Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics (Cambridge Texts in Applied Mathematics, Series Number 14)

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. Scaling Laws Reveal The Fundamental Property Of Phenomena, Namely Self-similarity - Repeating In Time And/or Space - Which Substantially Simplifies The Mathematical Modelling Of The Phenomena Themselves. This Book Begins From A Non-traditional Exposition Of Dimensional Analysis, Physical Similarity Theory, And General Theory Of Scaling Phenomena, Using Classical Examples To Demonstrate That The Onset Of Scaling Is Not Until The Influence Of Initial And/or Boundary Conditions Has Disappeared But When The System Is Still Far From Equilibrium. Numerous Examples From A Diverse Range Of Fields, Including Theoretical Biology, Fracture Mechanics, Atmospheric And Oceanic Phenomena, And Flame Propagation, Are Presented For Which The Ideas Of Scaling, Intermediate Asymptotics, Self-similarity, And Renormalisation Were Of Decisive Value In Modelling. Introduction -- 1. Dimensions, Dimensional Analysis And Similarity -- 2. The Construction Of Intermediate-asymptotic Solutions Using Dimensional Analysis. Self-similar Solutions -- 3. Self-similarities Of The Second Kind: First Examples -- 4. Self-similarities Of The Second Kind: Further Examples -- 5. Classification Of Similarity Rules And Self-similarity Solutions. A Recipe For The Application Of Similarity Analysis -- 6. Scaling And Transformation Groups. Renormalization Group -- 7. Self-similar Solutions And Travelling Waves -- 8. Invariant Solutions: Asymptotic Conservation Laws, Spectrum Of Eigenvalues, And Stability -- 9. Scaling In The Deformation And Fracture Of Solids -- 10. Scaling In Turbulence -- 11. Scaling In Geophysical Fluid Dynamics -- 12. Scaling: Miscellaneous Special Problems. Grigory Isaakovich Barenblatt. Includes Bibliographical References (p. [366]-382) And Index. 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 Publisher Description (unedited publisher data) Counter Scaling laws reveal the fundamental property of phenomena, namely self-similarity - repeating in time and/or space - which substantially simplifies the mathematical modelling of the phenomena themselves. This book begins from a non-traditional exposition of dimensional analysis, physical similarity theory, and general theory of scaling phenomena, using classical examples to demonstrate that the onset of scaling is not until the influence of initial and/or boundary conditions has disappeared but when the system is still far from equilibrium. Numerous examples from a diverse range of fields, including theoretical biology, fracture mechanics, atmospheric and oceanic phenomena, and flame propagation, are presented for which the ideas of scaling, intermediate asymptotics, self-similarity, and renormalisation were of decisive value in modelling