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

This book deals with a non-traditional exposition of dimensional analysis, physical similarity theory and general theory of scaling phenomena.

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.

Preface; Introduction; 1. Dimensions, dimensional analysis and similarity; 2. The application of dimensional analysis to the construction of intermediate asymptotic solutions to problems of mathematical physics. 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. Recipe for application of similarity analysis; 6. Scaling and transformation groups. Renormalization groups. 7. Self-similar solutions and travelling waves; 8. Invariant solutions: special problems of the theory; 9. Scaling in deformation and fracture in solids; 10. Scaling in turbulence; 11. Scaling in geophysical fluid dynamics; 12. Scaling: miscellaneous special problems.

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.

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.