Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics

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Cambridge University Press, Dec 12, 1996 - Mathematics - 386 pages
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Scaling (power-type) laws reveal the fundamental property of the phenomena--self similarity. Self-similar (scaling) phenomena repeat themselves in time and/or space. The property of self-similarity simplifies substantially the mathematical modeling of phenomena and its analysis--experimental, analytical and computational. The book begins from a non-traditional exposition of dimensional analysis, physical similarity theory and general theory of scaling phenomena. Classical examples of scaling phenomena are presented. It is demonstrated that scaling comes on a stage when the influence of fine details of initial and/or boundary conditions disappeared but the system is still far from ultimate equilibrium state (intermediate asymptotics). It is explained why the dimensional analysis as a rule is insufficient for establishing self-similarity and constructing scaling variables. Important examples of scaling phenomena for which the dimensional analysis is insufficient (self-similarities of the second kind) are presented and discussed. A close connection of intermediate asymptotics and self-similarities of the second kind with a fundamental concept of theoretical physics, the renormalization group, is explained and discussed. Numerous examples from various fields--from theoretical biology to fracture mechanics, turbulence, flame propagation, flow in porous strata, atmospheric and oceanic phenomena are presented for which the ideas of scaling, intermediate asymptotics, self-similarity and renormalization group were of decisive value in modeling.
 

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Contents

Dimensions Dimensional Analysis and Similarity
28
12 Dimensional Analysis
39
13 Similarity
52
The construction of intermediate asymptotic solutions using dimensional analysis Selfsimilar solutions
64
22 Phenomena at the initial stage of a nuclear explosion
76
23 Selfsimilarity Intermediate asymptotics
86
Selfsimilarities of the second kind first examples
95
the modified instantaneous heat source problem
104
83 Stability of invariant solutions
209
Scaling in the deformation and fracture of solids
220
92 Similarity laws for brittle and quasibrittle fracture
234
Scaling in turbulence
252
102 Turbulent shear flows
268
Scaling in geophysical fluid dynamics
296
112 Flows with strongly stable stratification
299
113 The regime of limiting saturation of a turbulent shear flow laden with sediment
301

Selfsimilarities of the second kind further examples
119
an impulsive loading
133
Classification of similarity rules and selfsimilar solutions A recipe for the application of similarity analysis
145
Scaling and transformation groups Renormalization group
161
62 The renormalisation group and incomplete similarity
171
Selfsimilar solutions and travelling waves
181
72 Burgers shock wave steady travelling wave of the first kind
183
steady travelling wave of the second kind
185
74 Nonlinear eigenvalue problem
192
an intermediate asymptotics
194
Invariant solutions asymptotic conservation laws spectrum of eigenvalues and stability
200
82 Spectrum of eigenvalues
203
114 Upper thermocline in the ocean the travelling thermal wave model
306
115 Strong interaction of turbulence with internal waves Deepening of the turbulent region
311
116 The breaking of internal waves and extension of mixedfluid patches in a stably stratified fluid
316
117 Several phenomena related to turbulence in a stably stratified fluid
329
Scaling miscellaneous special problems
334
scaling relationship between the breathing rate of animals and their mass Fractality of respiratory organs
342
123 The spreading of a groundwater mound
345
Afterword
360
References
366
Index
383
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About the author (1996)

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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