distribution in the intermediate region The vanishingviscosity asymptotics
150
85 Turbulent boundary layers
154
References
163
Index
170
Copyright

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About the author (2003)

G. I. Barenblatt is Emeritus G. I. Taylor Professor of Fluid Mechanics at the University of Cambridge, Emeritus Professor at the University of California, Berkeley, and Principal Scientist in the Institute of Oceanology of the Russian Academy of Sciences, Moscow.

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