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

✍ Scribed by Grigory Isaakovich Barenblatt

• Publisher: Cambridge University Press
• Year: 1996
• Tongue: English
• Leaves: 405
• Series: Cambridge Texts in Applied Mathematics
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

1 title......Page 1
2 contents......Page 6
3 preface......Page 9
4 C0 introduction......Page 20
5 C1 dimensions, dimensional analysis, and similarity......Page 48
6 C2 construct intermediate-asymptotic solutions using dimensional analysis. Self-similar soln......Page 84
7 C3 self-similarities of 2nd kind, examples......Page 114
8 C4 Self-similarities of 2nd kind, more examples......Page 138
9 C5 classify similarity rules and self-similar solns. Recipe......Page 164
10 C6 scaling, transformation groups, renormalization group......Page 180
11 C7 self-similar soln and traveling waves......Page 200
12 C8 invariant solns, asympt conservation laws, eigenvalues, stability......Page 220
13 C9 scaling in deformation and fracture of solids......Page 240
14 C10 scaling in turbulence......Page 272
15 C11 scaling in geophysical fluid dynamics......Page 316
16 C12 scaling, miscellaneous special problems......Page 354
17 afterword, bio, index......Page 380